Easy lambda-terms are not always simple

A closed λ -term M is easy if, for any other closed term N , the lambda theory generated by M  =  N is consistent. Recently, it has been introduced a general technique to prove the easiness of λ -terms through the semantical notion of simple easiness. Simple easiness implies easiness and allows to prove consistency results via construction of suitable filter models of λ -calculus living in the category of complete partial orderings: given a simple easy term M and an arbitrary closed term N , it is possible to build (in a canonical way) a non-trivial filter model which equates the interpretation of M and N . The question whether easiness implies simple easiness constitutes Problem 19 in the TLCA list of open problems. In this paper we negatively answer the question providing a non-empty co-r.e. (complement of a recursively enumerable) set of easy, but not simple easy, λ -terms.