High-Level Languages and Their Compilers (International by Des Watson

By Des Watson

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We enable proof of correspondence between types within the underlying model type system and types in the HOL type system. We then engineer a polymorphic value model that uses the HOL type system to expose this information; see Section 6. Modularity. UTP does not specify a particular value or type system, and hence it should be possible to reason about UTP predicates and theories independently of the underlying value model. This is important so that laws can be proved generically and then imported into different programming languages.

This function derives well-typedness: b b (x) x. We also formalise binding permutations, which are used to implement alpha conversion. Permutations, borrowed from Nominal Logic [24], are type-preserving 32 S. Foster et al. Table 2. A selection of predicate and expression operator definitions true p = UN IV ¬P p =− P p P ∨Q p = P ∪ Q P ⇒Q p = ¬P ∨ Q ∃vs. P p = {b ⊕b b on vs | b ∈ P ∀vs. f ( x e (b)) $x e = λb. b b (x) p} p} total bijections on variables. We define permutation composition, written π • b, which precomposes the binding function of b with the permutation π, yielding a binding where each variable takes on the value of the variable it was renamed from.

Substitution is defined as the set of bindings, such that updating the value of x in b with the evaluation of e under b yields a binding that is in P . We then prove the usual substitution laws; a selection is shown in Table 3. Likewise, many of these laws are subject to typing obligations, for instance to substitute e for x the types must match (written e e x). 4 Alphabetised Predicates We now extend our core predicate model with alphabets: the maximal set of variables to which a predicate can refer.

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