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

(0) Obligation:

Clauses:

countstack(empty, X) :- ','(!, eq(X, 0)).
countstack(S, X) :- ','(pop(S, nil), ','(!, ','(popped(S, Pd), countstack(Pd, X)))).
countstack(S, s(X)) :- ','(pop(S, P), ','(head(P, H), ','(tail(P, T), ','(popped(S, Pd), countstack(push(H, push(T, Pd)), X))))).
pop(empty, X1).
pop(push(P, X2), P).
popped(empty, empty).
popped(push(X3, Pd), Pd).
head(nil, X4).
head(cons(H, X5), H).
tail(nil, nil).
tail(cons(X6, T), T).
eq(X, X).

Query: countstack(g,a)

(1) PrologDeterminacyProcessorProof (EQUIVALENT transformation)

The root node satisfies the determinacy criterion.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="1: countstack(T1, T2)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
2 [label="2: countstack(T1, T2), SCOPE: 1, CLAUSE: 0 | countstack(T1, T2), SCOPE: 1, CLAUSE: 1 | countstack(T1, T2), SCOPE: 1, CLAUSE: 2\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: ','(!_1, eq(T5, 0)) | countstack(empty, T2), SCOPE: 1, CLAUSE: 1 | countstack(empty, T2), SCOPE: 1, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: countstack(T1, T2), SCOPE: 1, CLAUSE: 1 | countstack(T1, T2), SCOPE: 1, CLAUSE: 2\n\nT1 is ground\nX8 is free\ncountstack(T1, T2) !=? countstack(empty, X8)", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: eq(T5, 0)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: eq(T5, 0), SCOPE: 2, CLAUSE: 11\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: ','(pop(T11, nil), ','(!_1, ','(popped(T11, X16), countstack(X16, T13)))) | countstack(T11, T2), SCOPE: 1, CLAUSE: 2\n\nT11 is ground\nX8 is free\nX16 is free\ncountstack(T11, T2) !=? countstack(empty, X8)", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: ','(pop(T11, nil), ','(!_1, ','(popped(T11, X16), countstack(X16, T13)))), SCOPE: 3, CLAUSE: 3 | ','(pop(T11, nil), ','(!_1, ','(popped(T11, X16), countstack(X16, T13)))), SCOPE: 3, CLAUSE: 4 | ?_3 | countstack(T11, T2), SCOPE: 1, CLAUSE: 2\n\nT11 is ground\nX8 is free\nX16 is free\ncountstack(T11, T2) !=? countstack(empty, X8)", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: ','(pop(T11, nil), ','(!_1, ','(popped(T11, X16), countstack(X16, T13)))), SCOPE: 3, CLAUSE: 4 | ?_3 | countstack(T11, T2), SCOPE: 1, CLAUSE: 2\n\nT11 is ground\nX8 is free\nX16 is free\ncountstack(T11, T2) !=? countstack(empty, X8)", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: ','(!_1, ','(popped(push(nil, T19), X16), countstack(X16, T13))) | ?_3 | countstack(push(nil, T19), T2), SCOPE: 1, CLAUSE: 2\n\nT19 is ground\nX16 is free", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: ?_3 | countstack(T11, T2), SCOPE: 1, CLAUSE: 2\n\nT11 is ground\nX8 is free\nX22 is free\nX23 is free\ncountstack(T11, T2) !=? countstack(empty, X8)\npop(T11, nil) !=? pop(push(X22, X23), X22)", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: ','(popped(push(nil, T19), X16), countstack(X16, T13))\n\nT19 is ground\nX16 is free", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: ','(popped(push(nil, T19), X16), countstack(X16, T13)), SCOPE: 4, CLAUSE: 5 | ','(popped(push(nil, T19), X16), countstack(X16, T13)), SCOPE: 4, CLAUSE: 6\n\nT19 is ground\nX16 is free", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: ','(popped(push(nil, T19), X16), countstack(X16, T13)), SCOPE: 4, CLAUSE: 6\n\nT19 is ground\nX16 is free", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: countstack(T24, T13)\n\nT24 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: countstack(T11, T2), SCOPE: 1, CLAUSE: 2\n\nT11 is ground\nX8 is free\nX22 is free\nX23 is free\ncountstack(T11, T2) !=? countstack(empty, X8)\npop(T11, nil) !=? pop(push(X22, X23), X22)", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: ','(pop(T30, X43), ','(head(X43, X44), ','(tail(X43, X45), ','(popped(T30, X46), countstack(push(X44, push(X45, X46)), T32)))))\n\nT30 is ground\nX8 is free\nX22 is free\nX23 is free\nX43 is free\nX44 is free\nX45 is free\nX46 is free\ncountstack(T30, T2) !=? countstack(empty, X8)\npop(T30, nil) !