This paper presents 3MT, a framework for modular mechanized meta-theory of languages with effects. Using 3MT, individual language features and their corresponding definitions -semantic functions, theorem statements and proofs -can be built separately and then reused to create different languages with fully mechanized meta-theory. 3MT combines modular datatypes and monads to define denotational semantics with effects on a per-feature basis, without fixing the particular set of effects or language constructs.One well-established problem with type soundness proofs for denotational semantics is that they are notoriously brittle with respect to the addition of new effects. The statement of type soundness for a language depends intimately on the effects it uses, making it particularly challenging to achieve modularity. 3MT solves this long-standing problem by splitting these theorems into two separate and reusable parts: a feature theorem that captures the well-typing of denotations produced by the semantic function of an individual feature with respect to only the effects used, and an effect theorem that adapts well-typings of denotations to a fixed superset of effects. The proof of type soundness for a particular language simply combines these theorems for its features and the combination of their effects. To establish both theorems, 3MT uses two key reasoning techniques: modular induction and algebraic laws about effects. Several effectful language features, including references and errors, illustrate the capabilities of 3MT. A case study reuses these features to build fully mechanized definitions and proofs for 28 languages, including several versions of mini-ML with effects.

In order to lighten the burden of programming language mechanization many approaches have been developed that tackle the substantial boilerplate which arises from variable binders. Unfortunately, the existing approaches are limited in scope. They typically do not support complex binding forms (such as multi-binders) that arise in more advanced languages, or they do not tackle the boilerplate due to mentioning variables and binders in relations. As a consequence, the human mechanizer is still unnecessarily burdened with binder boilerplate and discouraged from taking on richer languages. This paper presents Knot, a new approach that substantially extends the support for binder boilerplate. Knot is a highly expressive language for natural and concise specification of syntax with binders. Its metatheory constructively guarantees for well-formed specifications the coverage of a considerable amount of binder boilerplate, including that for well-scoping of terms and context lookups. Knot also comes with a code generator, Needle, that specializes the generic boilerplate for convenient embedding in Coq and provides a tactic library for automatically discharging proof obligations that frequently come up in proofs of weakening and substitution lemmas of type-systems. Our evaluation shows, that Needle & Knot significantly reduce the size of language mechanizations (by 40% in our case study). Moreover, as far as we know, Knot enables the most concise mechanization of the POPLmark Challenge (1a + 2a) and is two-thirds the size of the next smallest. Finally, Knot allows us to mechanize for instance dependentlytyped languages, which is notoriously challenging because of dependent contexts and mutually-recursive sorts with variables.

In this paper we present a datatype-generic approach to syntax with variable binding. A universe specifies the binding and scoping structure of object languages, including binders that bind multiple variables as well as sequential and recursive scoping. Two interpretations of the universe are given: one based on parametric higher-order abstract syntax and one on well-typed de Bruijn indices. The former provides convenient interfaces to embedded domain-specific languages, but is awkward to analyse and manipulate directly, while the latter is a convenient representation in implementations, but is unusable as a surface language. We show how to generically convert from the parametric HOAS interpretation to the de Bruijn interpretation thereby taking the pain from DSL developer to write the conversion themselves.

This paper presents 3MT, a framework for modular mechanized meta-theory of languages with effects. Using 3MT, individual language features and their corresponding definitions -- semantic functions, theorem statements and proofs-- can be built separately and then reused to create different languages with fully mechanized meta-theory. 3MT combines modular datatypes and monads to define denotational semantics with effects on a per-feature basis, without fixing the particular set of effects or language constructs. One well-established problem with type soundness proofs for denotational semantics is that they are notoriously brittle with respect to the addition of new effects. The statement of type soundness for a language depends intimately on the effects it uses, making it particularly challenging to achieve modularity. 3MT solves this long-standing problem by splitting these theorems into two separate and reusable parts: a feature theorem that captures the well-typing of denotations produced by the semantic function of an individual feature with respect to only the effects used, and an effect theorem that adapts well-typings of denotations to a fixed superset of effects. The proof of type soundness for a particular language simply combines these theorems for its features and the combination of their effects. To establish both theorems, 3MT uses two key reasoning techniques: modular induction and algebraic laws about effects. Several effectful language features, including references and errors, illustrate the capabilities of 3MT. A case study reuses these features to build fully mechanized definitions and proofs for 28 languages, including several versions of mini-ML with effects

Formal reasoning in proof assistants, also known as mechanization, has high development costs. Building modular reusable components is a key issue in reducing these costs. A stumbling block for reuse is that inductive definitions and proofs are closed to extension. This is a manifestation of the expression problem that has been addressed by the Meta-Theoryà la Carte (MTC) framework in the context of programming language meta-theory. However, MTC's use of extensible Church-encodings is unsatisfactory.This paper takes a better approach to the problem with datatypegeneric programming (DGP). It applies well-known DGP techniques to represent modular datatypes, to build functions from functor algebras with folds and to compose proofs from proof algebras by means of induction. Moreover, for certain functionality and proofs our approach can achieve more reuse than MTC: instead of composing modular components we provide a single generic definition once and for all.