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

inductive 
Vector: Type u → Nat → Type u
Vector (
α: Type u
α : 
Type u: Type (u + 1)
Type u) : 
Nat: Type
Nat → 
Type u: Type (u + 1)
Type u
  | 
nil: {α : Type u} → Vector α 0
nil : 
Vector: Type u → Nat → Type u
Vector 
α: Type u
α 
0: Nat
0
  | 
cons: {α : Type u} → {n : Nat} → α → Vector α n → Vector α (n + 1)
cons : 
α: Type u
α → 
Vector: Type u → Nat → Type u
Vector 
α: Type u
α 
n: Nat
n → 
Vector: Type u → Nat → Type u
Vector 
α: Type u
α (
n: Nat
n+
1: Nat
1)

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

infix:67 " :: " => 
Vector.cons: {α : Type u} → {n : Nat} → α → Vector α n → Vector α (n + 1)
Vector.cons

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

inductive 
Ty: Type
Ty where
  | 
int: Ty
int
  | 
bool: Ty
bool
  | 
fn: Ty → Ty → Ty
fn (
a: Ty
a 
r: Ty
r : 
Ty: Type
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.

@[reducible] def 
Ty.interp: Ty → Type
Ty.interp : 
Ty: Type
Ty → 
Type: Type 1
Type
  | 
int: Ty
int    => 
Int: Type
Int
  | 
bool: Ty
bool   => 
Bool: Type
Bool
  | 
fn: Ty → Ty → Ty
fn 
a: Ty
a 
r: Ty
r => 
a: Ty
a.
interp: Ty → Type
interp → 
r: Ty
r.
interp: Ty → Type
interp

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

inductive 
HasType: {n : Nat} → Fin n → Vector Ty n → Ty → Type
HasType : 
Fin: Nat → Type
Fin 
n: Nat
n → 
Vector: Type → Nat → Type
Vector 
Ty: Type
Ty 
n: Nat
n → 
Ty: Type
Ty → 
Type: Type 1
Type where
  | 
stop: {ty : Ty} → {x : Nat} → {ctx : Vector Ty x} → HasType 0 (ty :: ctx) ty
stop : 
HasType: {n : Nat} → Fin n → Vector Ty n → Ty → Type
HasType 
0: Fin (?m.2729 + 1)
0 (
ty: Ty
ty :: 
ctx: Vector Ty ?m.2729
ctx) 
ty: Ty
ty
  | 
pop: {x : Nat} → {k : Fin x} → {ctx : Vector Ty x} → {ty u : Ty} → HasType k ctx ty → HasType k.succ (u :: ctx) ty
pop  : 
HasType: {n : Nat} → Fin n → Vector Ty n → Ty → Type
HasType 
k: Fin ?m.2913
k 
ctx: Vector Ty ?m.2913
ctx 
ty: Ty
ty → 
HasType: {n : Nat} → Fin n → Vector Ty n → Ty → Type
HasType 
k: Fin ?m.2913
k.
succ: {n : Nat} → Fin n → Fin n.succ
succ (
u: Ty
u :: 
ctx: Vector Ty ?m.2913
ctx) 
ty: Ty
ty