=? pop(push(X22, X23), X22)", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: ','(pop(T30, X43), ','(head(X43, X44), ','(tail(X43, X45), ','(popped(T30, X46), countstack(push(X44, push(X45, X46)), T32))))), SCOPE: 5, CLAUSE: 3 | ','(pop(T30, X43), ','(head(X43, X44), ','(tail(X43, X45), ','(popped(T30, X46), countstack(push(X44, push(X45, X46)), T32))))), SCOPE: 5, CLAUSE: 4\n\nT30 is ground\nX8 is free\nX22 is free\nX23 is free\nX43 is free\nX44 is free\nX45 is free\nX46 is free\ncountstack(T30, T2) !=? countstack(empty, X8)\npop(T30, nil) !=? pop(push(X22, X23), X22)", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: ','(pop(T30, X43), ','(head(X43, X44), ','(tail(X43, X45), ','(popped(T30, X46), countstack(push(X44, push(X45, X46)), T32))))), SCOPE: 5, CLAUSE: 4\n\nT30 is ground\nX8 is free\nX22 is free\nX23 is free\nX43 is free\nX44 is free\nX45 is free\nX46 is free\ncountstack(T30, T2) !=? countstack(empty, X8)\npop(T30, nil) !=? pop(push(X22, X23), X22)", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: ','(head(T37, X44), ','(tail(T37, X45), ','(popped(push(T37, T38), X46), countstack(push(X44, push(X45, X46)), T32))))\n\nT37 is ground\nT38 is ground\nX22 is free\nX23 is free\nX44 is free\nX45 is free\nX46 is free\npop(push(T37, T38), nil) !=? pop(push(X22, X23), X22)", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: ','(head(T37, X44), ','(tail(T37, X45), ','(popped(push(T37, T38), X46), countstack(push(X44, push(X45, X46)), T32)))), SCOPE: 6, CLAUSE: 7 | ','(head(T37, X44), ','(tail(T37, X45), ','(popped(push(T37, T38), X46), countstack(push(X44, push(X45, X46)), T32)))), SCOPE: 6, CLAUSE: 8\n\nT37 is ground\nT38 is ground\nX22 is free\nX23 is free\nX44 is free\nX45 is free\nX46 is free\npop(push(T37, T38), nil) !=? pop(push(X22, X23), X22)", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: ','(head(T37, X44), ','(tail(T37, X45), ','(popped(push(T37, T38), X46), countstack(push(X44, push(X45, X46)), T32)))), SCOPE: 6, CLAUSE: 8\n\nT37 is ground\nT38 is ground\nX22 is free\nX23 is free\nX44 is free\nX45 is free\nX46 is free\npop(push(T37, T38), nil) !=? pop(push(X22, X23), X22)", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: ','(tail(cons(T43, T44), X45), ','(popped(push(cons(T43, T44), T38), X46), countstack(push(T43, push(X45, X46)), T32)))\n\nT38 is ground\nT43 is ground\nT44 is ground\nX45 is free\nX46 is free", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: ','(tail(cons(T43, T44), X45), ','(popped(push(cons(T43, T44), T38), X46), countstack(push(T43, push(X45, X46)), T32))), SCOPE: 7, CLAUSE: 9 | ','(tail(cons(T43, T44), X45), ','(popped(push(cons(T43, T44), T38), X46), countstack(push(T43, push(X45, X46)), T32))), SCOPE: 7, CLAUSE: 10\n\nT38 is ground\nT43 is ground\nT44 is ground\nX45 is free\nX46 is free", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: ','(tail(cons(T43, T44), X45), ','(popped(push(cons(T43, T44), T38), X46), countstack(push(T43, push(X45, X46)), T32))), SCOPE: 7, CLAUSE: 10\n\nT38 is ground\nT43 is ground\nT44 is ground\nX45 is free\nX46 is free", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: ','(popped(push(cons(T49, T50), T38), X46), countstack(push(T49, push(T50, X46)), T32))\n\nT38 is ground\nT49 is ground\nT50 is ground\nX46 is free", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: ','(popped(push(cons(T49, T50), T38), X46), countstack(push(T49, push(T50, X46)), T32)), SCOPE: 8, CLAUSE: 5 | ','(popped(push(cons(T49, T50), T38), X46), countstack(push(T49, push(T50, X46)), T32)), SCOPE: 8, CLAUSE: 6\n\nT38 is ground\nT49 is ground\nT50 is ground\nX46 is free", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: ','(popped(push(cons(T49, T50), T38), X46), countstack(push(T49, push(T50, X46)), T32)), SCOPE: 8, CLAUSE: 6\n\nT38 is ground\nT49 is ground\nT50 is ground\nX46 is free", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: countstack(push(T57, push(T58, T59)), T32)\n\nT57 is ground\nT58 is ground\nT59 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 36 [arrowhead = none ];
36 -> 2;

