A brief summary of Lecture 3 (March 11):


(1) Refined type language for AEXP

Instead of

tau ::= Int

we define

tau' ::= IntE | IntO     -- we distinguish among even and odd integers



(2) We need to adjust AEXP's static semantics:


(OP)  |- e1 : IntE  |- e2 : IntE
     ----------------------------
         |- e1 op e2 : IntE

    
      |- e1 : IntE  |- e2 : IntO
     ----------------------------
         |- e1 op e2 : IntO


      where op in {+,-,*,/} and soon


(3) Further refinement:

a ::= i | a op a    where op in {+,-,*,/}

tau'' ::= Int(a)

(OP)  |- e1 : Int(a1)  |- e2 : Int(a2)
     ----------------------------------
        |- e1 op e2 : Int(a1 op a2)



(4) Soundness of static semantics wrt dynamic semantics

We say a system consituting of ( |- e : tau, |- e => v)
(note I use `=>` instead of the downarrow) is sound iff

for any expression and type tau if |- e : tau then |- e => v
for some value v