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

(0) Obligation:

Clauses:

e(X) :- ','(=(Y, .(e, .(o, Y))), c(Y, X)).
c(.(e, X1), 0).
c(.(X2, X), s(Y)) :- c(X, Y).

Query: e(g)

(1) PrologToTRSTransformerProof (SOUND transformation)

Transformed Prolog program to TRS.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="1: e(T1)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
4 [label="4: e(T1), SCOPE: 1, CLAUSE: 0\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: ','(=(X7, .(e, .(o, X7))), c(X7, T4))\n\nT4 is ground\nX7 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];
1 -> 13 [arrowhead = none ];
13 -> 4;

14 [label="ONLY EVAL with clause\ne(X6) :- ','(=(X7, .(e, .(o, X7))), c(X7, X6)).\nand substitutionT1 -> T4,\nX6 -> T4", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 14 [arrowhead = none ];
14 -> 11;

15 [label="UNIFY-FAIL\nbecause of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
11 -> 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