Programming with binders and indexed data-types

Abstract

We show how to combine a general purpose type system for an existing language with support for programming with binders and contexts by refining the type system of ML with a restricted form of dependent types where index objects are drawn from contextual LF. This allows the user to specify formal systems within the logical framework LF and index ML types with contextual LF objects. Our language design keeps the index language generic only requiring decidability of equality of the index language providing a modular design. To illustrate the elegance and effectiveness of our language, we give programs for closure conversion and normalization by evaluation.Our three key technical contribution are: 1) We give a bidirectional type system for our core language which is centered around refinement substitutions instead of constraint solving. As a consequence, type checking is decidable and easy to trust, although constraint solving may be undecidable. 2) We give a big-step environment based operational semantics with environments which lends itself to efficient implementation. 3) We prove our language to be type safe and have mechanized our theoretical development in the proof assistant Coq using the fresh approach to binding.

The Next 700 Challenge Problems for Reasoning with Higher-Order Abstract Syntax Representations

“…Each case of the proof corresponds to one branch in the function. Recently, Cave and Pientka have shown how to extend Beluga with support for inductive definitions (i.e., recursive datatypes) [8] and first-class substitutions [9] leading to Beluga μ . This allows us to define predicates about contexts and contextual objects, relate contexts via substitutions, and more importantly prove properties about them inductively.…”

“…However, Beluga μ [8] also supports indexed recursive datatypes and Type preservation for parallel reductions in Beluga first-class substitutions [9]. This allows us to define predicates about contexts and contextual objects, relate contexts via substitutions, and more importantly prove properties about them inductively.…”

“…We discuss this issue further in Section 8. 8 Using lists as the data-structure underlying contexts is not written in stone. In fact, the Isabelle/HOL version of Hybrid uses sets [16] and Abella may also offer multi-sets in the near future.…”

Mtac

“…In the last five years there has been increasing interest in languages that support safe tactics to manipulate proof terms of dependently typed logics. Delphin [23], Beluga [21,22,4], and VeriML [29,30] are languages that, like Mtac, fall into this category. By "safe" we mean that, if the execution of a tactic terminates, then the resulting proof term has the type specified by the tactic.…”

“…Indeed, our proof of type safety is completely straightforward, as it inherits from CIC all the relevant properties such as type preservation under substitution. Having a simple meta-theory is particularly important to avoid precluding future language extensions-indeed, extensions of the previous systems have often required a reworking of their meta-theory [30,4].…”