37 [label="EVAL with clause\ncountstack(empty, X8) :- ','(!_1, eq(X8, 0)).\nand substitutionT1 -> empty,\nT2 -> T5,\nX8 -> T5,\nT4 -> T5", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 37 [arrowhead = none ];
37 -> 3;

38 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
2 -> 38 [arrowhead = none ];
38 -> 4;

39 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
3 -> 39 [arrowhead = none ];
39 -> 5;

40 [label="ONLY EVAL with clause\ncountstack(X14, X15) :- ','(pop(X14, nil), ','(!_1, ','(popped(X14, X16), countstack(X16, X15)))).\nand substitutionT1 -> T11,\nX14 -> T11,\nT2 -> T13,\nX15 -> T13,\nT12 -> T13", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 40 [arrowhead = none ];
40 -> 10;

41 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
5 -> 41 [arrowhead = none ];
41 -> 6;

42 [label="EVAL with clause\neq(X11, X11).\nand substitutionT5 -> 0,\nX11 -> 0,\nT8 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 42 [arrowhead = none ];
42 -> 7;

43 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
6 -> 43 [arrowhead = none ];
43 -> 8;

44 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
7 -> 44 [arrowhead = none ];
44 -> 9;

45 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
10 -> 45 [arrowhead = none ];
45 -> 11;

46 [label="BACKTRACK\nfor clause: pop(empty, X1)\nwith clash: (countstack(T11, T2), countstack(empty, X8))", fontsize=14, style = filled, fillcolor = lightgreen];
11 -> 46 [arrowhead = none ];
46 -> 12;

47 [label="EVAL with clause\npop(push(X22, X23), X22).\nand substitutionX22 -> nil,\nX23 -> T19,\nT11 -> push(nil, T19),\nT18 -> nil", fontsize=14, style = filled, fillcolor = lightblue];
12 -> 47 [arrowhead = none ];
47 -> 13;

48 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
12 -> 48 [arrowhead = none ];
48 -> 14;

49 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
13 -> 49 [arrowhead = none ];
49 -> 15;

50 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
14 -> 50 [arrowhead = none ];
50 -> 19;

51 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
15 -> 51 [arrowhead = none ];
51 -> 16;

52 [label="BACKTRACK\nfor clause: popped(empty, empty)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
16 -> 52 [arrowhead = none ];
52 -> 17;