inductive 
Expr: {n : Nat} → Vector Ty n → Ty → Type
Expr : 
Vector: Type → Nat → Type
Vector 
Ty: Type
Ty 
n: Nat
n → 
Ty: Type
Ty → 
Type: Type 1
Type where
  | 
var: {n : Nat} → {i : Fin n} → {ctx : Vector Ty n} → {ty : Ty} → HasType i ctx ty → Expr ctx ty
var   : 
HasType: {n : Nat} → Fin n → Vector Ty n → Ty → Type
HasType 
i: Fin ?m.3680
i 
ctx: Vector Ty ?m.3680
ctx 
ty: Ty
ty → 
Expr: {n : Nat} → Vector Ty n → Ty → Type
Expr 
ctx: Vector Ty ?m.3680
ctx 
ty: Ty
ty
  | 
val: {n : Nat} → {ctx : Vector Ty n} → Int → Expr ctx Ty.int
val   : 
Int: Type
Int → 
Expr: {n : Nat} → Vector Ty n → Ty → Type
Expr 
ctx: Vector Ty ?m.3701
ctx 
Ty.int: Ty
Ty.int
  | 
lam: {n : Nat} → {a : Ty} → {ctx : Vector Ty n} → {ty : Ty} → Expr (a :: ctx) ty → Expr ctx (a.fn ty)
lam   : 
Expr: {n : Nat} → Vector Ty n → Ty → Type
Expr (
a: Ty
a :: 
ctx: Vector Ty ?m.3865
ctx) 
ty: Ty
ty → 
Expr: {n : Nat} → Vector Ty n → Ty → Type
Expr 
ctx: Vector Ty ?m.3865
ctx (
Ty.fn: Ty → Ty → Ty
Ty.fn 
a: Ty
a 
ty: Ty
ty)
  | 
app: {n : Nat} → {ctx : Vector Ty n} → {a ty : Ty} → Expr ctx (a.fn ty) → Expr ctx a → Expr ctx ty
app   : 
Expr: {n : Nat} → Vector Ty n → Ty → Type
Expr 
ctx: Vector Ty ?m.4020
ctx (
Ty.fn: Ty → Ty → Ty
Ty.fn 
a: Ty
a 
ty: Ty
ty) → 
Expr: {n : Nat} → Vector Ty n → Ty → Type
Expr 
ctx: Vector Ty ?m.4020
ctx 
a: Ty
a → 
Expr: {n : Nat} → Vector Ty n → Ty → Type
Expr 
ctx: Vector Ty ?m.4020
ctx 
ty: Ty
ty
  | 
op: {n : Nat} → {ctx : Vector Ty n} → {a b c : Ty} → (a.interp → b.interp → c.interp) → Expr ctx a → Expr ctx b → Expr ctx c
op    : (
a: Ty
a.
interp: Ty → Type
interp → 
b: Ty
b.
interp: Ty → Type
interp → 
c: Ty
c.
interp: Ty → Type
interp) → 
Expr: {n : Nat} → Vector Ty n → Ty → Type
Expr 
ctx: Vector Ty ?m.4110
ctx 
a: Ty
a → 
Expr: {n : Nat} → Vector Ty n → Ty → Type
Expr 
ctx: Vector Ty ?m.4110
ctx 
b: Ty
b → 
Expr: {n : Nat} → Vector Ty n → Ty → Type
Expr 
ctx: Vector Ty ?m.4110
ctx 
c: Ty
c
  | 
ife: {n : Nat} → {ctx : Vector Ty n} → {a : Ty} → Expr ctx Ty.bool → Expr ctx a → Expr ctx a → Expr ctx a
ife   : 
Expr: {n : Nat} → Vector Ty n → Ty → Type
Expr 
ctx: Vector Ty ?m.4231
ctx 
Ty.bool: Ty
Ty.bool → 
Expr: {n : Nat} → Vector Ty n → Ty → Type
Expr 
ctx: Vector Ty ?m.4231
ctx 
a: Ty
a → 
Expr: {n : Nat} → Vector Ty n → Ty → Type
Expr 
ctx: Vector Ty ?m.4231
ctx 
a: Ty
a → 
Expr: {n : Nat} → Vector Ty n → Ty → Type
Expr 
ctx: Vector Ty ?m.4231
ctx 
a: Ty
a
  | 
delay: {n : Nat} → {ctx : Vector Ty n} → {a : Ty} → (Unit → Expr ctx a) → Expr ctx a
delay : (
Unit: Type
Unit → 
Expr: {n : Nat} → Vector Ty n → Ty → Type
Expr 
ctx: Vector Ty ?m.4274
ctx 
a: Ty
a) → 
Expr: {n : Nat} → Vector Ty n → Ty → Type
Expr 
ctx: Vector Ty ?m.4274
ctx 
a: Ty
a

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

open HasType (
stop: {ty : Ty} → {x : Nat} → {ctx : Vector Ty x} → HasType 0 (ty :: ctx) ty
stop 
pop: {x : Nat} → {k : Fin x} → {ctx : Vector Ty x} → {ty u : Ty} → HasType k ctx ty → HasType k.succ (u :: ctx) ty
pop)

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.

inductive 
Env: {n : Nat} → Vector Ty n → Type
Env : 
Vector: Type → Nat → Type
Vector 
Ty: Type
Ty 
n: Nat
n → 
Type: Type 1
Type where
  | 
nil: Env Vector.nil
nil  : 
Env: {n : Nat} → Vector Ty n → Type
Env 
Vector.nil: {α : Type} → Vector α 0
Vector.nil
  | 
cons: {a : Ty} → {x : Nat} → {ctx : Vector Ty x} → a.interp → Env ctx → Env (a :: ctx)
cons : 
Ty.interp: Ty → Type
Ty.interp 
a: Ty
a → 
Env: {n : Nat} → Vector Ty n → Type
Env 
ctx: Vector Ty ?m.8584
ctx → 
Env: {n : Nat} → Vector Ty n → Type
Env (
a: Ty
a :: 
ctx: Vector Ty ?m.8584
ctx)

