# zbMATH — the first resource for mathematics

Type inference verified: Algorithm $${\mathcal W}$$ in Isabelle/HOL. (English) Zbl 0927.03025
Giménez, Eduardo (ed.) et al., Types for proofs and programs. International workshop TYPES ’96, Aussois, France, December 15–19, 1996. Selected papers. Berlin: Springer. Lect. Notes Comput. Sci. 1512, 317-332 (1998).
Most functional programming languages have a common core: the simply typed $$\lambda$$-calculus enriched with let-expressions, i.e. local definitions of polymorphic values. This language is called Mini-ML. The set of well-typed Mini-ML expressions is inductively defined by a set of inference rules. One of the key properties of Mini-ML is that every well-typed expression has a most general type. The computation of the most general type is called type inference. It was first studied by R. Hindley [Trans. Am. Math. Soc. 146, 29-60 (1969; Zbl 0196.01501)] in the context of combinatory logic and later independently by R. Milner [J. Comput. Syst. Sci. 17, 348-375 (1978; Zbl 0388.68003)] for Mini-ML. Milner’s type inference algorithm is known as algorithm $${\mathcal W}$$. L. Damas proved the completeness of $${\mathcal W}$$.
This paper presents the first machine-checked proof of correctness and completeness of $${\mathcal W}$$. It is an extension of the work by D. Nazareth and T. Nipkow [Lect. Notes Comput. Sci. 1125, 331-346 (1996)], who treated the monomorphic case (no let-expressions). A partial verification of $${\mathcal W}$$ in the proof checker Coq is reported by C. Dubois and V. Ménissier-Morain [in: J. von Wright et al. (eds.), Suppl. Proc. 9th Int. Conf. Theorem Proving in Higher-Order Logic, Turku Centre Comput. Sci., 15-30 (1996)]. Our paper provides the definition of (due to lack of space: almost) all concepts, the key lemmas, but almost no proofs. The complete development is accessible via http://www4.in.tum.de/~nipkow/.
For the entire collection see [Zbl 0898.00024].
##### MSC:
 03B35 Mechanization of proofs and logical operations 03B40 Combinatory logic and lambda calculus 68T15 Theorem proving (deduction, resolution, etc.) (MSC2010)
##### Software:
Coq; Isabelle; Isabelle/HOL; mini-ML