2006 Scheme and Functional Programming Papers, University of

let id (T:*) (x:T) : T = x;
The traditional limitation of both first-class types and
unrestricted refinement types is that they are not statically
decidable. Sage circumvents this difficulty by replacing
static type checking with hybrid type checking [14]. Sage
checks correctness properties and detects defects statically,
whenever possible. However, it resorts to dynamic checking
for particularly complicated specifications. The overall result
is that precise specifications can be enforced, with most
errors detected at compile time, and violations of some complicated
specifications detected at run time.
1.1 Hybrid Type Checking
We briefly illustrate the key idea of hybrid type checking by
considering the function application
(factorial t)
Suppose factorial has type Pos → Pos, where Pos =
{x : Int | x > 0} is the type of positive integers, and that
t has some type T . If the type checker can prove (or refute)
that T