Abstract. In this paper we define an inductive set that is bijective with the ff-equated lambda-terms. Unlike de-Bruijn indices, however, our inductive definition includes names and reasoning about this definition is very similar to informal reasoning on paper. For this we provide a structural induction principle that requires to prove the lambda-case for fresh binders only. The main technical novelty of this work is that it is compatible with the axiom-of-choice (unlike earlier nominal logic work by Pitts et al); thus we were able to implement all results in Isabelle/HOL and use them to formalise the standard proofs for Church-Rosser and strongnormalisation. Keywords. Lambda-calculus, nominal logic, structural induction, theoremassistants.

3 A complete example

Concurrent computation can be given an abstract mathematical treatment very similar to that provided for sequential computation by domain theory and denotational semantics of Scott and Strachey.

Variable binding is a prevalent feature of the syntax and proof theory of many logical systems. In this paper, we define a programming language that provides intrinsic support for both representing and computing with binding. This language is extracted as the Curry-Howard interpretation of a focused sequent calculus with two kinds of implication, of opposite polarity. The representational arrow extends systems of definitional reflection with a notion of scoped inference rules, which are used to represent binding. On the other hand, the usual computational arrow classifies recursive functions defined by pattern-matching. Unlike many previous approaches, both kinds of implication are connectives in a single logic, which serves as a rich logical framework capable of representing inference rules that mix binding and computation. 1

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The nominal datatype package implements an infrastructure in Isabelle/HOL for defining languages involving binders and for reasoning conveniently about alpha-equivalence classes. Pitts stated some general conditions under which functions over alpha-equivalence classes can be defined by a form of structural recursion and gave a clever proof for the existence of a primitive-recursion combinator. We give a version of this proof that works directly over nominal datatypes and does not rely upon auxiliary constructions. We further introduce proving tools and a heuristic that made the automation of our proof tractable. This automation is an essential prerequisite for the nominal datatype package to become useful.

Type-checking algorithms for dependent type theories often rely on the interpretation of terms in some semantic domain of values when checking equalities. Here we analyze a version of Coquand’s algorithm for checking the βη-equality of such semantic values in a theory with a predicative universe hierarchy and large elimination rules. Although this algorithm does not rely on normalization by evaluation explicitly, we show that similar ideas can be employed for its verification. In particular, our proof uses the new notions of contextual reification and strong semantic equality. The algorithm is part of a bi-directional type checking algorithm which checks whether a normal term has a certain semantic type, a technique notion of semantic domain in order to accommodate a variety of possible implementation techniques, such as normal forms, weak head normal forms, closures, and compiled code. Our aim is to get closer than previous work to verifying the type-checking algorithms which are actually used in practice.

Abstract. Combining higher-order abstract syntax and (co)-induction in a logical framework is well known to be problematic. Previous work [ACM02] described the implementation of a tool called Hybrid, within Isabelle HOL, 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 syntax, while offering tools for reasoning about them at the higher level. In this paper we describe how to use it in a multi-level reasoning fashion, similar in spirit to other meta-logics such as Linc and Twelf. 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

Powerful proof techniques, such as logical relation arguments, have been developed for establishing the strong normalisation property of term-rewriting systems. The first author used such a logical relation argument to establish strong normalising for a cut-elimination procedure in classical logic. He presented a rather complicated, but informal, proof establishing this property. The difficulties in this proof arise from a quite subtle substitution operation. We have formalised this proof in the theorem prover Isabelle/HOL using the Nominal Datatype Package, closely following the first authors PhD. In the process, we identified and resolved a gap in one central lemma and a number of smaller problems in others. We also needed to make one informal definition rigorous. We thus show that the original proof is indeed a proof and that present automated proving technology is adequate for formalising such difficult proofs.

Abstract. Placing our result in a web of related mechanised results, we give a direct proof that the de Bruijn λ-calculus (à la Huet, Nipkow and Shankar) is isomorphic to an α-quotiented λ-calculus. In order to establish the link, we introduce an “index-carrying ” abstraction mechanism over de Bruijn terms, and consider it alongside a simplified substitution mechanism. Relating the new notions to those of the α-quotiented and the proper de Bruijn formalisms draws on techniques from the theory of nominal sets. 1