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31
Formalized proof, computation, and the construction problem in algebraic geometry
, 2004
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Towards observational type theory
 In preparation
, 2006
"... Observational Type Theory (OTT) combines beneficial aspects of Intensional and Extensional Type Theory (ITT/ETT). It separates definitional equality, decidable as in ITT, and a substitutive propositional equality, capturing extensional equality of functions, as in ETT. Moreover, canonicity holds: an ..."
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Observational Type Theory (OTT) combines beneficial aspects of Intensional and Extensional Type Theory (ITT/ETT). It separates definitional equality, decidable as in ITT, and a substitutive propositional equality, capturing extensional equality of functions, as in ETT. Moreover, canonicity holds: any closed term is definitionally reducible to a canonical value. Building on previous work by each author, this article reports substantial progress in the form of a simplified theory with a straightforward syntactic presentation, which we have implemented. As well as simplifying reasoning about functions, OTT offers potential foundational benefits, e.g. it gives rise to a closed type theory encoding inductive datatypes. 1.
Experiments in Formalizing Basic Category Theory in Higher Order Logic and Set Theory
, 1995
"... this paper is the product category, defined by ..."
Situation Calculus in Coq
, 1998
"... 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 ..."
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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...
Relative monads formalised
 UNDER CONSIDERATION FOR PUBLICATION IN MATH. STRUCT. IN COMP. SCIENCE
, 2010
"... 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 req ..."
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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.
Formalisation of General Logics in the Calculus of Inductive Constructions: Towards an Abstract . . .
, 1999
"... 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 in ..."
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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...
The Interpretation of Inuitionistic . . .
, 2008
"... We give an intuitionistic view of Seely’s interpretation of MartinLöf’s intuitionistic type theory in locally cartesian closed categories. The idea is to use MartinLöf type theory itself as metalanguage, and Ecategories, the appropriate notion of categories when working in this metalanguage. As a ..."
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We give an intuitionistic view of Seely’s interpretation of MartinLöf’s intuitionistic type theory in locally cartesian closed categories. The idea is to use MartinLöf type theory itself as metalanguage, and Ecategories, the appropriate notion of categories when working in this metalanguage. As an Ecategorical substitute for the formal system of MartinLöf type theory we use Ecategories with families (Ecwfs). These come in two flavours: groupoidstyle Ecwfs and proofirrelevant Ecwfs. We then analyze Seely’s interpretation as consisting of three parts. The first part is purely categorical: the interpretation of groupoidstyle Ecwfs in Elocally cartesian closed categories. (The key part of this interpretation has been typechecked in the Coq system.) The second is a coherence problem which relates groupoidstyle Ecwfs with proofirrelevant ones. The third is a purely syntactic problem: that proofirrelevant Ecwfs are equivalent to traditional lambda calculus based formulations of MartinLöf type theory.
Machine assisted proofs in the theory of monads
"... In this paper (and the ongoing development it describes) we formalise some aspects of the theory of monads in Agda. In doing so we give motivation for our work, relate it to previous efforts, explain why it pushes the theory and implementation of type theoretic theorem provers to breaking point, and ..."
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In this paper (and the ongoing development it describes) we formalise some aspects of the theory of monads in Agda. In doing so we give motivation for our work, relate it to previous efforts, explain why it pushes the theory and implementation of type theoretic theorem provers to breaking point, and suggest some future directions.