The decidability of higher-order matching is a longstanding open problem. This paper contributes to this question by investigating the naive method of generating arity-bounded unifiers in η-long lifted form and then testing whether they are solutions. Arity-bounded means that the arity of types and subtypes of terms in unifiers in lifted form are bounded by some given number N. It is shown that this generation can be enforced to terminate without losing unifiers, if compared using a variant of Padovani's behavioral equivalence. The same method also shows decidability of higher-order matching under the restriction that every subterm of a unifier in η-long β-normal form (or in β-normal form, respectively) contains at most N different free variables. The paper also gives an improved account of a variant of Padovani's behavioral equivalence.
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