Type inference verified: Algorithm W in Isabelle/HOL

A Certified Implementation of ML with Structural Polymorphism

“…Again, correctness has been proved before for Core ML [13,6,19], but to our knowledge never for a system containing equi-recursive types.…”

“…Typed Scheme [18] has a type system remarkably similar to ours, and part of the soundness proof was mechanized in Isabelle/HOL, but the mechanized part does not contain recursive types. Concerning unification and type inference, we have already mentioned the works of Paulson in LCF [16], Dubois and Ménissier-Morain in Coq [6], and Naraschewski and Nipkow in Isabelle [13], and the more recent Isabelle/Nominal proof by Urban and Nipkow [19]. The main difference is the introduction of structural polymorphism, which results in much extended statements to handle admissible substitutions.…”

“…Even in the absence of structural polymorphism, just handling equi-recursive types makes type inference more complex, and we are aware of no proof of principality including them. It might be interesting to compare these different proofs of W in more detail, as the first two use de Bruijn indices [6,13], the latter nominal datatypes [19], and ours cofinite quantification. However, as Urban and Nipkow already observed, while there are clear differences between the different approaches, in the case of type inference lots of lowlevel handling of type variables has to be done, and as a result clever encodings do not seem to be that helpful.…”

“…While many results have already been obtained in the mechanization of metatheory for ML [13,6,5,11,19] and pure type systems [3,1], Objective Caml [12] has unique features which are not covered by existing works. For instance, polymorphic object and variant types require some form of structural polymorphism [8], combined with recursive types, and both of them do not map directly to usual type systems.…”

General Bindings and Alpha-Equivalence in Nominal Isabelle

“…To our knowledge the earliest usage of general binders in a theorem prover is described by Naraschewski and Nipkow [15] with a formalisation of the algorithm W. This formalisation implements binding in type-schemes using a de-Bruijn indices representation. Since type-schemes in W contain only a single place where variables are bound, different indices do not refer to different binders (as in the usual de-Bruijn representation), but to different bound variables.…”