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

(0) Obligation:

Clauses:

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

Query: app(g,a,a)

(1) PrologDeterminacyProcessorProof (EQUIVALENT transformation)

The root node satisfies the determinacy criterion.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="1: app(T1, T2, T3)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
2 [label="2: app(T1, T2, T3), SCOPE: 1, CLAUSE: 0 | app(T1, T2, T3), SCOPE: 1, CLAUSE: 1\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: true | app([], T2, T3), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: app(T1, T2, T3), SCOPE: 1, CLAUSE: 1\n\nT1 is ground\nX6 is free\napp(T1, T2, T3) !=? app([], X6, X6)", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: app([], T2, T3), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: ','(no(empty([])), ','(head([], T12), ','(tail([], X15), app(X15, T13, T14))))\n\nX15 is free", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: ','(no(empty([])), ','(head([], T12), ','(tail([], X15), app(X15, T13, T14)))), SCOPE: 2, CLAUSE: 7 | ','(no(empty([])), ','(head([], T12), ','(tail([], X15), app(X15, T13, T14)))), SCOPE: 2, CLAUSE: 8\n\nX15 is free", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: ','(','(call(empty([])), ','(!_2, failure(a))), ','(head([], T12), ','(tail([], X15), app(X15, T13, T14)))) | ','(no(empty([])), ','(head([], T12), ','(tail([], X15), app(X15, T13, T14)))), SCOPE: 2, CLAUSE: 8\n\nX15 is free", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: ','(empty([]), ','(','(!_2, failure(a)), ','(head([], T12), ','(tail([], X15), app(X15, T13, T14))))) | ?_3 | ','(no(empty([])), ','(head([], T12), ','(tail([], X15), app(X15, T13, T14)))), SCOPE: 2, CLAUSE: 8\n\nX15 is free", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: ','(empty([]), ','(','(!_2, failure(a)), ','(head([], T12), ','(tail([], X15), app(X15, T13, T14))))), SCOPE: 4, CLAUSE: 6 | ?_4 | ?_3 | ','(no(empty([])), ','(head([], T12), ','(tail([], X15), app(X15, T13, T14)))), SCOPE: 2, CLAUSE: 8\n\nX15 is free", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: ','(','(!_2, failure(a)), ','(head([], T12), ','(tail([], X15), app(X15, T13, T14)))) | ?_4 | ?_3 | ','(no(empty([])), ','(head([], T12), ','(tail([], X15), app(X15, T13, T14)))), SCOPE: 2, CLAUSE: 8\n\nX15 is free", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: ','(failure(a), ','(head([], T12), ','(tail([], X15), app(X15, T13, T14))))\n\nX15 is free", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: ','(failure(a), ','(head([], T12), ','(tail([], X15), app(X15, T13, T14)))), SCOPE: 5, CLAUSE: 9\n\nX15 is free", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: ','(no(empty(T21)), ','(head(T21, T25), ','(tail(T21, X29), app(X29, T26, T27))))\n\nT21 is ground\nX6 is free\nX29 is free\napp(T21, T2, T3) !=? app([], X6, X6)", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: ','(no(empty(T21)), ','(head(T21, T25), ','(tail(T21, X29), app(X29, T26, T27)))), SCOPE: 6, CLAUSE: 7 | ','(no(empty(T21)), ','(head(T21, T25), ','(tail(T21, X29), app(X29, T26, T27)))), SCOPE: 6, CLAUSE: 8\n\nT21 is ground\nX6 is free\nX29 is free\napp(T21, T2, T3) !=? app([], X6, X6)", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: ','(','(call(empty(T30)), ','(!_6, failure(a))), ','(head(T30, T25), ','(tail(T30, X29), app(X29, T26, T27)))) | ','(no(empty(T30)), ','(head(T30, T25), ','(tail(T30, X29), app(X29, T26, T27)))), SCOPE: 6, CLAUSE: 8\n\nT30 is ground\nX6 is free\nX29 is free\napp(T30, T2, T3) !=? app([], X6, X6)", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: ','(empty(T30), ','(','(!