Nonfree datatypes in Isabelle/HOL: animating a many-sorted metatheory

Datatypes freely generated by their constructors are well supported in mainstream proof assistants. Algebraic specification languages offer more expressive datatypes on axiomatic means: nonfree datatypes generated from constructors modulo equations. We have implemented an Isabelle/HOL package for nonfree datatypes, without compromising foundations. The use of the package, and its nonfree iterator in particular, is illustrated with examples: bags, polynomials and λ-terms modulo α-equivalence. The many-sorted metatheory of nonfree datatypes is formalized as an ordinary Isabelle theory and is animated by the package into user-specified instances. HOL lacks a type of types, so we employ an ad hoc construction of a universe embedding the relevant parameter types

A formalized general theory of syntax with bindings: extended version

We present the formalization of a theory of syntax with bindings that has been developed and refined over the last decade to support several large formalization efforts. Terms are defined for an arbitrary number of constructors of varying numbers of inputs, quotiented to alpha-equivalence and sorted according to a binding signature. The theory contains a rich collection of properties of the standard operators on terms, including substitution, swapping and freshness—namely, there are lemmas showing how each of the operators interacts with all the others and with the syntactic constructors. The theory also features induction and recursion principles and support for semantic interpretation, all tailored for smooth interaction with the bindings and the standard operators