infix:67 " :: " => 
Env.cons: {a : Ty} → {x : Nat} → {ctx : Vector Ty x} → a.interp → Env ctx → Env (a :: ctx)
Env.cons

def 
Env.lookup: {n : Nat} → {i : Fin n} → {ctx : Vector Ty n} → {ty : Ty} → HasType i ctx ty → Env ctx → ty.interp
Env.lookup : 
HasType: {n : Nat} → Fin n → Vector Ty n → Ty → Type
HasType 
i: Fin ?m.11068
i 
ctx: Vector Ty ?m.11068
ctx 
ty: Ty
ty → 
Env: {n : Nat} → Vector Ty n → Type
Env 
ctx: Vector Ty ?m.11068
ctx → 
ty: Ty
ty.
interp: Ty → Type
interp
  | 
stop: {ty : Ty} → {x : Nat} → {ctx : Vector Ty x} → HasType 0 (ty :: ctx) ty
stop,  
x: ty.interp
x :: 
xs: Env ctx✝
xs => 
x: ty.interp
x
  | 
pop: {x : Nat} → {k : Fin x} → {ctx : Vector Ty x} → {ty u : Ty} → HasType k ctx ty → HasType k.succ (u :: ctx) ty
pop 
k: HasType k✝ ctx✝ ty
k, 
x: u✝.interp
x :: 
xs: Env ctx✝
xs => 
lookup: {n : Nat} → {i : Fin n} → {ctx : Vector Ty n} → {ty : Ty} → HasType i ctx ty → Env ctx → ty.interp
lookup 
k: HasType k✝ ctx✝ ty
k 
xs: Env ctx✝
xs

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

def 
Expr.interp: {a : Nat} → {ctx : Vector Ty a} → {ty : Ty} → Env ctx → Expr ctx ty → ty.interp
Expr.interp (
env: Env ctx
env : 
Env: {n : Nat} → Vector Ty n → Type
Env 
ctx: Vector Ty ?m.13905
ctx) : 
Expr: {n : Nat} → Vector Ty n → Ty → Type
Expr 
ctx: Vector Ty ?m.13905
ctx 
ty: Ty
ty → 
ty: Ty
ty.
interp: Ty → Type
interp
  | 
var: {n : Nat} → {i : Fin n} → {ctx : Vector Ty n} → {ty : Ty} → HasType i ctx ty → Expr ctx ty
var 
i: HasType i✝ ctx ty
i     => 
env: Env ctx
env.
lookup: {n : Nat} → {i : Fin n} → {ctx : Vector Ty n} → {ty : Ty} → HasType i ctx ty → Env ctx → ty.interp
lookup 
i: HasType i✝ ctx ty
i
  | 
val: {n : Nat} → {ctx : Vector Ty n} → Int → Expr ctx Ty.int
val 
x: Int
x     => 
x: Int
x
  | 
lam: {n : Nat} → {a : Ty} → {ctx : Vector Ty n} → {ty : Ty} → Expr (a :: ctx) ty → Expr ctx (a.fn ty)
lam 
b: Expr (a✝ :: ctx) ty✝
b     => fun 
x: a✝.interp
x => 
b: Expr (a✝ :: ctx) ty✝
b.
interp: {a : Nat} → {ctx : Vector Ty a} → {ty : Ty} → Env ctx → Expr ctx ty → ty.interp
interp (
Env.cons: {a : Ty} → {x : Nat} → {ctx : Vector Ty x} → a.interp → Env ctx → Env (a :: ctx)
Env.cons 
x: a✝.interp
x 
env: Env ctx
env)
  | 
app: {n : Nat} → {ctx : Vector Ty n} → {a ty : Ty} → Expr ctx (a.fn ty) → Expr ctx a → Expr ctx ty
app 
f: Expr ctx (a✝.fn ty)
f 
a: Expr ctx a✝
a   => 
f: Expr ctx (a✝.fn ty)
f.
interp: {a : Nat} → {ctx : Vector Ty a} → {ty : Ty} → Env ctx → Expr ctx ty → ty.interp
interp 
env: Env ctx
env (
a: Expr ctx a✝
a.
interp: {a : Nat} → {ctx : Vector Ty a} → {ty : Ty} → Env ctx → Expr ctx ty → ty.interp
interp 
env: Env ctx
env)
  | 
op: {n : Nat} → {ctx : Vector Ty n} → {a b c : Ty} → (a.interp → b.interp → c.interp) → Expr ctx a → Expr ctx b → Expr ctx c
op 
o: a✝.interp → b✝.interp → ty.interp
o 
x: Expr ctx a✝
x 
y: Expr ctx b✝
y  => 
o: a✝.interp → b✝.interp → ty.interp
o (
x: Expr ctx a✝
x.
interp: {a : Nat} → {ctx : Vector Ty a} → {ty : Ty} → Env ctx → Expr ctx ty → ty.interp
interp 
env: Env ctx
env) (
y: Expr ctx b✝
y.
interp: {a : Nat} → {ctx : Vector Ty a} → {ty : Ty} → Env ctx → Expr ctx ty → ty.interp
interp 
env: Env ctx
env)
  | 
ife: {n : Nat} → {ctx : Vector Ty n} → {a : Ty} → Expr ctx Ty.bool → Expr ctx a → Expr ctx a → Expr ctx a
ife 
c: Expr ctx Ty.bool
c 
t: Expr ctx ty
t 
e: Expr ctx ty
e => if 
c: Expr ctx Ty.bool
c.
interp: {a : Nat} → {ctx : Vector Ty a} → {ty : Ty} → Env ctx → Expr ctx ty → ty.interp
interp 
env: Env ctx
env then 
t: Expr ctx ty
t.
interp: {a : Nat} → {ctx : Vector Ty a} → {ty : Ty} → Env ctx → Expr ctx ty → ty.interp
interp 
env: Env ctx
env else 
e: Expr ctx ty
e.
interp: {a : Nat} → {ctx : Vector Ty a} → {ty : Ty} → Env ctx → Expr ctx ty → ty.interp
interp 
env: Env ctx
env
  | 
delay: {n : Nat} → {ctx : Vector Ty n} → {a : Ty} → (Unit → Expr ctx a) → Expr ctx a
delay 
a: Unit → Expr ctx ty
a   => (
a: Unit → Expr ctx ty
a 
(): Unit
()).
interp: {a : Nat} → {ctx : Vector Ty a} → {ty : Ty} → Env ctx → Expr ctx ty → ty.interp
interp 
env: Env ctx
env

