The Well-Typed Interpreter

The Well-Typed Interpreter

In this example, we build an interpreter for a simple functional programming language, with variables, function application, binary operators and an if...then...else construct. We will use the dependent type system to ensure that any programs which can be represented are well-typed.

Remark: this example is based on an example found in the Idris manual.

Vectors

A Vec is a list of size n whose elements belong to a type α.

We can overload the List.cons notation :: and use it to create Vecs.

Now, we define the types of our simple functional language. We have integers, booleans, and functions, represented by Ty.

We can write a function to translate Ty values to a Lean type — remember that types are first class, so can be calculated just like any other value. We mark Ty.interp as [reducible] to make sure the typeclass resolution procedure can unfold/reduce it. For example, suppose Lean is trying to synthesize a value for the instance Add (Ty.interp Ty.int). Since Ty.interp is marked as [reducible], the typeclass resolution procedure can reduce Ty.interp Ty.int to Int, and use the builtin instance for Add Int as the solution.

Expressions are indexed by the types of the local variables, and the type of the expression itself.

We use the command open to create the aliases stop and pop for HasType.stop and HasType.pop respectively.

Since expressions are indexed by their type, we can read the typing rules of the language from the definitions of the constructors. Let us look at each constructor in turn.

We use a nameless representation for variables — they are de Bruijn indexed. Variables are represented by a proof of their membership in the context, HasType i ctx ty, which is a proof that variable i in context ctx has type ty.

We can treat stop as a proof that the most recently defined variable is well-typed, and pop n as a proof that, if the nth most recently defined variable is well-typed, so is the n+1th. In practice, this means we use stop to refer to the most recently defined variable, pop stop to refer to the next, and so on, via the Expr.var constructor.

A value Expr.val carries a concrete representation of an integer.

A lambda Expr.lam creates a function. In the scope of a function of type Ty.fn a ty, there is a new local variable of type a.

A function application Expr.app produces a value of type ty given a function from a to ty and a value of type a.

The constructor Expr.op allows us to use arbitrary binary operators, where the type of the operator informs what the types of the arguments must be.

Finally, the constructor Exp.ife represents a if-then-else expression. The condition is a Boolean, and each branch must have the same type.

The auxiliary constructor Expr.delay is used to delay evaluation.

When we evaluate an Expr, we’ll need to know the values in scope, as well as their types. Env is an environment, indexed over the types in scope. Since an environment is just another form of list, albeit with a strongly specified connection to the vector of local variable types, we overload again the notation :: so that we can use the usual list syntax. Given a proof that a variable is defined in the context, we can then produce a value from the environment.

Given this, an interpreter is a function which translates an Expr into a Lean value with respect to a specific environment.

We can make some simple test functions. Firstly, adding two inputs fun x y => y + x is written as follows.

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More interestingly, a factorial function fact (e.g. fun x => if (x == 0) then 1 else (fact (x-1) * x)), can be written as. Note that this is a recursive (non-terminating) definition. For every input value, the interpreter terminates, but the definition itself is non-terminating. We use two tricks to make sure Lean accepts it. First, we use the auxiliary constructor Expr.delay to delay its unfolding. Second, we add the annotation decreasing_by sorry which can be viewed as "trust me, this recursive definition makes sense". Recall that sorry is an unsound axiom in Lean.

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