Abstract
AbstractWe develop the Scott model of the programming language PCF in univalent type theory. Moreover, we work constructively and predicatively. To account for the non-termination in PCF, we use the lifting monad (also known as the partial map classifier monad) from topos theory, which has been extended to univalent type theory by Escardó and Knapp. Our results show that lifting is a viable approach to partiality in univalent type theory. Moreover, we show that the Scott model can be constructed in a predicative and constructive setting. Other approaches to partiality either require some form of choice or quotient inductive-inductive types. We show that one can do without these extensions.
Highlights
We develop the Scott model of the programming language PCF in constructive predicative univalent mathematics
We introduce the theory of dcpos with ⊥ in predicative constructive univalent type theory
Basic Domain Theory We introduce basic domain theory in the setting of constructive predicative univalent mathematics
Summary
We develop the Scott model of the programming language PCF in constructive predicative univalent mathematics. The essential difference (for our development) between univalent type theory on the one hand, and set theory or systems like Coq on the other, is the treatment of truth values (propositions). 1.1.3 Univalent type theory As mentioned at the beginning of Section 1, an essential difference between univalent type theory on the one hand and set theory or systems like Coq on the other is the treatment of truth values (propositions). To illustrate this difference, consider the definition of a poset (cf Definition 2). Sets, and propositional truncation in univalent type theory, see The Univalent Foundations Program (2013, Chapter 3)
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