Library Extensional

Last chapter's constant folding example was particularly easy to verify, because that transformation used the same source and target language. In this chapter, we verify a different translation, illustrating the added complexities in translating between languages.

Program transformations can be classified as intensional, when they require some notion of inequality between variables; or extensional, otherwise. This chapter's example is extensional, and the next chapter deals with the trickier intensional case.

CPS Conversion for Simply-Typed Lambda Calculus

A convenient method for compiling functional programs begins with conversion to continuation-passing style, or CPS. In this restricted form, function calls never return; instead, we pass explicit return pointers, much as in assembly language. Additionally, we make order of evaluation explicit, breaking complex expressions into sequences of primitive operations.

Our translation will operate over the same source language that we used in the first part of last chapter, so we omit most of the language definition. However, we do make one significant change: since we will be working with multiple languages that involve similar constructs, we use Coq's notation scope mechanism to disambiguate. For instance, the span of code dealing with type notations looks like this:

  Notation "'Nat'" := TNat : source_scope.
  Infix "-->" := Arrow (right associativity, at level 60) : source_scope.

  Open Scope source_scope.
  Bind Scope source_scope with type.
  Delimit Scope source_scope with source.

We explicitly place our notations inside a scope named source_scope, and we associate a delimiting key source with source_scope. Without further commands, our notations would only be used in expressions like (...)%source. We also open our scope locally within this module, so that we avoid repeating %source in many places. Further, we bind our scope to type. In some circumstances where Coq is able to infer that some subexpression has type type, that subexpression will automatically be parsed in source_scope.

The other critical new ingredient is a generalization of the Closed relation from two chapters ago. The new relation exp_equiv characters when two expressions may be considered syntactically equal. We need to be able to handle cases where each expression uses a different var type. Intuitively, we will want to compare expressions that use their variables to store source-level and target-level values. We express pairs of equivalent variables using a list parameter to the relation; variable expressions will be considered equivalent if and only if their variables belong to this list. The rule for function abstraction extends the list in a higher-order way. The remaining rules just implement the obvious congruence over expressions.

  Section exp_equiv.
    Variables var1 var2 : type -> Type.

    Inductive exp_equiv : list { t : type & var1 t * var2 t }%type
      -> forall t, exp var1 t -> exp var2 t -> Prop :=
    | EqVar : forall G t (v1 : var1 t) v2,
      In (existT _ t (v1, v2)) G
      -> exp_equiv G (#v1) (#v2)

    | EqConst : forall G n,
      exp_equiv G (*) n) (*) n)
    | EqPlus : forall G x1 y1 x2 y2,
      exp_equiv G x1 x2
      -> exp_equiv G y1 y2
      -> exp_equiv G (x1 +^ y1) (x2 +^ y2)

    | EqApp : forall G t1 t2 (f1 : exp _ (t1 --> t2)) (x1 : exp _ t1) f2 x2,
      exp_equiv G f1 f2
      -> exp_equiv G x1 x2
      -> exp_equiv G (f1 @ x1) (f2 @ x2)
    | EqAbs : forall G t1 t2 (f1 : var1 t1 -> exp var1 t2) f2,
      (forall v1 v2, exp_equiv (existT _ t1 (v1, v2) :: G) (f1 v1) (f2 v2))
      -> exp_equiv G (Abs f1) (Abs f2).
  End exp_equiv.

It turns out that, for any parametric expression E, any two instantiations of E with particular var types must be equivalent, with respect to an empty variable list. The parametricity of Gallina guarantees this, in much the same way that it guaranteed the truth of the axiom about Closed. Thus, we assert an analogous axiom here.

  Axiom Exp_equiv : forall t (E : Exp t) var1 var2,
    exp_equiv nil (E var1) (E var2).
End Source.

Now we need to define the CPS language, where binary function types are replaced with unary continuation types, and we add product types because they will be useful in our translation.

Module CPS.
  Inductive type : Type :=
  | TNat : type
  | Cont : type -> type
  | Prod : type -> type -> type.

  Notation "'Nat'" := TNat : cps_scope.
  Notation "t --->" := (Cont t) (at level 61) : cps_scope.
  Infix "**" := Prod (right associativity, at level 60) : cps_scope.

