2 research outputs found
On Irrelevance and Algorithmic Equality in Predicative Type Theory
Dependently typed programs contain an excessive amount of static terms which
are necessary to please the type checker but irrelevant for computation. To
separate static and dynamic code, several static analyses and type systems have
been put forward. We consider Pfenning's type theory with irrelevant
quantification which is compatible with a type-based notion of equality that
respects eta-laws. We extend Pfenning's theory to universes and large
eliminations and develop its meta-theory. Subject reduction, normalization and
consistency are obtained by a Kripke model over the typed equality judgement.
Finally, a type-directed equality algorithm is described whose completeness is
proven by a second Kripke model.Comment: 36 pages, superseds the FoSSaCS 2011 paper of the first author,
titled "Irrelevance in Type Theory with a Heterogeneous Equality Judgement
A MODULAR TYPE-CHECKING ALGORITHM FOR TYPE THEORY WITH SINGLETON TYPES AND PROOF IRRELEVANCE
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