A Modular Type-Checking Algorithm for Type Theory with Singleton Types and Proof Irrelevance

Abstract

Abstract. We define a logical framework with singleton types and one universe of small types. We give the semantics using a PER model; it is used for constructing a normalisation-by-evaluation algorithm. We prove completeness and soundness of the algorithm; and get as a corollary the injectivity of type constructors. Then we give the definition of a correct and complete type-checking algorithm for terms in normal form. We extend the results to proof-irrelevant propositions.

Irrelevance in Type Theory with a Heterogeneous Equality Judgement

“…This preserves provability while erasing the proof terms. Conservativity of this rule can be proven as in joint work of the author with Coquand and Pagano [5].…”

“…△(T x). Such a modality has been present in Nuprl as Squash type [21] and it is also known as the type of proofs of (proposition) T [5,9]. Using the extensions of Example 1, we can encode it as △T = ( ÷T ) × 1.…”

Towards Normalization by Evaluation for the βη-Calculus of Constructions

“…Also we will skip all congruence rules expressing that equality is an equivalence relation and that it is closed under all syntactic constructions. We have taken a similar approach before [ACP09] which is justified by Cartmell's work on generalized algebraic theories [Car86].…”

“…We conceive normalization by evaluation as the composition of a standard interpreter : Exp → D mapping expressions into a semantics D and a reifier which computes a long normal form from a value in D. Coquand [ACP09] observed that the η-expansion part of reification can be carried out entirely within the semantics which splits reification into an η-expansion phase ↓ : D → D nf and a read-back phase R nf :…”

“…In particular, the model of closures in weak head normal form we will use in Section 3. Recent meta theoretic studies involving explicit substitutions include [Dan07,Cha09,Gra09,ACP09]. In the presence of explicit substitutions, variables are most naturally represented as de Bruijn indices [ACCL91].…”