_6, failure(a)), ','(head(T30, T25), ','(tail(T30, X29), app(X29, T26, T27))))) | ?_7 | ','(no(empty(T30)), ','(head(T30, T25), ','(tail(T30, X29), app(X29, T26, T27)))), SCOPE: 6, CLAUSE: 8\n\nT30 is ground\nX6 is free\nX29 is free\napp(T30, T2, T3) !=? app([], X6, X6)", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: ','(empty(T30), ','(','(!_6, failure(a)), ','(head(T30, T25), ','(tail(T30, X29), app(X29, T26, T27))))), SCOPE: 8, CLAUSE: 6 | ?_8 | ?_7 | ','(no(empty(T30)), ','(head(T30, T25), ','(tail(T30, X29), app(X29, T26, T27)))), SCOPE: 6, CLAUSE: 8\n\nT30 is ground\nX6 is free\nX29 is free\napp(T30, T2, T3) !=? app([], X6, X6)", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: ','(','(!_6, failure(a)), ','(head([], T25), ','(tail([], X29), app(X29, T26, T27)))) | ?_8 | ?_7 | ','(no(empty([])), ','(head([], T25), ','(tail([], X29), app(X29, T26, T27)))), SCOPE: 6, CLAUSE: 8\n\nX29 is free", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: ?_8 | ?_7 | ','(no(empty(T30)), ','(head(T30, T25), ','(tail(T30, X29), app(X29, T26, T27)))), SCOPE: 6, CLAUSE: 8\n\nT30 is ground\nX6 is free\nX29 is free\napp(T30, T2, T3) !=? app([], X6, X6)\nempty(T30) !=? empty([])", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: ','(failure(a), ','(head([], T25), ','(tail([], X29), app(X29, T26, T27))))\n\nX29 is free", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: ','(failure(a), ','(head([], T25), ','(tail([], X29), app(X29, T26, T27)))), SCOPE: 9, CLAUSE: 9\n\nX29 is free", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: ?_7 | ','(no(empty(T30)), ','(head(T30, T25), ','(tail(T30, X29), app(X29, T26, T27)))), SCOPE: 6, CLAUSE: 8\n\nT30 is ground\nX6 is free\nX29 is free\napp(T30, T2, T3) !=? app([], X6, X6)\nempty(T30) !=? empty([])", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: ','(no(empty(T30)), ','(head(T30, T25), ','(tail(T30, X29), app(X29, T26, T27)))), SCOPE: 6, CLAUSE: 8\n\nT30 is ground\nX6 is free\nX29 is free\napp(T30, T2, T3) !=? app([], X6, X6)\nempty(T30) !=? empty([])", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: ','(head(T37, T25), ','(tail(T37, X29), app(X29, T26, T27)))\n\nT37 is ground\nX6 is free\nX29 is free\napp(T37, T2, T3) !=? app([], X6, X6)\nempty(T37) !=? empty([])", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: ','(head(T37, T25), ','(tail(T37, X29), app(X29, T26, T27))), SCOPE: 10, CLAUSE: 2 | ','(head(T37, T25), ','(tail(T37, X29), app(X29, T26, T27))), SCOPE: 10, CLAUSE: 3\n\nT37 is ground\nX6 is free\nX29 is free\napp(T37, T2, T3) !=? app([], X6, X6)\nempty(T37) !=? empty([])", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: ','(head(T37, T25), ','(tail(T37, X29), app(X29, T26, T27))), SCOPE: 10, CLAUSE: 3\n\nT37 is ground\nX6 is free\nX29 is free\napp(T37, T2, T3) !=? app([], X6, X6)\nempty(T37) !=? empty([])", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: ','(tail(.(T45, T46), X29), app(X29, T47, T48))\n\nT45 is ground\nT46 is ground\nX29 is free", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: ','(tail(.(T45, T46), X29), app(X29, T47, T48)), SCOPE: 11, CLAUSE: 4 | ','(tail(.(T45, T46), X29), app(X29, T47, T48)), SCOPE: 11, CLAUSE: 5\n\nT45 is ground\nT46 is ground\nX29 is free", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: ','(tail(.(T45, T46), X29), app(X29, T47, T48)), SCOPE: 11, CLAUSE: 5\n\nT45 is ground\nT46 is ground\nX29 is free", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: app(T58, T47, T48)\n\nT58 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 37 [arrowhead = none ];
37 -> 2;

