Abstractions for Multi-Sorted Substitutions

Formalizing a typed programming language in a proof assistant requires to choose representations for variables and typing. Variables are often represented as de Bruijn indices, where substitution is usually defined in terms of renamings to allow for proofs by structural induction. Typing can be represented extrinsically by defining untyped terms and a typing relation, or intrinsically by combining syntax and typing into a single definition of well-typed terms. For extrinsic typing, there is again a choice between extrinsic scoping, where terms and the notion of free variables are defined separately, and intrinsic scoping, where terms are indexed by their free variables. This paper describes an Agda framework for formalizing programming languages with extrinsic typing, intrinsic scoping, and de Bruijn Indices for variables. The framework supports object languages with arbitrary many variable sorts and dependencies, making it suitable for polymorphic languages and dependent types. Given an Agda definition of syntax and typing, the framework derives substitution operations and lemmas for untyped terms, and provides an abstraction to prove type preservation of these operations with just a single lemma. The key insights behind the framework are the use of multi-sorted syntax definitions, which enable parallel substitutions that replace all variables of all sorts simultaneously, and abstractions that unify the definitions, compositions, typings, and type preservation lemmas of multi-sorted renamings and substitutions. Case studies have been conducted to prove subject reduction for System F with subtyping, dependently typed lambda calculus, and lambda calculus with pattern matching.