nat and function type.
LF tp : type =
| nat : tp
| arr : tp → tp → tp;
LF exp : tp → type =
| lam : (exp T1 → exp T2) → exp (arr T1 T2)
| app : exp (arr T2 T) → exp T2 → exp T;
Next, we define the context schema expressing the fact that all declarations must be instances of the type exp T for some T.
schema ctx = exp T;
Finally, we write a function which traverses a lambda-term and normalizes it. In the type of the function norm we leave context variables implicit by writing (g:ctx). As a consequence, we can omit passing a concrete context for it when calling norm. In the program, we distinguish between three different cases: the variable case [ ⊢ #p ], the abstraction case [g ⊢ lam λx. M], and the application case [g ⊢ app M N]. In the variable case, we simply return the variable. In the abstraction case, we recursively normalize [g, x:exp _ ⊢ M] extending the context with the additional declaration x:exp _. Since we do not have a concrete name for the type of x, we simply write an underscore and let Beluga's reconstruction algorithm infer the argument. In the application case, we first normalize recursively [g ⊢ M]. If this results in an abstraction [g ⊢ lam λx. M'], then we continue to normalize [g ⊢ M'[ ,N] ] substituting for x the term N. Otherwise, we normalize recursively [g ⊢ N] and rebuild the application.
Recall that all meta-variables are associated with the identity substitution by default which may be omitted.
rec norm : (g:ctx) [g ⊢ exp T] → [g ⊢ exp T] =
fn e ⇒ case e of
| [g ⊢ #p] ⇒ [g ⊢ #p]
| [g ⊢ lam (λx. M)] ⇒
let [g, x : exp _ ⊢ M'] = norm [g, x : exp _ ⊢ M] in [g ⊢ lam (λx. M')]
| [g ⊢ app M N] ⇒
case norm [g ⊢ M] of
| [g ⊢ lam (λx. M')] ⇒ norm [g ⊢ M'[…, N]]
| [g ⊢ M'] ⇒ let [g ⊢ N'] = norm [g ⊢ N] in [g ⊢ app M' N'];