Needle & Knot: Binder Boilerplate Tied Up

Abstract

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.

Bindings as bounded natural functors

“…The literature on specification mechanisms for syntax with bindings offers a wide range of syntactic formats of various levels of sophistication, including those underlying CαMl [Pottier 2006], Ott [Sewell et al 2010], Unbound [Weirich et al 2011], Isabelle Nominal2 [Urban and Kaliszyk 2012], and Needle & Knot [Keuchel et al 2016]. By contrast, we axiomatize binders semantically, via a class of functors.…”

“…An expressive type theory such as Agda's or Coq's caters for generic developments using universes populated by codes for types. Besides containers, universe-based developments include the works of Keuchel and Jeuring [2012] (using HOAS and nameless representations), Keuchel et al [2016], Licata and Harper [2009], Allais et al [2018], Lee et al [2012] (using nameless representations), and Copello et al [2018] (using a nameful representation). Some of these support formats for complex binder patterns and feature a substantial amount of infrastructure lemmas and formalized case studies.…”

Scopes as types

“…Various libraries and DSLs have been developed to automate the support for name binding in proof assistants such as Coq. For example, AutoSubst [Schäfer et al 2015] is a Coq library that derives substitution and renaming functions and lemmas about their properties from annotations on an inductive type definition; Ott [Sewell et al 2010] is a DSL to define type systems and reduction rules for languages with name binding, from which it automatically generates data types and substitution functions for different proof assistant backends; Lem [Mulligan et al 2014] and Needle & Knot [Keuchel et al 2016] provide similar support. These tools follow similar schemas to define bindings: an annotation in the constructor signature indicates that a binding occurrence is bound in one or more sub-terms of the binding construct.…”

Visitors unchained

“…In Talcott's binding structures [1993], each operator carries a fixed number of variables as well as a fixed number of subterms, and a fixed binding relation tells which variables are considered bound within which subterms. More generally, Ott [Sewell et al 2010] and Knot [Keuchel et al 2016] allow defining auxiliary name-collecting functions, which are then used to specify which sets of names are in scope in which subterms. Still more generally, it seems, InBound [Keuchel and Schrijvers 2015] exploits inherited and synthesized attributes to express the flow of bound variables and contexts.…”