Hybrid

Abstract

Combining higher-order abstract syntax and (co)-induction in a logical framework is well known to be problematic. We describe the theory and the practice of a tool called Hybrid, within Isabelle/HOL and Coq, which aims to address many of these difficulties. It allows object logics to be represented using higher-order abstract syntax, and reasoned about using tactical theorem proving and principles of (co)induction. Moreover, it is definitional, which guarantees consistency within a classical type theory. The idea is to have a de Bruijn representation of λ-terms providing a definitional layer that allows the user to represent object languages using higher-order abstract syntax, while offering tools for reasoning about them at the higher level. In this paper we describe how to use Hybrid in a multi-level reasoning fashion, similar in spirit to other systems such as Twelf and Abella. By explicitly referencing provability in a middle layer called a specification logic, we solve the problem of reasoning by (co)induction in the presence of non-stratifiable hypothetical judgments, which allow very elegant and succinct specifications of object logic inference rules. We first demonstrate the method on a simple example, formally proving type soundness (subject reduction) for a fragment of a pure functional language, using a minimal intuitionistic logic as the specification logic. We then prove an analogous result for a continuation-machine presentation of the operational semantics of the same language, encoded this time in an ordered linear logic that serves as the specification layer. This example demonstrates the ease with which we can incorporate new specification logics, and also illustrates a significantly more complex object logic whose encoding is elegantly expressed using features of the new specification logic.

QED at Large: A Survey of Engineering of Formally Verified Software

“…Other notable frameworks for language design include the Fiat Delaware et al, 2015) framework for Coq, as well as the Hybrid (Felty and Momigliano, 2012) framework for Isabelle/HOL and Coq, which addresses the difficulties of using HOAS with inductive and coinductive proofs.…”

Rapid Prototyping Formal Systems in MMT: 5 Case Studies

“…We declare the side condition Val as an untyped constant and a rule ValRule of type ExternalConditionRule. The Scala code of ValRule is given in Listing 6 8 . It checks the condition Val(t, a), i.e., that t is of the form free n or λ F; if that fails, it registers an error message that the IDE will show to the user.…”

“…Features realized generically in logical frameworks include reasoning about object logics (Twelf [17], Abella [9]), interactive theorem proving (Isabelle [16]), concurrency (CLF [24]), reasoning about contexts (Beluga [18]), rewriting (Dedukti [2]), side conditions (LLF P [12]), or integration with proof assistants (Hybrid [8]). Moreover, many logic-specific systems are investigating how to allow users to experiment with system behavior inside the system, e.g., via meta-programming (for Idris in [4], for Lean in [7]) or unification hints (for Coq in [10], for Matita in [1]).…”

A Case Study on Logical Relations using Contextual Types

“…There are two main approaches to support reasoning about HOAS encodings: 1) In the prooftheoretic approaches, we adopt a two-level system where we implement a specification logic (similar to LF) inside a higher-order logic supporting (co)inductive definitions, the approach taken in Abella (9), or type theory, the aproach taken in Hybrid (8). To distinguish in the proof theory between quantification over variables and quantification over terms, (10) introduce a new quantifier, ∇, to describe nominal abstraction logically.…”