Left Termination of the query pattern p_in_3(a, a, a) w.r.t. the given Prolog program could successfully be proven:

(0) Obligation:

Clauses:

p(X, Y, Z) :- ','(=(X, Y), ','(=(Z, 1), !)).
p(X, Y, Z) :- ','(=(Z, 1), ','(=(Y, X), p(X, Y, Z))).
=(X, X).

Queries:

p(a,a,a).

(1) BuiltinConflictTransformerProof (SOUND transformation)

Renamed defined predicates conflicting with built-in predicates [PROLOG].

(2) Obligation:

Clauses:

p(X, Y, Z) :- ','(user_defined_=(X, Y), ','(user_defined_=(Z, 1), !)).
p(X, Y, Z) :- ','(user_defined_=(Z, 1), ','(user_defined_=(Y, X), p(X, Y, Z))).
user_defined_=(X, X).

Queries:

p(a,a,a).

(3) PrologToDTProblemTransformerProof (SOUND transformation)

Built DT problem from termination graph.

(4) Obligation:

Triples:

p1(T51, T51, 1) :- p19(T51).
p1(T92, T92, 1) :- p19(T92).

Clauses:

pc19(T60).

Afs:

p1(x1, x2, x3)  =  p1

(5) UndefinedPredicateInTriplesTransformerProof (SOUND transformation)

Deleted triples and predicates having undefined goals [UNKNOWN].

(6) Obligation:

Triples:


Clauses:

pc19(T60).

Afs:

p1(x1, x2, x3)  =  p1

(7) TPisEmptyProof (EQUIVALENT transformation)

There are no more dependency triples. Hence, the dependency triple problem trivially terminates.

(8) YES