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Simply typed lambda-calculus

Write an interpreter of the simply typed lambda-calculus with bool and nat base types, according to the following guidelines.

Abstract syntax

The abstract syntax of the language is defined in ast.ml as follows:

type term =
    Var of string
  | Abs of string * ty * term
  | App of term * term
  | True
  | False
  | Not of term
  | And of term * term
  | Or of term * term
  | If of term * term * term
  | Zero
  | Succ of term
  | Pred of term
  | IsZero of term

Note that, unlike in the untyped lambda-calculus, in the simply typed lambda-calculus the variables in abstractions are explicitly typed. The possible types are defined by the following abstract syntax:

type ty = TBool | TNat | TFun of ty * ty

where the type TBool represents booleans, TNat is for naturals, and TFun for function types. For instance, the term:

fun f: nat->bool. f 0

represents a function that takes as argument a function f from nat to bool, and applies it to 0. Its abstract syntax is:

"fun f: nat->bool. f 0" |> parse;;
- : term = Abs ("f", TFun (TNat, TBool), App (Var "f", Zero))

Type checking

Write a function with type

typecheck : (string -> ty) -> term -> ty

such that typecheck gamma t returns the type of the term t in the type environment gamma, or raises a TypeError exception if the term is not typeable.

For instance, in the empty type environment (bot) we expect to have:

(fun x:nat. iszero x) 0" |> parse |> typecheck bot;;
- : ty = TBool

fun x:nat. iszero x" |> parse |> typecheck bot;;
- : ty = TFun (TNat, TBool)

Small-step semantics

The call-by-value evaluation strategy extends that of the untyped lambda calculus. The base rules are the following rules:

v ::= fun x . t | bv | nv       (values)
bv ::= true | false             (bool values)
nv ::= 0 | succ nv              (nat values)

------------------------------- [CBV-AppAbs]
(fun x:T . t1) v2 -> [x -> v2] t1

t1 -> t1'
------------------------------- [CBV-App1]
t1 t2 -> t1' t2

t2 -> t2'
------------------------------- [CBV-App2]
v1 t2 -> v1 t2'

Besides these rules, the semantics of the simply typed lambda calculus include rules for operating on the base types. These rules are similar to those given for the language of arithmetic expressions.

Implement the call-by-value strategy as a function with the following type:

trace : term -> term list

Note that, unlike in the untyped lambda-calculus, the well-typed terms of the simply typed lambda calculus always terminate into a value. Therefore, there is no need for using feeding trace with the number of steps.

Frontend and testing

Run the frontend with the commands:

dune exec simplytyped trace
dune exec simplytyped typecheck

For instance, we should obtain:

dune exec simplytyped typecheck
(fun x:bool. fun y:bool. x and y) true true
bool

dune exec simplytyped trace
(fun x:bool. fun y:bool. x and y) true true
((fun x : bool . fun y : bool . x and y) (true)) (true)
 -> (fun y : bool . true and y) (true)
 -> true and true
 -> true