Syntax Involving Binders Murdoch Gabbay Cambridge University DPMMS Cambridge CB2 1SB, UK M.J.Gabbay@cantab.com Andrew Pitts Cambridge University Computer Laboratory Cambridge CB2 3QG, UK ap@cl.cam.ac.uk Abstract The Fraenkel-Mostowski permutation model of set theory with atoms (FM-sets) can serve as the semantic basis of meta-logics for specifying and reasoning about formal systems involving name binding, ff-conversion, capture avoiding substitution, and so on. We show that in FM-set theory one can express statements quantifying over `fresh' names and we use this to give a novel set-theoretic interpretation of name abstraction. Inductively defined FM-sets involving this name-abstraction set former (together with cartesian product and disjoint union) can correctly encode object-level syntax modulo ff-conversion. In this way, the standard theory of algebraic data types can be extended to encompass signatures involving binding operators. In particular, there is an associated n...

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A logic for specification and verification is derived from the axioms of Zermelo-Fraenkel set theory. The proofs are performed using the proof assistant Isabelle. Isabelle is generic, supporting several different logics. Isabelle has the flexibility to adapt to variants of set theory. Its higher-order syntax supports the definition of new binding operators. Unknowns in subgoals can be instantiated incrementally. The paper describes the derivation of rules for descriptions, relations and functions, and discusses interactive proofs of Cantor’s Theorem, the Composition of Homomorphisms challenge [9], and Ramsey’s Theorem [5]. A generic proof assistant can stand up against provers dedicated to particular logics. Key words. Isabelle, set theory, generic theorem proving, Ramsey’s Theorem,

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We define the notion of weak relative pseudo-complement on meet semi-lattices, and we show that it is strictly weaker than relative pseudo-complementation, but stronger than pseudo-complementation. Our main result is that if a complete lattice L is meet-continuous, then every closure operator on L admits weak relative pseudo-complements with respect to continuous closure operators on L.

We advocate a pragmatic approach to constructive set theory, using axioms based solely on set-theoretic principles that are directly relevant to (constructive) mathematical practice. Following this approach, we present theories ranging in power from weaker predicative theories to stronger impredicative ones. The theories we consider all have sound and complete classes of category-theoretic models, obtained by axiomatizing the structure of an ambient category of classes together with its subcategory of sets. In certain special cases, the categories of sets have independent characterizations in familiar category-theoretic terms, and one thereby obtains a rich source of naturally occurring mathematical models for (both predicative and impredicative) constructive set theories.

The open calculus of constructions integrates key features of Martin-Löf's type theory, the calculus of constructions, Membership Equational Logic, and Rewriting Logic into a single uniform language. The two key ingredients are dependent function types and conditional rewriting modulo equational theories. We explore the open calculus of constructions as a uniform framework for programming, specification and interactive verification in an equational higher-order style. By having equational logic and rewriting logic as executable sublogics we preserve the advantages of a first-order semantic and logical framework and especially target applications involving symbolic computation and symbolic execution of nondeterministic and concurrent systems.

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This paper investigates an alternative set theory (due to Peter Aczel) called Hyperset Theory. Aczel uses a graphical representation for sets and thereby allows the representation of non-well-founded sets. A program, called HYPERSOLVER, which can solve systems of equations defined in terms of sets in the universe of this new theory is presented. This may be a useful tool for commonsense reasoning. 1 INTRODUCTION Set theory has long occupied a unique place in mathematics since it allows various other branches of mathematics to be formally defined within it. The theory has ignited many debates on its nature and a number of different axiomatizations were developed to formalize its underlying `philosophical' principles. Collecting entities into an abstraction for further thought (i.e., set construction) is an important process in mathematics, and this brings in assorted problems [5]. The theory had many ground-shaking crises (like the discovery of the Russell's Paradox [6]) throughout it...

Abstract. Although higher-order rewrite systems (HRS) seem to have a first-order flavor, the direct translation into first-order rewrite systems, using e.g. explicit substitutions, is by no means trivial. In this paper, we explore a two-stage approach, by showing how higher-order pattern rewrite systems, and in fact a somewhat more general class, can be expressed by conditional first-order rewriting in the open calculus of constructions (OCC), which itself has been presented and implemented using explicit substitutions. The key feature of OCC that we exploit is that conditions are allowed to contain quantifiers and equations which can be solved using first-order matching. The way we express HRS works in spite of the fact that structural equality of OCC does not subsume α-conversion. Another topic that we touch upon in this paper is the use of higher-order abstract syntax in a classical framework like OCC, because it is often used in connection with higher-order rewriting. 1