2 research outputs found
Refinement Types for Logical Frameworks and Their Interpretation as Proof Irrelevance
Refinement types sharpen systems of simple and dependent types by offering
expressive means to more precisely classify well-typed terms. We present a
system of refinement types for LF in the style of recent formulations where
only canonical forms are well-typed. Both the usual LF rules and the rules for
type refinements are bidirectional, leading to a straightforward proof of
decidability of typechecking even in the presence of intersection types.
Because we insist on canonical forms, structural rules for subtyping can now be
derived rather than being assumed as primitive. We illustrate the expressive
power of our system with examples and validate its design by demonstrating a
precise correspondence with traditional presentations of subtyping. Proof
irrelevance provides a mechanism for selectively hiding the identities of terms
in type theories. We show that LF refinement types can be interpreted as
predicates using proof irrelevance, establishing a uniform relationship between
two previously studied concepts in type theory. The interpretation and its
correctness proof are surprisingly complex, lending support to the claim that
refinement types are a fundamental construct rather than just a convenient
surface syntax for certain uses of proof irrelevance
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