Operational and Denotational Semantics

The types are , ::= Nat ∣ → . All the terms in the language come with explicit types. In particular, we assume an infinite, countable set of variables of each type , and write x for a variable of type . The terms t of type (in short, t : ) are defined by induction on their size by : – every variable x is of type ; – if N : , then x ⋅N is a term of type → ; – if M : → and N : , then MN is a term of type ; – for each n ∈ N, there is a distinct constant n : Nat ; – there are distinct constants pred : Nat → Nat (subtract one), succ : Nat (add one), ifz : Nat → → → (test if first argument equals 0), and Y : ( → ) → (fixpoint, recursion). As in the -calculus, the terms are understood up to -renaming. This takes the special form that x can be replaced by any fresh variable y of the same type in x ⋅ t. We drop type indices whenever they are irrelevant or can be reconstructed from context.