  Bind Scope cps_scope with type.
  Delimit Scope cps_scope with cps.

  Section vars.
    Variable var : type -> Type.

A CPS program is a series of bindings of primitive operations (primops), followed by either a halt with a final program result or by a call to a continuation. The arguments to these program-ending operations are enforced to be variables. To use the values of compound expressions instead, those expressions must be decomposed into bindings of primops. The primop language itself similarly forces variables for all arguments besides bodies of function abstractions.

    Inductive prog : Type :=
    | PHalt :
      var Nat
      -> prog
    | App : forall t,
      var (t --->)
      -> var t
      -> prog
    | Bind : forall t,
      primop t
      -> (var t -> prog)
      -> prog

    with primop : type -> Type :=
    | Const : nat -> primop Nat
    | Plus : var Nat -> var Nat -> primop Nat
    | Abs : forall t,
      (var t -> prog)
      -> primop (t --->)

    | Pair : forall t1 t2,
      var t1
      -> var t2
      -> primop (t1 ** t2)
    | Fst : forall t1 t2,
      var (t1 ** t2)
      -> primop t1
    | Snd : forall t1 t2,
      var (t1 ** t2)
      -> primop t2.
  End vars.

  Implicit Arguments PHalt [var].
  Implicit Arguments App [var t].

  Implicit Arguments Const [var].
  Implicit Arguments Plus [var].
  Implicit Arguments Abs [var t].
  Implicit Arguments Pair [var t1 t2].
  Implicit Arguments Fst [var t1 t2].
  Implicit Arguments Snd [var t1 t2].

  Notation "'Halt' x" := (PHalt x) (no associativity, at level 75) : cps_scope.
  Infix "@@" := App (no associativity, at level 75) : cps_scope.
  Notation "x <- p ; e" := (Bind p (fun x => e))
    (right associativity, at level 76, p at next level) : cps_scope.
  Notation "! <- p ; e" := (Bind p (fun _ => e))
    (right associativity, at level 76, p at next level) : cps_scope.

  Notation "^ n" := (Const n) (at level 70) : cps_scope.
  Infix "+^" := Plus (left associativity, at level 79) : cps_scope.

  Notation "\ x , e" := (Abs (fun x => e)) (at level 78) : cps_scope.
  Notation "\ ! , e" := (Abs (fun _ => e)) (at level 78) : cps_scope.

  Notation "[ x1 , x2 ]" := (Pair x1 x2) : cps_scope.
  Notation "#1 x" := (Fst x) (at level 72) : cps_scope.
  Notation "#2 x" := (Snd x) (at level 72) : cps_scope.

  Bind Scope cps_scope with prog primop.

  Open Scope cps_scope.

In interpreting types, we treat continuations as functions with codomain nat, choosing nat as our arbitrary program result type.

  Fixpoint typeDenote (t : type) : Set :=
    match t with
      | Nat => nat
      | t' ---> => typeDenote t' -> nat
      | t1 ** t2 => (typeDenote t1 * typeDenote t2)%type

A mutually-recursive definition establishes the meanings of programs and primops.

  Fixpoint progDenote (e : prog typeDenote) : nat :=
    match e with
      | PHalt n => n
      | App _ f x => f x
      | Bind _ p x => progDenote (x (primopDenote p))

  with primopDenote t (p : primop typeDenote t) : typeDenote t :=
    match p with
      | Const n => n
      | Plus n1 n2 => n1 + n2

      | Abs _ e => fun x => progDenote (e x)

      | Pair _ _ v1 v2 => (v1, v2)
      | Fst _ _ v => fst v
      | Snd _ _ v => snd v

  Definition Prog := forall var, prog var.
  Definition Primop t := forall var, primop var t.
  Definition ProgDenote (E : Prog) := progDenote (E _).
  Definition PrimopDenote t (P : Primop t) := primopDenote (P _).
End CPS.

Import Source CPS.

The translation itself begins with a type-level compilation function. We change every function into a continuation whose argument is a pair, consisting of the translation of the original argument and of an explicit return pointer.

