Bedwyr is a generalization of logic programming that allows model checking directly on syntactic expressions possibly containing bindings. This system, written in OCaml, is a direct implementation of two recent advances in the theory of proof search. The first is centered on the fact that both finite success and finite failure can be captured in the sequent calculus by incorporating inference rules for definitions that allow fixed points to be explored. As a result, proof search in such a sequent calculus can capture simple model checking problems as well as may and must behavior in operational semantics. The second is that higherorder abstract syntax is directly supported using term-level λ-binders and the quantifier known as ∇. These features allow reasoning directly on expressions containing bound variables. 2

Many semantical aspects of programming languages are specified through calculi for constructing proofs: consider, for example, the specification of structured operational semantics, labeled transition systems, and typing systems. Recent proof theory research has identified two features that allow direct, logic-based reasoning about such descriptions: the treatment of atomic judgments as fixed points (recursive definitions) and an encoding of binding constructs via generic judgments. However, the logics encompassing these two features have thus far treated them orthogonally. In particular, they have not contained the ability to form definitions of object-logic properties that themselves depend on an intrinsic treatment of binding. We propose a new and simple integration of these features within an intuitionistic logic enhanced with induction over natural numbers and we show that the resulting logic is consistent. The pivotal part of the integration allows recursive definitions to define generic judgments in general and not just the simpler atomic judgments that are traditionally allowed. The usefulness of this logic is illustrated by showing how it can provide elegant treatments of object-logic contexts that appear in proofs involving typing calculi and arbitrarily cascading substitutions in reducibility arguments.

An extension of the simply-typed lambda-calculus allowing iteration and case reasoning over terms defined by means of higher order abstract syntax has recently been introduced by Joëlle Despeyroux, Frank Pfenning and Carsten Schürmann. This thorny mixing is achieved thanks to the help of the operator ` ' of modal logic IS4. Here we give a new presentation of their system, with reduction rules, instead of evaluation judgments, that compute the canonical forms of terms. Our presentation is based on a modal lambda-calculus that is better from the user's point of view, is more concise and we do not impose a particular strategy of reduction during the computation. Our system enjoys the decidability of typability, soundness of typed reduction with respect to typing rules, the Church-Rosser and strong normalization properties. Finally it is a conservative extension of the simply-typed lambda-calculus.

The Hybrid system (Ambler et al., 2002b), implemented within Isabelle/HOL, allows object logics to be represented using higher order abstract syntax (HOAS), and reasoned about using tactical theorem proving in general and principles of (co)induction in particular. The form of HOAS provided by Hybrid is essentially a lambda calculus with constants. Of fundamental interest is the form of the lambda abstractions provided by Hybrid. The user has the convenience of writing lambda abstractions using names for the binding variables. However each abstraction is actually a definition of a de Bruijn expression, and Hybrid can unwind the user’s abstractions (written with names) to machine friendly de Bruijn expressions (without names). In this sense the formal system contains a hybrid of named and nameless bound variable notation. In this paper, we present a formal theory in a logical framework which can be viewed as a model of core Hybrid, and state and prove that the model is representationally adequate for HOAS. In particular, it is the canonical translation function from λ-expressions to Hybrid that witnesses adequacy. We also prove two results that characterise how Hybrid represents certain classes of λ-expressions. The Hybrid system contains a number of different syntactic classes of expression, and associated abstraction mechanisms. Hence this paper also aims to provide a self-contained theoretical introduction to both the syntax and key ideas of the system; background in automated theorem proving is not essential, although this paper will be of considerable interest to those who wish to work with Hybrid in Isabelle/HOL.

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