In this chapter we show that the encodings of the previous chapter can be extended to encodings of typed λ-calculi. To do this we have to define translations on types to match those on terms. We analyse the case of the simply-typed λ-calculus in detail, and discuss subtyping and recursive types. For studies of other type systems, see the notes at the end of the Part.

16.1 Typed λ-calculus

In a core simplv-typed λ-calculus, types are built from basic types, such as integers and booleans, using the function type constructor. The syntax of terms is that of the untyped λ-calculus plus basic constants. Each constant has a unique predefined type. We use only constants of basic types: this is sufficient to have a non-empty set of (closed) well-typed terms. This simplv-typed λ-calculus is presented in Table 16.1. We write Λ{c} for the set of terms of the calculus. As usual, function type associates to the right, so T → S → U should be read T → (S → U). Basic types are ranged over by t, basic constants by c. The reduction relation and the reduction strategies are defined as for the untyped calculus; the only difference is that the set of values for a reduction strategy normally contains the constants. We call the typed versions of λV and λN simply-typed call-by-value (λV→) and simply-typed call-by-name (λN→), respectively.

We add the same basic constants and basic types to HOπ and π-calculus and repeat the diagram of Figure 15.1, this time for λV→ and λN→. We show how to extend the encodings of λV and λN (from Sections 15.3 and 15.4), and their correctness results, to take account of types. The encodings in Sections 15.5-15.7 can be extended similarly.

Lemma 16.1.1 In the simplv-typed λ-calculus, for every Γ and M there is at most one T such that Γ M : T .

16.2 The interpretation of typed call-by-value

We begin with the left part of Figure 15.1, which concerns λV We follow a schema similar to that of Section 15.3, pointing out the main additions.