Fixpoint cpsType (t : Source.type) : CPS.type :=
  match t with
    | Nat => Nat%cps
    | t1 --> t2 => (cpsType t1 ** (cpsType t2 ---> ) ---> )%cps

Now we can define the expression translation. The notation x <-- e1; e2 stands for translating source-level expression e1, binding x to the CPS-level result of running the translated program, and then evaluating CPS-level expression e2 in that context.

Reserved Notation "x <-- e1 ; e2" (right associativity, at level 76, e1 at next level).

Section cpsExp.
  Variable var : CPS.type -> Type.

  Import Source.
  Open Scope cps_scope.

We implement a well-known variety of higher-order, one-pass CPS translation. The translation cpsExp is parameterized not only by the expression e to translate, but also by a meta-level continuation. The idea is that cpsExp evaluates the translation of e and calls the continuation on the result. With this convention, cpsExp itself is a natural match for the notation we just reserved.

  Fixpoint cpsExp t (e : exp (fun t => var (cpsType t)) t)
    : (var (cpsType t) -> prog var) -> prog var :=
    match e with
      | Var _ v => fun k => k v

      | Const n => fun k =>
        x <- *) n;
        k x
      | Plus e1 e2 => fun k =>
        x1 <-- e1;
        x2 <-- e2;
        x <- x1 +^ x2;
        k x

      | App _ _ e1 e2 => fun k =>
        f <-- e1;
        x <-- e2;
        kf <- \ r, k r;
        p <- [x, kf];
        f @@ p
      | Abs _ _ e' => fun k =>
        f <- CPS.Abs (var := var) (fun p =>
          x <- #1 p;
          kf <- #2 p;
          r <-- e' x;
          kf @@ r);
        k f

    where "x <-- e1 ; e2" := (cpsExp e1 (fun x => e2)).
End cpsExp.

Since notations do not survive the closing of sections, we redefine the notation associated with cpsExp.

Notation "x <-- e1 ; e2" := (cpsExp e1 (fun x => e2)) : cps_scope.

Implicit Arguments cpsExp [var t].

We wrap cpsExp into the parametric version CpsExp, passing an always-halt continuation at the root of the recursion.

Definition CpsExp (E : Exp Nat) : Prog :=
  fun _ => cpsExp (E _) (PHalt (var := _)).

Eval compute in CpsExp zero.

     = fun var : type -> Type => x <- ^0; Halt x
     : Prog

Eval compute in CpsExp one.

     = fun var : type -> Type => x <- ^1; Halt x
     : Prog

Eval compute in CpsExp zpo.

     = fun var : type -> Type => x <- ^0; x0 <- ^1; x1 <- (x +^ x0); Halt x1
     : Prog

Eval compute in CpsExp app_ident.

     = fun var : type -> Type =>
       f <- (\ p, x <- #1 p; kf <- #2 p; kf @@ x);
       x <- ^0;
       x0 <- ^1; x1 <- (x +^ x0); kf <- (\ r, Halt r); p <- [x1, kf]; f @@ p
     : Prog

Eval compute in CpsExp app_ident'.

     = fun var : type -> Type =>
       f <-
       (\ p,
        x <- #1 p;
        kf <- #2 p;
        f <-
        (\ p0,
         x0 <- #1 p0;
         kf0 <- #2 p0; kf1 <- (\ r, kf0 @@ r); p1 <- [x0, kf1]; x @@ p1);
        kf @@ f);
       f0 <- (\ p, x <- #1 p; kf <- #2 p; kf @@ x);
       kf <-
       (\ r,
        x <- ^0;
        x0 <- ^1;
        x1 <- (x +^ x0); kf <- (\ r0, Halt r0); p <- [x1, kf]; r @@ p);
       p <- [f0, kf]; f @@ p
     : Prog

Eval compute in ProgDenote (CpsExp zero).

     = 0
     : nat

Eval compute in ProgDenote (CpsExp one).

     = 1
     : nat

Eval compute in ProgDenote (CpsExp zpo).

     = 1
     : nat

Eval compute in ProgDenote (CpsExp app_ident).

     = 1
     : nat

Eval compute in ProgDenote (CpsExp app_ident').

