A Machine Checked Model of Idempotent MGU Axioms For Lists of Equational Constraints

Abstract

We present formalized proofs verifying that the first-order unification algorithm defined over lists of satisfiable constraints generates a most general unifier (MGU), which also happens to be idempotent. All of our proofs have been formalized in the Coq theorem prover. Our proofs show that finite maps produced by the unification algorithm provide a model of the axioms characterizing idempotent MGUs of lists of constraints. The axioms that serve as the basis for our verification are derived from a standard set by extending them to lists of constraints. For us, constraints are equalities between terms in the language of simple types. Substitutions are formally modeled as finite maps using the Coq library Coq.FSets.FMapInterface. Coq's method of functional induction is the main proof technique used in proving many of the axioms

A Mechanized Proof of a Textbook Type Unification Algorithm

“…Kothari [11] describes an implementation of a unification function in Coq and proves some properties of most general unifiers. Such properties are used to postulate that unification function does produce most general unifiers on some formalizations of type inference algorithms in type theory [33].…”

“…Formalizations of unification algorithms: Formalization of unification algorithms has been the subject of several research works [8,9,10,11]. In Paulson's work [8] the representation of terms, built by using a binary operator, uses equivalence classes of finite lists where order and multiplicity of elements is considered irrelevant, deviating from simple textbook unification algorithms ( [13,12]).…”

“…The soundness and completeness proofs have a constructive nature, and can thus be formalized in proof assistant systems based on type theory, like Coq [6] and Agda [7]. Formalizations of unification have been reported before in the literature [8,9,10,11], using different proof assistants, but none of them follows the style of textbook proofs (cf. e.g.…”

Certified Semantics for Relational Programming

“…The standard approach is to define a generally-recursive function and a well-founded order for its arguments. This route is taken in [24,4,17,26], where the descriptions of unification algorithms are given in LCF, Alf, Coq and Coq respectively. As a well-founded order lexicographically ordered tuples, containing the information about the number of different free variables and the sizes of the arguments, is used.…”

Mechanized metatheory for a $$\lambda $$-calculus with trust types

“…Allowing non-annotated λ-abstractions characterises a type inference problem that would require a formalisation of a unification algorithm. The formalisation of a unification algorithm has been studied elsewhere [26,32]. We let a formalisation of the type inference problem for this trust-calculus for future work.…”