By Peter Aczel, Nicola Gambino (auth.), Paul Callaghan, Zhaohui Luo, James McKinna, Robert Pollack, Robert Pollack (eds.)
This publication constitutes the completely refereed post-proceedings of the foreign Workshop of the kinds operating team, varieties 2000, held in Durham, united kingdom in December 2000.
The 15 revised complete papers provided have been conscientiously reviewed and chosen in the course of rounds of refereeing and revision. All present matters on kind conception and kind structures and their purposes to programming, structures layout, and evidence concept are addressed.
Read or Download Types for Proofs and Programs: International Workshop, TYPES 2000 Durham, UK, December 8–12, 2000 Selected Papers PDF
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Additional resources for Types for Proofs and Programs: International Workshop, TYPES 2000 Durham, UK, December 8–12, 2000 Selected Papers
Sample text
The ¬¬-interpretation of ML(CZF) + (REM) in ML(CZF)+ (DNSP) is sound. The type theory ML(CZF) + (REM) and the set theory CZF + REM are mutually interpretable. Recall that the set theory CZF + REM has proof theoretic strength at least above that of Bounded Zermelo set theory, which is obtained from Zermelo set theory by limiting the separation axiom scheme to restricted formulas. This is because the power set axiom is derivable in CZF + REM and Bounded Zermelo set theory has a double-negation interpretation into its intuitionistic counterpart.
6. Peter Aczel and Michael Rathjen, Notes on Constructive Set Theory, Preprint no. 40, Institut Mittag-Leffler, The Royal Swedish Academy of Sciences, 2001. se. 7. Thierry Coquand and Erik Palmgren, Intuitionistic Choice and Classical Logic, Archive for Mathematical Logic, vol. 39 (2000) 53 – 74. 8. Thierry Coquand, Giovanni Sambin, Jan Smith and Silvio Valentini, Inductively Generated Formal Topologies, Submitted for publication, 2000. 9. Harvey M. Friedman, The Consistency of Classical Set Theory Relative to a Set Theory with Intuitionistic Logic, Journal of Symbolic Logic, vol.
A solution σ is a mapping of variables to closed solution terms. A term t is called a solution term iff • t is of function type, or • t = c t1 . . tn , where the ti are solution terms and c ∈ C. Let solve be a function that returns for each query a set of solutions. We distinguish two kinds of queries: Functional queries have the form t = X, where t is a closed executable term and X is a variable. g. solve(add 0 (Suc 0) = X) = {[X → Suc 0]} Relational queries have the form (t1 , . . , tn ) ∈ r, where r is an executable inductively defined relation and ti is either a closed executable term or a variable.
