Alpha Conversion in Simply Typed Lambda Calculus

In the usual presentations of simply typed-calculus, it is usual to identify terms that are-convertible. However, this is not at all a practise in most (typed) functional languages, for which simply typed-calculus is a theoretical foundation. Here, ve well known variants of the type system for simply-calculus which work with variable names are presented. Essentially, these formulations diier in the way variable declarations are handled and all of them are equivalent if-convertible terms are identiied. However, if-convertible terms are not identiied, some of the systems turn out to type less terms than the others. The main aim of this paper is to relate these systems by comparing their set of typable terms and study the property of closure under-conversion for each system.