Foundational, Compositional (Co)datatypes for Higher-Order Logic: Category Theory Applied to Theorem Proving

Abstract

Interactive theorem provers based on higher-order logic (HOL) traditionally follow the definitional approach, reducing high-level specifications to logical primitives. This also applies to the support for datatype definitions. However, the internal datatype construction used in HOL4, HOL Light, and Isabelle/ HOL is fundamentally noncompositional, limiting its efficiency and flexibility, and it does not cater for codatatypes.We present a fully modular framework for constructing (co)datatypes in HOL, with support for mixed mutual and nested (co)recursion. Mixed (co)recursion enables type definitions involving both datatypes and codatatypes, such as the type of finitely branching trees of possibly infinite depth. Our framework draws heavily from category theory. The key notion is that of a bounded natural functor-an enriched type constructor satisfying specific properties preserved by interesting categorical operations. Our ideas are implemented as a definitional package in Isabelle, addressing a frequent request from users.

Friends with Benefits

“…Our earlier papers [17,67] discuss further theoretical sources. AmiCo implements the first general, provably sound, and fully automatic method for mixing recursive and corecursive calls in function definitions.…”

“…It continues a program of integrating category theory insight into proof assistant technology [16][17][18]67]. There is a lot of related work on corecursion and productivity, both theoretical and applied to proof assistants and functional programming languages.…”

“…The tool satisfies this requirement by employing Isabelle's infrastructure for bounded natural functors (BNFs) [16,67]. For example, the codatatype α stream is defined as the greatest solution to the fixpoint equation β ∼ = α × β, where both the right-hand side α × β and the resulting type α stream are BNFs.…”

“…Isabelle/HOL is the first of these systems to also offer codatatypes. It took two years, and about 24 000 lines of Standard ML, to move from an understanding of the mathematics [18,67] to an implementation that automates the process of checking high-level user specifications and producing the necessary corecursion and coinduction theorems [16].…”

Comprehending Isabelle/HOL’s Consistency

“…4 By contrast, Isabelle/HOL allows first declaring a constant, and at later times overloading it by defining some of its instances, as in the following example: 5 In between the declaration and the instance definitions, arbitrary commands may occur, including type definitions ("typedef") and (co)datatype definitions (which are derived from typedef [7,45]). For example, the following definition introduces a type of polynomials over an arbitrary domain α, where ∀ ∞ is the "for all but finitely many" quantifier: typedef α poly ≡ {f : nat → α | ∀ ∞ n. f n = 0}…”