f :: A -> B, x :: A --------------------- (App) f x :: BThis rule, called "App" for application, says that if expression f has type A -> B and expression x has type A then we can deduce that expression (f x) has type B. The expressions above the line are the premises and below, the conclusion. An alternative notation often used is:
G |- x : Awhere "|-" is the turnstile symbol (LaTeX \vdash) and G is a type assignment for the free variables of expression x. The above can be read "under assumptions G, expression x has type A". (As in Haskell, we use a double "::" for type declarations and a single ":" for the infix list constructor, cons). Given an expression
plus (head l) 1we can label each subexpression with a type, using type variables X, Y, etc. for unknown types:
(plus :: Int -> Int -> Int) (((head :: [a] -> a) (l :: Y)) :: X) (1 :: Int)We then use unification on type variables to match the partial application of plus to its first argument against the App rule, yielding a type (Int -> Int) and a substitution X = Int. Re-using App for the application to the second argument gives an overall type Int and no further substitutions. Similarly, matching App against the application (head l) we get Y = [X]. We already know X = Int so therefore Y = [Int]. This process is used both to infer types for expressions and to check that any types given by the user are consistent. See also generic type variable, principal type.
Last updated: 1995-02-03