53 [label="ONLY EVAL with clause\npopped(push(X30, X31), X31).\nand substitutionX30 -> nil,\nT19 -> T24,\nX31 -> T24,\nX16 -> T24", fontsize=14, style = filled, fillcolor = lightblue];
17 -> 53 [arrowhead = none ];
53 -> 18;

54 [label="INSTANCE with matching:\nT1 -> T24\nT2 -> T13", fontsize=14, style = filled, fillcolor = lightgrey];
18 -> 54 [arrowhead = none , style=dashed];
54 -> 1[style=dashed];

55 [label="EVAL with clause\ncountstack(X41, s(X42)) :- ','(pop(X41, X43), ','(head(X43, X44), ','(tail(X43, X45), ','(popped(X41, X46), countstack(push(X44, push(X45, X46)), X42))))).\nand substitutionT11 -> T30,\nX41 -> T30,\nX42 -> T32,\nT2 -> s(T32),\nT31 -> T32", fontsize=14, style = filled, fillcolor = lightblue];
19 -> 55 [arrowhead = none ];
55 -> 20;

56 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
19 -> 56 [arrowhead = none ];
56 -> 21;

57 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
20 -> 57 [arrowhead = none ];
57 -> 22;

58 [label="BACKTRACK\nfor clause: pop(empty, X1)\nwith clash: (countstack(T30, T2), countstack(empty, X8))", fontsize=14, style = filled, fillcolor = lightgreen];
22 -> 58 [arrowhead = none ];
58 -> 23;

59 [label="EVAL with clause\npop(push(X54, X55), X54).\nand substitutionX54 -> T37,\nX55 -> T38,\nT30 -> push(T37, T38),\nX43 -> T37", fontsize=14, style = filled, fillcolor = lightblue];
23 -> 59 [arrowhead = none ];
59 -> 24;

60 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
23 -> 60 [arrowhead = none ];
60 -> 25;

61 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
24 -> 61 [arrowhead = none ];
61 -> 26;

62 [label="BACKTRACK\nfor clause: head(nil, X4)\nwith clash: (pop(push(T37, T38), nil), pop(push(X22, X23), X22))", fontsize=14, style = filled, fillcolor = lightgreen];
26 -> 62 [arrowhead = none ];
62 -> 27;

63 [label="EVAL with clause\nhead(cons(X63, X64), X63).\nand substitutionX63 -> T43,\nX64 -> T44,\nT37 -> cons(T43, T44),\nX44 -> T43", fontsize=14, style = filled, fillcolor = lightblue];
27 -> 63 [arrowhead = none ];
63 -> 28;

64 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
27 -> 64 [arrowhead = none ];
64 -> 29;

65 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
28 -> 65 [arrowhead = none ];
65 -> 30;

66 [label="BACKTRACK\nfor clause: tail(nil, nil)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
30 -> 66 [arrowhead = none ];
66 -> 31;

67 [label="ONLY EVAL with clause\ntail(cons(X71, X72), X72).\nand substitutionT43 -> T49,\nX71 -> T49,\nT44 -> T50,\nX72 -> T50,\nX45 -> T50", fontsize=14, style = filled, fillcolor = lightblue];
31 -> 67 [arrowhead = none ];
67 -> 32;

68 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
32 -> 68 [arrowhead = none ];
68 -> 33;

69 [label="BACKTRACK\nfor clause: popped(empty, empty)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
33 -> 69 [arrowhead = none ];
69 -> 34;

70 [label="ONLY EVAL with clause\npopped(push(X79, X80), X80).\nand substitutionT49 -> T57,\nT50 -> T58,\nX79 -> cons(T57, T58),\nT38 -> T59,\nX80 -> T59,\nX46 -> T59", fontsize=14, style = filled, fillcolor = lightblue];
34 -> 70 [arrowhead = none ];
70 -> 35;

71 [label="INSTANCE with matching:\nT1 -> push(T57, push(T58, T59))\nT2 -> T32", fontsize=14, style = filled, fillcolor = lightgrey];
35 -> 71 [arrowhead = none , style=dashed];
71 -> 1[style=dashed];

}

(2) YES