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

(0) Obligation:

Clauses:

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

Query: even(g)

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph ICLP10.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="1: even(T1)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
2 [label="2: even(T1), SCOPE: 1, CLAUSE: 0\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: ','(eq(X5, f(e, f(o, X5))), c(X5, T3))\n\nT3 is ground\nX5 is free", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: ','(eq(X5, f(e, f(o, X5))), c(X5, T3)), SCOPE: 2, CLAUSE: 3\n\nT3 is ground\nX5 is free", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 6 [arrowhead = none ];
6 -> 2;

7 [label="ONLY EVAL with clause\neven(X4) :- ','(eq(X5, f(e, f(o, X5))), c(X5, X4)).\nand substitutionT1 -> T3,\nX4 -> T3", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 7 [arrowhead = none ];
7 -> 3;

8 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
3 -> 8 [arrowhead = none ];
8 -> 4;

9 [label="BACKTRACK\nfor clause: eq(X, X)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
4 -> 9 [arrowhead = none ];
9 -> 5;

}

(2) TRUE