Classical Realizability in the CPS Target Language

Formalizing Implicative Algebras in Coq

“…Any ordered combinatory algebra also induces an implicative structure: Similarly, we can define an implication on the complete lattice P(A) which give rise to an implicative structure: Proposition 8. If A is an ordered combinatory algebra, then the complete lattice P(A) equipped with the implication 9 :…”

“…From now on, let A = (A, , →) denotes an arbitrary implicative structure. 9 This definition is related with the consttruction of a realizability tripos from an OCA A. Indeed, given a set X, the ordering on predicates of P(A) X is defined by:…”

“…Algebraization of classical realizability During the last decade, the study of the algebraic structure of the models that classical realizability induces have been an active research topic. This line of work was first initiated by Streicher, who proposed the concept of abstract Krivine structure [24], followed by Ferrer, Frey, Guillermo, Malherbe and Miquel who introduced other structures peculiar to classical realizability [6,7,5,8,9]. Aside from the algebraic study of classical realizability models, these works had the interest of building the bridge with the algebraic structures arising from intuitionistic realizability.…”