scholarly journals A definitional approach to primitivexs recursion over higher order abstract syntax

2003 ◽  
Author(s):  
S. J. Ambler ◽  
R. L. Crole ◽  
Alberto Momigliano
Keyword(s):  
Higher Order ◽  
Abstract Syntax

scholarly journals Translating Higher-Order Specifications to Coq Libraries Supporting Hybrid Proofs

10.29007/jqtz ◽  
2018 ◽  
Author(s):  
Nada Habli ◽  
Amy P. Felty
Keyword(s):  
Programming Languages ◽  
Automatic Generation ◽  
Higher Order ◽  
Specification Language ◽  
Proof Assistants ◽  
Abstract Syntax ◽  
Formal Systems ◽  
Ongoing Work ◽  
General Specification ◽  
Automated Proof

We describe ongoing work on building an environment to support reasoning in proof assistants that represent formal systems using higher-order abstract syntax (HOAS). We use a simple and general specification language whose syntax supports HOAS. Using this language, we can encode the syntax and inference rules of a variety of formal systems, such as programming languages and logics. We describe our tool, implemented in OCaml, which parses this syntax, and translates it to a Coq library that includes definitions and hints for aiding automated proof in the Hybrid system. Hybrid itself is implemented in Coq, and designed specifically to reason about such formal systems. Given an input specification, the library that is automatically generated by our tool imports the general Hybrid library and adds definitions and hints for aiding automated proof in Hybrid about the specific programming language or logic defined in the specification. This work is part of a larger project to compare reasoning in systems supporting HOAS. Our current work focuses on Hybrid, Abella, Twelf, and Beluga, and the specification language is designed to be general enough to allow the automatic generation of libraries for all of these systems from a single specification.

scholarly journals Primitive recursion for higher-order abstract syntax

2001 ◽  
Vol 266 (1-2) ◽  
pp. 1-57 ◽  
Author(s):  
Carsten Schürmann ◽  
Joëlle Despeyroux ◽  
Frank Pfenning
Keyword(s):  
Higher Order ◽  
Abstract Syntax ◽  
Primitive Recursion

Mechanizing proofs with logical relations – Kripke-style

2018 ◽  
Vol 28 (9) ◽  
pp. 1606-1638 ◽  
Author(s):  
ANDREW CAVE ◽  
BRIGITTE PIENTKA
Keyword(s):  
Case Studies ◽  
Operational Semantics ◽  
Lambda Calculus ◽  
Equational Theory ◽  
Higher Order ◽  
Abstract Syntax ◽  
Completeness Proof ◽  
Typed Lambda Calculus ◽  
Contextual Equivalence ◽  
The Right

Proofs with logical relations play a key role to establish rich properties such as normalization or contextual equivalence. They are also challenging to mechanize. In this paper, we describe two case studies using the proof environmentBeluga: First, we explain the mechanization of the weak normalization proof for the simply typed lambda-calculus; second, we outline how to mechanize the completeness proof of algorithmic equality for simply typed lambda-terms where we reason about logically equivalent terms. The development of these proofs inBelugarelies on three key ingredients: (1) we encode lambda-terms together with their typing rules, operational semantics, algorithmic and declarative equality using higher order abstract syntax (HOAS) thereby avoiding the need to manipulate and deal with binders, renaming and substitutions, (2) we take advantage ofBeluga's support for representing derivations that depend on assumptions and first-class contexts to directly state inductive properties such as logical relations and inductive proofs, (3) we exploitBeluga's rich equational theory for simultaneous substitutions; as a consequence, users do not need to establish and subsequently use substitution properties, and proofs are not cluttered with references to them. We believe these examples demonstrate thatBelugaprovides the right level of abstractions and primitives to mechanize challenging proofs using HOAS encodings. It also may serve as a valuable benchmark for other proof environments.

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