Bridging Curry and Church's typing style

Fairouz Dib Kamareddine*, Jonathan P. Seldin, Joseph Brian Wells

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

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There are two versions of type assignment in the λ-calculus: Church-style, in which the type of each variable is fixed, and Curry-style (also called "domain free"), in which it is not. As an example, in Church-style typing, λx:A.x is the identity function on type A, and it has type A→A but not B→B for a type B different from A. In Curry-style typing, λx.x is a general identity function with type C→C for every type C. In this paper, we will show how to interpret in a Curry-style system every Pure Type System (PTS) in the Church-style without losing any typing information. We will also prove a kind of conservative extension result for this interpretation, a result which implies that for most consistent PTSs of the Church-style, the corresponding Curry-style system is consistent. We will then show how to interpret in a system of the Church-style (a modified PTS, stronger than a PTS) every PTS-like system in the Curry style.

Original languageEnglish
Pages (from-to)42-70
Number of pages29
JournalJournal of Applied Logic
Early online date16 Jun 2016
Publication statusPublished - Nov 2016


  • Church-style typing
  • Curry-style typing
  • Domain-free typing
  • Domain-full typing

ASJC Scopus subject areas

  • Logic
  • Applied Mathematics


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