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ECC, an Extended Calculus of Constructions
, 1989
"... We present a higherorder calculus ECC which can be seen as an extension of the calculus of constructions [CH88] by adding strong sum types and a fully cumulative type hierarchy. ECC turns out to be rather expressive so that mathematical theories can be abstractly described and abstract mathematics ..."
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We present a higherorder calculus ECC which can be seen as an extension of the calculus of constructions [CH88] by adding strong sum types and a fully cumulative type hierarchy. ECC turns out to be rather expressive so that mathematical theories can be abstractly described and abstract mathematics may be adequately formalized. It is shown that ECC is strongly normalizing and has other nice prooftheoretic properties. An !\GammaSet (realizability) model is described to show how the essential properties of the calculus can be captured settheoretically.
Formalising mathematics in UTT: fundamentals and case studies
, 1994
"... We give a detailed account of the use of type theory as a foundational language to formalise mathematics. We develop in the type system UTT a coherent approach to naive set theory and elementary mathematical notions. In the second part of the paper, we present a fullychecked example based on our re ..."
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We give a detailed account of the use of type theory as a foundational language to formalise mathematics. We develop in the type system UTT a coherent approach to naive set theory and elementary mathematical notions. In the second part of the paper, we present a fullychecked example based on our representation of naive set theory. Contents 1 Introduction 1 2 Fundamentals 3 2.1 Naive set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.1.1 Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.1.2 Discrete sets . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.1.3 Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.1.4 The category of sets . . . . . . . . . . . . . . . . . . . . . 5 2.1.5 Multivariate maps . . . . . . . . . . . . . . . . . . . . . . 6 2.1.6 Predicates and relations . . . . . . . . . . . . . . . . . . . 7 2.1.7 Subsets and powerset . . . . . . . . . . . . . . . . . . . . 7 2.1.8 Quotients . . . . . . . . . . . . . . . ...