Consider the axioms
forall a. S [a] = [S a] (R1) T Int = Int (R2)
S and T are type functions of kind *->* For convenience, I drop the `redundant' forall a. on R1's lhs.
Suppose some type annotations/pattern matchings give rise to the local assumptions
T [Int] = S [Int] (R3) T Int = S Int (R4)
and under these assumptions we need to verify
T [Int] = [Int]
Logically, we can express the above as follows:
(forall a. S [a] = [S a]) /\ -- axioms (T Int = Int) |= (T [Int] = S [Int]) /\ -- local assumptions (T Int = S Int) implies (T [Int] = [Int]) -- (local) property
That is, any model (in the first-order sense) which is a model of the axioms and local assumptions is also a model of the property.
NOTE: There are further axioms such as reflexitivity of = etc. We'll leave them our for simplicitiy.
The all important question: How can we algorithmically check the above statement? Roughly, we perform the following two steps.
- Generate the appropriate implication constraint out of the program text. That's easy cause GHC supports now implication constraints. (There are some potential subtleties, see GENERATEIMP below).
- Solve the implication constraint by applying axioms and local assumptions until the (local) property is verified. That's the hard part.
We assume that (implication) constraints consist of equality constraints only. In general, we'll also find type class constraints. We ignore such constraints for the moment.
In the following, we assume that symbols t refer to types and symbols C refer to conjunctions of equality constraints and Ax refers to an axiom set.
We'll restrict ourselves to simple implication constraints of the form
C implies t1=t2
In general, implication constraints may be nested, e.g
C1 implies (C2 implies C3) and may contain conjunctions
of implications, e.g.
C1 implies (F1 /\ F2) where F1 and F2 are arbitrary implication constraints. Implication constraints may be universally quantified, e.g.
forall a (S a = T a implies ...)
These universal quantifiers arise from universal type annotations, e.g.
f :: S a = T a => ...., and
pattern matchings over data types with abstract components, e.g. data Foo where
K :: S a = T a => a -> Foo
We can operationally deal with universally quantified variables by skolemizing them (and we must ensure that skolemized/universal variables do not escape).
End of NOTE