15-312 Recitation #9: Subtyping
2002-10-23
Joshua Dunfield (joshuad@cs)
Carnegie Mellon University

+ Subtyping:

  - subset interp:  odd, even, int, float, earlier recitation

  - coercion interp:  string, float, int
         coherence

  Subsumption

  Contravariant rule for ->

  Rule for sums

  Rule for unit

  Rule for void; the void coercions

  Rule for polymorphism; coercions

+ Intersection types:

  - Basic idea: e : A & B means e : A and e : B

  - Fairly old, but only introduced into a practical language around 1990 [Reynolds]

  - Various uses, including operator overloading:

       + : (int * int -> int) & (float * float -> float)

  - The "obvious" 3 subtyping rules

  - The "obvious" introduction rule

    e : A   e : B
    -------------
      e : A & B

   ...is wrong if we have store!  [Davies & Pfenning '00]

   let x = (ref 2) : int ref & even ref
   in
       x := 3;
       !x
   end : even
 
    Need to add a condition  e value.

  - Typing these is hard.   --> bidirectional typing