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

(0) Obligation:

Clauses:

e(X) :- ','(=(Y, f(e, f(o, Y))), ','(!, c(Y, X))).
e(X) :- e(X).
c(f(e, X1), 0).
c(f(X2, X), s(Y)) :- c(X, Y).

Query: e(g)

(1) PrologDeterminacyProcessorProof (EQUIVALENT transformation)

The root node satisfies the determinacy criterion.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="1: e(T1)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
2 [label="2: e(T1), SCOPE: 1, CLAUSE: 0 | e(T1), SCOPE: 1, CLAUSE: 1\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: ','(=(X5, f(e, f(o, X5))), ','(!_1, c(X5, T3))) | e(T3), SCOPE: 1, CLAUSE: 1\n\nT3 is ground\nX5 is free", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: e(T3), SCOPE: 1, CLAUSE: 1\n\nT3 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: e(T5)\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: e(T5), SCOPE: 2, CLAUSE: 0 | e(T5), SCOPE: 2, CLAUSE: 1\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: e(T5), SCOPE: 2, CLAUSE: 0\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: e(T5), SCOPE: 2, CLAUSE: 1\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: ','(=(X16, f(e, f(o, X16))), ','(!_2, c(X16, T10)))\n\nT10 is ground\nX16 is free", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: e(T14)\n\nT14 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 12 [arrowhead = none ];
12 -> 2;

13 [label="ONLY EVAL with clause\ne(X4) :- ','(=(X5, f(e, f(o, X5))), ','(!_1, c(X5, X4))).\nand substitutionT1 -> T3,\nX4 -> T3", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 13 [arrowhead = none ];
13 -> 3;

14 [label="UNIFY-FAIL\nbecause of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
3 -> 14 [arrowhead = none ];
14 -> 4;

15 [label="ONLY EVAL with clause\ne(X8) :- e(X8).\nand substitutionT3 -> T5,\nX8 -> T5", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 15 [arrowhead = none ];
15 -> 5;

16 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
5 -> 16 [arrowhead = none ];
16 -> 6;

17 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
6 -> 17 [arrowhead = none ];
17 -> 7;

18 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
6 -> 18 [arrowhead = none ];
18 -> 8;

19 [label="ONLY EVAL with clause\ne(X15) :- ','(=(X16, f(e, f(o, X16))), ','(!_2, c(X16, X15))).\nand substitutionT5 -> T10,\nX15 -> T10", fontsize=14, style = filled, fillcolor = lightblue];
7 -> 19 [arrowhead = none ];
19 -> 9;

20 [label="ONLY EVAL with clause\ne(X22) :- e(X22).\nand substitutionT5 -> T14,\nX22 -> T14", fontsize=14, style = filled, fillcolor = lightblue];
8 -> 20 [arrowhead = none ];
20 -> 11;

21 [label="UNIFY-FAIL\nbecause of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
9 -> 21 [arrowhead = none ];
21 -> 10;

22 [label="INSTANCE with matching:\nT1 -> T14", fontsize=14, style = filled, fillcolor = lightgrey];
11 -> 22 [arrowhead = none , style=dashed];
22 -> 1[style=dashed];

}

(2) YES