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

(0) Obligation:

Clauses:

f(0, 1, X) :- f(X, X, X).

Query: f(a,a,g)

(1) PrologToTRSTransformerProof (SOUND transformation)

Transformed Prolog program to TRS.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
3 [label="3: f(T1, T2, T3)\n\nT3 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
6 [label="6: f(T1, T2, T3), SCOPE: 1, CLAUSE: 0\n\nT3 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: f(T6, T6, T6)\n\nT6 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: f(T6, T6, T6), SCOPE: 2, CLAUSE: 0\n\nT6 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
3 -> 19 [arrowhead = none ];
19 -> 6;

20 [label="EVAL with clause\nf(0, 1, X3) :- f(X3, X3, X3).\nand substitutionT1 -> 0,\nT2 -> 1,\nT3 -> T6,\nX3 -> T6", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 20 [arrowhead = none ];
20 -> 7;

21 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
6 -> 21 [arrowhead = none ];
21 -> 12;

22 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
7 -> 22 [arrowhead = none ];
22 -> 17;

23 [label="BACKTRACK\nfor clause: f(0, 1, X) :- f(X, X, X)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
17 -> 23 [arrowhead = none ];
23 -> 18;

}

(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