     = 1
     : nat

Our main inductive lemma about cpsExp needs a notion of compatibility between source-level and CPS-level values. We express compatibility with a logical relation; that is, we define a binary relation by recursion on type structure, and the function case of the relation considers functions related if they map related arguments to related results. In detail, the function case is slightly more complicated, since it must deal with our continuation-based calling convention.

Fixpoint lr (t : Source.type)
  : Source.typeDenote t -> CPS.typeDenote (cpsType t) -> Prop :=
  match t with
    | Nat => fun n1 n2 => n1 = n2
    | t1 --> t2 => fun f1 f2 =>
      forall x1 x2, lr _ x1 x2
        -> forall k, exists r,
          f2 (x2, k) = k r
          /\ lr _ (f1 x1) r

The main lemma is now easily stated and proved. The most surprising aspect of the statement is the presence of two versions of the expression to be compiled. The first, e1, uses a var choice that makes it a suitable argument to expDenote. The second expression, e2, uses a var choice that makes its compilation, cpsExp e2 k, a suitable argument to progDenote. We use exp_equiv to assert that e1 and e2 have the same underlying structure, up to a variable correspondence list G. A hypothesis about G ensures that all of its pairs of variables belong to the logical relation lr. We also use lr, in concert with some quantification over continuations and program results, in the conclusion of the lemma.

The lemma's proof should be unsurprising by now. It uses our standard bag of Ltac tricks to help out with quantifier instantiation; crush and eauto can handle the rest.

Lemma cpsExp_correct : forall G t (e1 : exp _ t) (e2 : exp _ t),
  exp_equiv G e1 e2
  -> (forall t v1 v2, In (existT _ t (v1, v2)) G -> lr t v1 v2)
  -> forall k, exists r,
    progDenote (cpsExp e2 k) = progDenote (k r)
    /\ lr t (expDenote e1) r.
  induction 1; crush;
    repeat (match goal with
              | [ H : forall k, exists r, progDenote (cpsExp ?E k) = _ /\ _
                  |- context[cpsExp ?E ?K] ] =>
                generalize (H K); clear H
              | [ |- exists r, progDenote (_ ?R) = progDenote (_ r) /\ _ ] =>
                exists R
              | [ t1 : Source.type |- _ ] =>
                match goal with
                  | [ Hlr : lr t1 ?X1 ?X2, IH : forall v1 v2, _ |- _ ] =>
                    generalize (IH X1 X2); clear IH; intro IH;
                      match type of IH with
                        | ?P -> _ => assert P
            end; crush); eauto.

A simple lemma establishes the degenerate case of cpsExp_correct's hypothesis about G.

Lemma vars_easy : forall t v1 v2,
  In (existT (fun t0 => (Source.typeDenote t0 * typeDenote (cpsType t0))%type) t
    (v1, v2)) nil -> lr t v1 v2.

A manual application of cpsExp_correct proves a version applicable to CpsExp. This is where we use the axiom Exp_equiv.

Theorem CpsExp_correct : forall (E : Exp Nat),
  ProgDenote (CpsExp E) = ExpDenote E.
  unfold ProgDenote, CpsExp, ExpDenote; intros;
    generalize (cpsExp_correct (e1 := E _) (e2 := E _)
      (Exp_equiv _ _ _) vars_easy (PHalt (var := _))); crush.


  1. When in the last chapter we implemented constant folding for simply-typed lambda calculus, it may have seemed natural to try applying beta reductions. This would have been a lot more trouble than is apparent at first, because we would have needed to convince Coq that our normalizing function always terminated.

    It might also seem that beta reduction is a lost cause because we have no effective way of substituting in the exp type; we only managed to write a substitution function for the parametric Exp type. This is not as big of a problem as it seems. For instance, for the language we built by extending simply-typed lambda calculus with products and sums, it also appears that we need substitution for simplifying case expressions whose discriminees are known to be inl or inr, but the function is still implementable.

    For this exercise, extend the products and sums constant folder from the last chapter so that it simplifies case expressions as well, by checking if the discriminee is a known inl or known inr. Also extend the correctness theorem to apply to your new definition. You will probably want to assert an axiom relating to an expression equivalence relation like the one defined in this chapter. Any such axiom should only mention syntax; it should not mention any compilation or denotation functions. Following the format of the axiom from the last chapter is the safest bet to avoid proving a worthless theorem.