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

(0) Obligation:

Clauses:

n :- ','(=(X, f(1, f(0, X))), g(X)).
g(f(X, f(X, X1))) :- !.
g(f(X2, X)) :- g(X).

Query: n()

(1) PrologDeterminacyProcessorProof (EQUIVALENT transformation)

The root node satisfies the determinacy criterion.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="1: n\n\n", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
2 [label="2: n, SCOPE: 1, CLAUSE: 0\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: ','(=(X3, f(1, f(0, X3))), g(X3))\n\nX3 is free", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 5 [arrowhead = none ];
5 -> 2;

6 [label="ONLY EVAL with clause\nn :- ','(=(X3, f(1, f(0, X3))), g(X3)).\nand substitution", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 6 [arrowhead = none ];
6 -> 3;

7 [label="UNIFY-FAIL\nbecause of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
3 -> 7 [arrowhead = none ];
7 -> 4;

}

(2) YES