From realizability to induction via dependent intersection

Abstract

In this paper, it is shown that induction is derivable in a type-assignment formulation of the second-order dependent type theory λP2, extended with the implicit product type of Miquel, dependent intersection type of Kopylov, and a built-in equality type. The crucial idea is to use dependent intersections to internalize a result of Leivant's showing that Church-encoded data may be seen as realizing their own type correctness statements, under the Curry–Howard isomorphism.