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Inÿnite trees form a free completely iterative theory over any given signature—this fact, proved by Elgot, Bloom and Tindell, turns out to be a special case of a much more general categorical result exhibited in the present paper. We prove that whenever an endofunctor H of a category has ÿnal coalgebras for all functors H () + X , then those coalgebras, TX(More)
Introduction The original motivation 1 for the work described in this paper was to determine the proof theoretic strength of the type theories implemented in the proof development systems Lego and Coq, Luo and Pollack 92, Barras et al 96]. These type theories combine the impredicative type of propositions 2 , from the calculus of constructions, Coquand 90],(More)
We present a generalisation of the type-theoretic interpretation of constructive set theory into Martin-Löf type theory. The original interpretation treated logic in Martin-Löf type theory via the propositions-as-types interpretation. The generalisation involves replacing Martin-Löf type theory with a new type theory in which logic is treated as primitive.(More)
This paper gives a generalisation of a version of order sorted logic and an abstract axiomatic setting for the treatment of substitution. The two ideas are shown to be related, an equational speciication of term declaration logic being a presentation of a nitary generalised compositum. The axiomatic approach to substitution is compared with some others.