A Proof of Strong Normalisation using Domain Theory

Abstract

International audienceU. Berger, significantly simplified Tait's normalisation proof for bar recursion, replacing Tait's introduction of infinite terms by the construction of a domain having the property that a term is strongly normalizing if its semantics is not bottom. The goal of this paper is to show that, using ideas from the theory of intersection types and Martin-Löf's domain interpretation of type theory, we can in turn simplify U. Berger's argument in the construction of such a domain model. We think that our domain model can be used to give modular proofs of strong normalization for various type theory. As an example, we show in some details how it can be used to prove strong normalization for Martin-Löf dependent type theory extended with bar recursion, and with some form of proof-irrelevance

Constructive Type Theory, An Appetizer

“…However, it also limits in some respects the expressive capabilities of CTT compared with stronger impredicative type theories. 26 In fact, the form of CTT I have considered so 25 See (Girard 1972) and (Coquand 1986, Coquand 1989) for insightful analysis of Girard's paradox. In particular, Coquand clarifies the significance of the strong form of Curry-Howard correspondence that CTT implements.…”

Intuitionistic fixed point logic

“…The first is essentially an untyped version of Plotkin's Adequacy Theorem for the simply typed language PCF [58]. Its proof uses compact elements of the untyped domain model as a replacement for types, a technique introduced by Coquand and Spiwack [26], and follows roughly the lines of [8]. The second Adequacy Theorem concerns the computation of infinite data.…”

“…We prove the 'if' part of Thm. 5 (b) following [8], which uses ideas from [58] and [26]. Let D 0 be the set of compact elements of D. To every a ∈ D 0 we assign a set of closed programs Pr(a) by induction on rk(a) (Sect.…”

Size-based termination of higher-order rewriting

“…Giménez (1998) himself later proposed a similar system but provided no termination proof. Note that Plotkin's result was later extended to higher order types and rewriting-based function definitions by Berger (2005), Coquand & Spiwack (2007) and Berger (2008).…”