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Typed arithmetic expressions with dynamic type checking

Extend the language of boolean expressions with arithmetic expressions on natural numbers, according to the following AST:

type expr =
    True
  | False
  | Not of expr
  | And of expr * expr
  | Or of expr * expr
  | If of expr * expr * expr
  | Zero
  | Succ of expr
  | Pred of expr
  | IsZero of expr

The meaning of the new constructs is the following:

  • Zero represents the natural number 0;
  • Succ e evaluates to the successor of e;
  • Pred e evaluates to the predecessor of e (defined only if e is not zero);
  • IsZero e evaluates to true iff e evaluates to 0.

Concrete syntax

Follow the unit tests in arithexpr.ml for the concrete syntax of the language. To run the tests, execute the following command from the project directory:

dune test

For example, the following is a syntactically correct expression:

iszero pred succ 0 and not iszero succ pred succ 0

You can check its AST via dune utop lib as follows:

"iszero pred succ 0 and not iszero succ pred succ 0" |> ArithexprLib.Main.parse;;

Big-step semantics

The big-step semantics extends that of boolean expressions with the following rules:

---------------------------- [B-Zero]
Zero => 0

e => n
---------------------------- [B-Succ]
Succ(e) => n+1

e => n   n>0
---------------------------- [B-Pred]
Pred(e) => n-1

e => 0
---------------------------- [B-IsZeroZero]
IsZero(e) => true

e => n   n>0
---------------------------- [B-IsZeroSucc]
IsZero(e) => false

The eval function must have the following type:

eval : expr -> exprval

where the type exprval, used to wrap booleans and naturals, must be defined as follows (in lib/main.ml):

type exprval = Bool of bool | Nat of int

Note that some expressions in this language are not well-typed, because they improperly mix natural numbers with booleans. For instance, this is the case for the expressions:

iszero true
succ iszero 0
not 0

In all these cases, the evaluation should produce a run-time error specifying the cause of the error. Run-time errors should also be raised for the expressions which evaluate to negative values, like the following:

pred 0
pred pred succ 0

Small-step semantics

The small-step semantics extends that of boolean expressions with the following rules:

nv ::= Zero | Succ(nv)

e -> e'
----------------------------- [S-Succ]
Succ(e) -> Succ(e') 

----------------------------- [S-PredSucc]
Pred(Succ(nv)) -> nv 

e -> e'
----------------------------- [S-Pred]
Pred(e) -> Pred(e') 

----------------------------- [S-IsZeroZero]
IsZero(Zero) -> True

----------------------------- [S-IsZeroSucc]
IsZero(Succ(nv)) -> False 

e -> e'
----------------------------- [S-IsZero]
IsZero(e) -> IsZero(e')

For example, this semantics should give rise to the following execution trace:

dune exec arithexpr trace test/test1
iszero(pred(succ(pred(succ(0)))))   
 -> iszero(pred(succ(0)))
 -> iszero(0)
 -> true