The main goal of this work is to implement some versions of the Situation Calculus in Coq. This is done through various stages. First we implement the Situation and State Calculus, then the Linear Situation and State Calculus and finally the Object Situation and State Calculus. In addition we verify the soundness, with respect to standard interpretation structures, of the proposed axiomatizations. Finally we are able to specify and verify systems. 1 Acknowledgments To Prof. Am'ilcar Sernadas, my supervisor, for offering me the opportunity to do this work, for his motivation and many suggestions. To Jaime for his patience and generous help which were invaluable. To Carlos and Paulo for their suggestions and positive presence. To Alexandra and Nuno for their companionship and constant support. My thanks also go to all section 84. This work was partially supported by the PRAXIS XXI Program and FCT, as well as by PRAXIS XXI Projects 2/2.1/MAT/262/94 SitCalc, PCEX/P/MAT/46/96 ACL pl...

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this paper is but a tiny initial fragment of the theory of categories. However, it is quite promising, in that the power of dependent types and inductive types (or at least \Sigma-types) is put to full use; note in particular the dependent equality between morphisms of possibly non-convertible types.

Relative monads are a generalisation of ordinary monads where the underlying functor need not be an endofunctor. In this paper, we describe a formalisation of the basic theory of relative monads in the interactive theorem prover and dependently typed programming language Agda. This comprises the requisite basic category theory, the central concepts of the theory of relative monads and adjunctions, compared to their ordinary counterparts, and two running examples from programming theory.

Formal specifications of logics share many standard concepts and in order to avoid repetitious works, it seems desirable to express these specifications in a uniform framework. General logics à la J. Meseguer provide an uniform and modular way of encoding a logical language, its semantics and its inference system. Hence, we describe here a formalisation of general logics in the calculus of inductive constructions thus providing a generic and modular set of speci cations (with the proofs of s...

This paper has something new and positive to say about propositional equality in programming and proof systems based on the Curry-Howard correspondence between propositions and types. We have found a way to present a propositional equality type • which is substitutive, allowing us to reason by replacing equal for equal in propositions; • which reflects the observable behaviour of values rather than their construction: in particular, we have extensionality— functions are equal if they take equal inputs to equal outputs; • which retains strong normalisation, decidable typechecking and canonicity—the property that closed normal forms inhabiting datatypes have canonical constructors; • which allows inductive data structures to be expressed in terms of a standard characterisation of well-founded trees; • which is presented syntactically—you can implement it directly, and we are doing so—this approach stands at the core of Epigram 2; • which you can play with now: we have simulated our system by a shallow embedding in Agda 2, shipping as part of the standard examples package for that system [20]. Until now, it has always been necessary to sacrifice some of these aspects. The closest attempt in the literature is Altenkirch’s construction of a setoid-model for a system with canonicity and extensionality on top of an intensional type theory with proof-irrelevant propositions [4]. Our new proposal simplifies Altenkirch’s construction by adopting McBride’s heterogeneous approach to equality [18]. 1.

We give an intuitionistic view of Seely’s interpretation of Martin-Löf’s intuitionistic type theory in locally cartesian closed categories. The idea is to use Martin-Löf type theory itself as metalanguage, and E-categories, the appropriate notion of categories when working in this metalanguage. As an E-categorical substitute for the formal system of Martin-Löf type theory we use E-categories with families (E-cwfs). These come in two flavours: groupoid-style E-cwfs and proofirrelevant E-cwfs. We then analyze Seely’s interpretation as consisting of three parts. The first part is purely categorical: the interpretation of groupoid-style E-cwfs in E-locally cartesian closed categories. (The key part of this interpretation has been type-checked in the Coq system.) The second is a coherence problem which relates groupoid-style E-cwfs with proofirrelevant ones. The third is a purely syntactic problem: that proof-irrelevant E-cwfs are equivalent to traditional lambda calculus based formulations of Martin-Löf type theory.