We present a logical framework # for reasoning on a very general class of languages featuring binding operators, called nominal algebras, presented in higher-order abstract syntax (HOAS). # is based on an axiomatic syntactic standpoint and it consists of a simple types theory a la Church extended with a set of axioms called the Theory of Contexts, recursion operators and induction principles. This framework is rather expressive and, most notably, the axioms of the Theory of Contexts allow for a smooth reasoning of schemata in HOAS. An advantage of this framework is that it requires a very low mathematical and logical overhead. Some case studies and comparison with related work are briefly discussed.

We present and discuss various formalizations of Modal Logics in Logical Frameworks based on Type Theories. We consider both Hilbert- and Natural Deductionstyle proof systems for representing both truth (local) and validity (global) consequence relations for various Modal Logics. We introduce several techniques for encoding the structural peculiarities of necessitation rules, in the typed -calculus metalanguage of the Logical Frameworks. These formalizations yield readily proof-editors for Modal Logics when implemented in Proof Development Environments, such as Coq or LEGO. Keywords: Hilbert and Natural-Deduction proof systems for Modal Logics, Logical Frameworks, Typed -calculus, Proof Assistants. Introduction In this paper we address the issue of designing proof development environments (i.e. "proof editors" or, even better, "proof assistants") for Modal Logics, in the style of [11, 12]. To this end, we explore the possibility of using Logical Frameworks (LF's) based on Type Theory...

The Theory of Contexts is a type-theoretic axiomatization which has been recently proposed by some of the authors for giving a metalogical account of the fundamental notions of variable and context as they appear in Higher Order Abstract Syntax. In this paper, we prove that this theory is consistent by building a model based on functor categories. By means of a suitable notion of forcing, we prove that this model validates Classical Higher Order Logic, the Theory of Contexts, and also (parametrised) structural induction and recursion principles over contexts. The approach we present in full detail should be useful also for reasoning on other models based on functor categories. Moreover, the construction could be adopted, and possibly generalized, also for validating other theories of names and binders. Contents 1 The object language 4 2 The metalanguage (Framework System #) 6 2.1 Syntax 6 2.2 Typing and logical judgements 7 2.3 Adequacy of the encoding 8 2.4 Remarks on the design of # 9 3 Category-theoretic preliminaries 11 4.1 The ambient categories 4.2 Interpreting types 16 4.3 Interpreting environments 18 4.4 Interpreting the typing judgement of terms 19 4.5 Interpreting logical judgements 21 is a model of # 22 5.1 Forcing 22 5.2 Characterisation of Leibniz equality 23 models logical axioms and rules 26 models the Theory of Contexts 27 6 Recursion 28 6.1 First-order recursion 28 6.2 Higher-order recursion 31 7 Induction 33 7.1 First-order induction 34 7.2 Higher-order induction 37 8 Connections with tripos theory 38 9 Related work 41 9.1 Semantics based on functor categories 41 9.2 Logics for nominal calculi 44 10 Conclusions 45 A Proofs 46 A.1 Proof of Proposition 4.2 46 A.2 Proof of Proposition 4.3 47 A.3 Proof of Theorem 5.1 48 A.4 Proof of...

Abstract. We present an implementation of a program logic of objects, extending that (AL) of Abadi and Leino. In particular, the implementation uses higher-order abstract syntax (HOAS) and—unlike previous approaches using HOAS—at the same time uses the built-in higher-order logic of the theorem prover to formulate specifications. We give examples of verifications, extending those given in [1], that have been attempted with the implementation. Due to the mixing of HOAS and built-in logic the soundness of the encoding is nontrivial. In particular, unhke in other HOAS encodings of program logics, it is not possible to directly reduce normal proofs in the higher-order system to proofs in the first-order object logic. 1

We present a framework for machine implementation of families of non-classical logics with Kripke-style semantics. We decompose a logic into two interacting parts, each a natural deduction system: a base logic of labelled formulae, and a theory of labels characterizing the properties of the Kripke models. By appropriate combinations we capture both partial and complete fragments of large families of non-classical logics such as modal, relevance, and intuitionistic logics. Our approach is modular and supports uniform proofs of correctness and proof normalization. We have implemented our work in the Isabelle Logical Framework.

This paper is part of an ongoing research programme at the Computer Science Department of the University of Udine on proof editors, started in 1992, based on HOAS encodings in dependent typed #-calculus for program logics [15, 21, 16]. In this paper, we investigate the applicability of this approach to the modal -calculus. Due to its expressive power, we adopt the Calculus of Inductive Constructions (CIC), implemented in the system Coq. Beside its importance in the theory and verification of processes, the modal -calculus is interesting also for its syntactic and proof theoretic peculiarities. These idiosyncrasies are mainly due to a) the negative arity of "" (i.e., the bound variable x ranges over the same syntactic class of x#); b) a context-sensitive grammar due the condition on x#; c) rules with complex side conditions (sequent-style "proof " rules). These anomalies escape the "standard" representation paradigm of CIC; hence, we need to accommodate special techniques for enforcing these peculiarities. Moreover, since generated editors allow the user to reason "under assumptions", the designer of a proof editor for a given logic is urged to look for a Natural Deduction formulation of the system. Hence, we introduce a new proof system N # K in Natural Deduction style for K. This system should be more natural to use than traditional Hilbert-style systems; moreover, it takes best advantage of the possibility of manipulating assumptions o#ered by CIC in order to implement the problematic substitution of formul for variables. In fact, substitutions are delayed as much as possible, and are kept in the derivation context by means of assumptions. This mechanism fits perfectly the stack discipline of assumptions of Natural Deduction, and it is neatly formalized in CIC. Bes...

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this paper to advocate the use of functor categories as a semantic foundation of higher-order abstract syntax (HOAS). By way of example, we will show how functor categories can be used for at least the following applications:

avoids explicit