38 [label="EVAL with clause\napp([], X6, X6).\nand substitutionT1 -> [],\nT2 -> T5,\nX6 -> T5,\nT3 -> T5", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 38 [arrowhead = none ];
38 -> 3;

39 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
2 -> 39 [arrowhead = none ];
39 -> 4;

40 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
3 -> 40 [arrowhead = none ];
40 -> 5;

41 [label="EVAL with clause\napp(X25, X26, .(X27, X28)) :- ','(no(empty(X25)), ','(head(X25, X27), ','(tail(X25, X29), app(X29, X26, X28)))).\nand substitutionT1 -> T21,\nX25 -> T21,\nT2 -> T26,\nX26 -> T26,\nX27 -> T25,\nX28 -> T27,\nT3 -> .(T25, T27),\nT23 -> T25,\nT22 -> T26,\nT24 -> T27", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 41 [arrowhead = none ];
41 -> 16;

42 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
4 -> 42 [arrowhead = none ];
42 -> 17;

43 [label="EVAL with clause\napp(X11, X12, .(X13, X14)) :- ','(no(empty(X11)), ','(head(X11, X13), ','(tail(X11, X15), app(X15, X12, X14)))).\nand substitutionX11 -> [],\nT2 -> T13,\nX12 -> T13,\nX13 -> T12,\nX14 -> T14,\nT3 -> .(T12, T14),\nT10 -> T12,\nT9 -> T13,\nT11 -> T14", fontsize=14, style = filled, fillcolor = lightblue];
5 -> 43 [arrowhead = none ];
43 -> 6;

44 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
5 -> 44 [arrowhead = none ];
44 -> 7;

45 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
6 -> 45 [arrowhead = none ];
45 -> 8;

46 [label="ONLY EVAL with clause\nno(X18) :- ','(call(X18), ','(!_2, failure(a))).\nand substitutionX18 -> empty([])", fontsize=14, style = filled, fillcolor = lightblue];
8 -> 46 [arrowhead = none ];
46 -> 9;

47 [label="CALL", fontsize=14, style = filled, fillcolor = lightcyan];
9 -> 47 [arrowhead = none ];
47 -> 10;

48 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
10 -> 48 [arrowhead = none ];
48 -> 11;

49 [label="ONLY EVAL with clause\nempty([]).\nand substitution", fontsize=14, style = filled, fillcolor = lightblue];
11 -> 49 [arrowhead = none ];
49 -> 12;

50 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
12 -> 50 [arrowhead = none ];
50 -> 13;

51 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
13 -> 51 [arrowhead = none ];
51 -> 14;

52 [label="BACKTRACK\nfor clause: failure(b)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
14 -> 52 [arrowhead = none ];
52 -> 15;

53 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
16 -> 53 [arrowhead = none ];
53 -> 18;

54 [label="ONLY EVAL with clause\nno(X32) :- ','(call(X32), ','(!_6, failure(a))).\nand substitutionT21 -> T30,\nX32 -> empty(T30)", fontsize=14, style = filled, fillcolor = lightblue];
18 -> 54 [arrowhead = none ];
54 -> 19;

55 [label="CALL", fontsize=14, style = filled, fillcolor = lightcyan];
19 -> 55 [arrowhead = none ];
55 -> 20;

56 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
20 -> 56 [arrowhead = none ];
56 -> 21;

57 [label="EVAL with clause\nempty([]).\nand substitutionT30 -> []", fontsize=14, style = filled, fillcolor = lightblue];
21 -> 57 [arrowhead = none ];
57 -> 22;

58 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
21 -> 58 [arrowhead = none ];
58 -> 23;

59 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
22 -> 59 [arrowhead = none ];
59 -> 24;

60 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
23 -> 60 [arrowhead = none ];
60 -> 27;

61 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
24 -> 61 [arrowhead = none ];
61 -> 25;

62 [label="BACKTRACK\nfor clause: failure(b)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
25 -> 62 [arrowhead = none ];
62 -> 26;

63 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
27 -> 63 [arrowhead = none ];
63 -> 28;

64 [label="ONLY EVAL with clause\nno(X39).\nand substitutionT30 -> T37,\nX39 -> empty(T37)", fontsize=14, style = filled, fillcolor = lightblue];
28 -> 64 [arrowhead = none ];
64 -> 29;

65 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
29 -> 65 [arrowhead = none ];
65 -> 30;

66 [label="BACKTRACK\nfor clause: head([], X1)\nwith clash: (empty(T37), empty([]))", fontsize=14, style = filled, fillcolor = lightgreen];
30 -> 66 [arrowhead = none ];
66 -> 31;

67 [label="EVAL with clause\nhead(.(X47, X48), X47).\nand substitutionX47 -> T45,\nX48 -> T46,\nT37 -> .(T45, T46),\nT25 -> T45,\nT26 -> T47,\nT27 -> T48", fontsize=14, style = filled, fillcolor = lightblue];
31 -> 67 [arrowhead = none ];
67 -> 32;

68 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
31 -> 68 [arrowhead = none ];
68 -> 33;

69 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
32 -> 69 [arrowhead = none ];
69 -> 34;

70 [label="BACKTRACK\nfor clause: tail([], [])because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
34 -> 70 [arrowhead = none ];
70 -> 35;

71 [label="ONLY EVAL with clause\ntail(.(X57, X58), X58).\nand substitutionT45 -> T57,\nX57 -> T57,\nT46 -> T58,\nX58 -> T58,\nX29 -> T58", fontsize=14, style = filled, fillcolor = lightblue];
35 -> 71 [arrowhead = none ];
71 -> 36;

72 [label="INSTANCE with matching:\nT1 -> T58\nT2 -> T47\nT3 -> T48", fontsize=14, style = filled, fillcolor = lightgrey];
36 -> 72 [arrowhead = none , style=dashed];
72 -> 1[style=dashed];

}

(2) YES