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) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph ICLP10.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="1: f(T1, T2, T3)\n\nT3 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
2 [label="2: f(T1, T2, T3), SCOPE: 1, CLAUSE: 0\n\nT3 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: f(T5, T5, T5)\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: f(T5, T5, T5), SCOPE: 2, CLAUSE: 0\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 7 [arrowhead = none ];
7 -> 2;

8 [label="EVAL with clause\nf(0, 1, X2) :- f(X2, X2, X2).\nand substitutionT1 -> 0,\nT2 -> 1,\nT3 -> T5,\nX2 -> T5", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 8 [arrowhead = none ];
8 -> 3;

9 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
2 -> 9 [arrowhead = none ];
9 -> 4;

10 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
3 -> 10 [arrowhead = none ];
10 -> 5;

11 [label="BACKTRACK\nfor clause: f(0, 1, X) :- f(X, X, X)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
5 -> 11 [arrowhead = none ];
11 -> 6;

}

(2) TRUE