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

(0) Obligation:

Clauses:

p(X) :- ','(q(f(Y)), p(Y)).
q(g(Y)).

Query: p(a)

(1) PrologToTRSTransformerProof (SOUND transformation)

Transformed Prolog program to TRS.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="1: p(T1)\n\n", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
6 [label="6: p(T1), SCOPE: 1, CLAUSE: 0\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: ','(q(f(X8)), p(X8))\n\nX8 is free", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: ','(q(f(X8)), p(X8)), SCOPE: 2, CLAUSE: 1\n\nX8 is free", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 14 [arrowhead = none ];
14 -> 6;

15 [label="ONLY EVAL with clause\np(X7) :- ','(q(f(X8)), p(X8)).\nand substitutionT1 -> T4,\nX7 -> T4", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 15 [arrowhead = none ];
15 -> 8;

16 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
8 -> 16 [arrowhead = none ];
16 -> 11;

17 [label="BACKTRACK\nfor clause: q(g(Y))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
11 -> 17 [arrowhead = none ];
17 -> 13;

}

(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