open Expr

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

def 
add: {a : Nat} → {ctx : Vector Ty a} → Expr ctx (Ty.int.fn (Ty.int.fn Ty.int))
add : 
Expr: {n : Nat} → Vector Ty n → Ty → Type
Expr 
ctx: Vector Ty ?m.18226
ctx (
Ty.fn: Ty → Ty → Ty
Ty.fn 
Ty.int: Ty
Ty.int (
Ty.fn: Ty → Ty → Ty
Ty.fn 
Ty.int: Ty
Ty.int 
Ty.int: Ty
Ty.int)) :=
  
lam: {n : Nat} → {a : Ty} → {ctx : Vector Ty n} → {ty : Ty} → Expr (a :: ctx) ty → Expr ctx (a.fn ty)
lam (
lam: {n : Nat} → {a : Ty} → {ctx : Vector Ty n} → {ty : Ty} → Expr (a :: ctx) ty → Expr ctx (a.fn ty)
lam (
op: {n : Nat} → {ctx : Vector Ty n} → {a b c : Ty} → (a.interp → b.interp → c.interp) → Expr ctx a → Expr ctx b → Expr ctx c
op 
(·+·): Ty.int.interp → Ty.int.interp → Ty.int.interp
(·+·) (
var: {n : Nat} → {i : Fin n} → {ctx : Vector Ty n} → {ty : Ty} → HasType i ctx ty → Expr ctx ty
var 
stop: {ty : Ty} → {x : Nat} → {ctx : Vector Ty x} → HasType 0 (ty :: ctx) ty
stop) (
var: {n : Nat} → {i : Fin n} → {ctx : Vector Ty n} → {ty : Ty} → HasType i ctx ty → Expr ctx ty
var (
pop: {x : Nat} → {k : Fin x} → {ctx : Vector Ty x} → {ty u : Ty} → HasType k ctx ty → HasType k.succ (u :: ctx) ty
pop 
stop: {ty : Ty} → {x : Nat} → {ctx : Vector Ty x} → HasType 0 (ty :: ctx) ty
stop))))

