Left Termination 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) PrologToTRSTransformerProof (SOUND transformation)

Transformed Prolog program to TRS.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
5 [label="5: n\n\n", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
6 [label="6: n, SCOPE: 1, CLAUSE: 0\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: ','(=(X4, f(1, f(0, X4))), g(X4))\n\nX4 is free", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
5 -> 13 [arrowhead = none ];
13 -> 6;

14 [label="ONLY EVAL with clause\nn :- ','(=(X4, f(1, f(0, X4))), g(X4)).\nand substitution", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 14 [arrowhead = none ];
14 -> 8;

15 [label="UNIFY-FAIL\nbecause of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
8 -> 15 [arrowhead = none ];
15 -> 12;

}

(2) Obligation:

Q restricted rewrite system:
R is empty.
Q is empty.

(3) RisEmptyProof (EQUIVALENT transformation)

The TRS R is empty. Hence, termination is trivially proven.

(4) YES