The Essence of Principal Typings

Let S be some type system. A typing in S for a typable term M is the collection of all of the information other than M which appears in the final judgement of a proof derivation showing that M is typable. For example, suppose there is a derivation in S ending with the judgement A ` M : τ meaning that M has result type τ when assuming the types of free variables are given by A. Then (A, τ ) is a typing for M . A principal typing in S for a term M is a typing for M which somehow represents all other possible typings in S for M . It is important not to confuse this notion with the weaker notion of principal type often mentioned in connection with the Hindley/Milner type system. Previous definitions of principal typings for specific type systems have involved various syntactic operations on typings such as substitution of types for type variables, expansion, lifting,