#eval
30
 
add: {a : Nat} → {ctx : Vector Ty a} → Expr ctx (Ty.int.fn (Ty.int.fn Ty.int))
add.
interp: {a : Nat} → {ctx : Vector Ty a} → {ty : Ty} → Env ctx → Expr ctx ty → ty.interp
interp 
Env.nil: Env Vector.nil
Env.nil 
10: Ty.int.interp
10 
20: Ty.int.interp
20

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.

def 
fact: {a : Nat} → {ctx : Vector Ty a} → Expr ctx (Ty.int.fn Ty.int)
fact
Warning: declaration uses 'sorry' : 
Expr: {n : Nat} → Vector Ty n → Ty → Type
Expr 
ctx: Vector Ty ?m.18601
ctx (
Ty.fn: Ty → Ty → Ty
Ty.fn 
Ty.int: Ty
Ty.int 
Ty.int: Ty
Ty.int) :=
  
lam: {n : Nat} → {a : Ty} → {ctx : Vector Ty n} → {ty : Ty} → Expr (a :: ctx) ty → Expr ctx (a.fn ty)
lam (
ife: {n : Nat} → {ctx : Vector Ty n} → {a : Ty} → Expr ctx Ty.bool → Expr ctx a → Expr ctx a → Expr ctx a
ife (
op: {n : Nat} → {ctx : Vector Ty n} → {a b c : Ty} → (a.interp → b.interp → c.interp) → Expr ctx a → Expr ctx b → Expr ctx c
op 
(·==·): Ty.int.interp → Ty.int.interp → Bool
(·==·) (
var: {n : Nat} → {i : Fin n} → {ctx : Vector Ty n} → {ty : Ty} → HasType i ctx ty → Expr ctx ty
var 
stop: {ty : Ty} → {x : Nat} → {ctx : Vector Ty x} → HasType 0 (ty :: ctx) ty
stop) (
val: {n : Nat} → {ctx : Vector Ty n} → Int → Expr ctx Ty.int
val 
0: Int
0))
           (
val: {n : Nat} → {ctx : Vector Ty n} → Int → Expr ctx Ty.int
val 
1: Int
1)
           (
op: {n : Nat} → {ctx : Vector Ty n} → {a b c : Ty} → (a.interp → b.interp → c.interp) → Expr ctx a → Expr ctx b → Expr ctx c
op 
(·*·): Ty.int.interp → Ty.int.interp → Ty.int.interp
(·*·) (
delay: {n : Nat} → {ctx : Vector Ty n} → {a : Ty} → (Unit → Expr ctx a) → Expr ctx a
delay fun 
_: Unit
_ => 
app: {n : Nat} → {ctx : Vector Ty n} → {a ty : Ty} → Expr ctx (a.fn ty) → Expr ctx a → Expr ctx ty
app 
fact: {a : Nat} → {ctx : Vector Ty a} → Expr ctx (Ty.int.fn Ty.int)
fact (
op: {n : Nat} → {ctx : Vector Ty n} → {a b c : Ty} → (a.interp → b.interp → c.interp) → Expr ctx a → Expr ctx b → Expr ctx c
op 
(·-·): Ty.int.interp → Ty.int.interp → Ty.int.interp
(·-·) (
var: {n : Nat} → {i : Fin n} → {ctx : Vector Ty n} → {ty : Ty} → HasType i ctx ty → Expr ctx ty
var 
stop: {ty : Ty} → {x : Nat} → {ctx : Vector Ty x} → HasType 0 (ty :: ctx) ty
stop) (
val: {n : Nat} → {ctx : Vector Ty n} → Int → Expr ctx Ty.int
val 
1: Int
1))) (
var: {n : Nat} → {i : Fin n} → {ctx : Vector Ty n} → {ty : Ty} → HasType i ctx ty → Expr ctx ty
var 
stop: {ty : Ty} → {x : Nat} → {ctx : Vector Ty x} → HasType 0 (ty :: ctx) ty
stop)))
  decreasing_by
a✝: Nat
ctx: Vector Ty a✝
(invImage (fun x => PSigma.casesOn x fun a ctx => a) instWellFoundedRelationOfSizeOf).1 ⟨a✝ + 1, Ty.int :: ctx⟩
  ⟨a✝, ctx⟩ sorry
Goals accomplished! 🐙

#eval
3628800
 
fact: {a : Nat} → {ctx : Vector Ty a} → Expr ctx (Ty.int.fn Ty.int)
fact.
interp: {a : Nat} → {ctx : Vector Ty a} → {ty : Ty} → Env ctx → Expr ctx ty → ty.interp
interp 
Env.nil: Env Vector.nil
Env.nil 
10: Ty.int.interp
10