446 lines
16 KiB
Coq
446 lines
16 KiB
Coq
Require Import MyTactics.
|
|
Require Import FixExtra.
|
|
Require Import LambdaCalculusSyntax.
|
|
Require Import LambdaCalculusValues.
|
|
|
|
(* This is a definition of the CPS transformation. *)
|
|
|
|
(* This CPS transformation is "one-pass" in the sense that it does not produce
|
|
any administrative redexes. (In other words, there is no need for a second
|
|
pass, whose purpose would be to remove administrative redexes.)
|
|
|
|
To achieve this, instead of defining [cps t k], where the continuation [k]
|
|
is a term, we define [cps t c], where the continuation [c] is either a term
|
|
(also known as an object-level continuation) or a term-with-a-hole [K]
|
|
(also known as a meta-level continuation).
|
|
|
|
This formulation of the CPS transformation is analogous to Danvy and
|
|
Filinski's higher-order formulation. Yet, it is still technically
|
|
first-order, because we represent a term-with-a-hole [K] as a term,
|
|
where the variable 0 denotes the hole. *)
|
|
|
|
(* This CPS transformation is defined by recursion on the size of terms. This
|
|
allows recursive calls of the form [cps (lift 1 t)], which would be illegal
|
|
if [cps] was defined by structural induction. Definitions by well-founded
|
|
recursion in Coq are somewhat tricky, requiring the use of the fixed point
|
|
combinator [Fix] and the tactic [refine]. For explanations, see the chapter
|
|
on general recursion in Chlipala's book at
|
|
http://adam.chlipala.net/cpdt/html/GeneralRec.html *)
|
|
|
|
(* The situation could be complicated by the fact that we wish to define two
|
|
functions simultaneously:
|
|
|
|
[cpsv v] is the translation of a value [v].
|
|
|
|
[cps t c] is the translation of a term [t] with continuation [c].
|
|
|
|
To avoid this problem, we follow Danvy and Filinski's approach, which is to
|
|
first define [cps] directly (as this does not cause much duplication), then
|
|
define [cpsv] in terms of [cps]. In the latter step, no case analysis is
|
|
required: [cpsv] is obtained by invoking [cps] with an identity meta-level
|
|
continuation.
|
|
|
|
Regardless of how [cps] and [cpsv] are defined, we prove that the they
|
|
satisfy the desired recurrence equations, so, in the end, everything is
|
|
just as if they had been defined in a mutually recursive manner. *)
|
|
|
|
(* -------------------------------------------------------------------------- *)
|
|
|
|
(* As explained above, a continuation [c] is
|
|
|
|
either [O k], where [k] is a term (in fact, a value)
|
|
(an object-level continuation)
|
|
|
|
or [M K], where [K] is term-with-a-hole
|
|
(a meta-level continuation).
|
|
|
|
A term-with-a-hole [K] is represented as a term where the variable 0 denotes
|
|
the hole (and, of course, all other variables are shifted up). *)
|
|
|
|
Inductive continuation :=
|
|
| O (k : term)
|
|
| M (K : term).
|
|
|
|
(* The term [apply c v] is the application of the continuation [c] to the
|
|
value [v]. If [c] is an object-level continuation [k] (that is, a term),
|
|
then it is just the object-level application [App k v]. If [c] is a
|
|
meta-level continuation [K], then it is the meta-level operation of filling
|
|
the hole with the value [v], which in fact is just a substitution,
|
|
[K.[v/]]. *)
|
|
|
|
Definition apply (c : continuation) (v : term) : term :=
|
|
match c with
|
|
| O k =>
|
|
App k v
|
|
| M K =>
|
|
K.[v/]
|
|
end.
|
|
|
|
(* The term [reify c] is the reification of the continuation [c] as an
|
|
object-level continuation (that is, a term). If [c] is an object-level
|
|
continuation [k], then it is just [k]. If [c] is a meta-level continuation
|
|
[K], then [reify c] is the term [\x.K x]. (This is usually known as a
|
|
two-level eta-expansion.) Because the hole is already represented by the
|
|
variable 0, filling the hole with the variable [x] is a no-op. Therefore,
|
|
it suffices to write [Lam K] to obtain the desired lambda-abstraction. *)
|
|
|
|
Definition reify (c : continuation) : term :=
|
|
match c with
|
|
| O k =>
|
|
k
|
|
| M K =>
|
|
Lam K
|
|
end.
|
|
|
|
(* The continuation [substc sigma c] is the result of applying the
|
|
substitution [sigma] to the continuation [c]. *)
|
|
|
|
Definition substc sigma (c : continuation) : continuation :=
|
|
match c with
|
|
| O k =>
|
|
O k.[sigma]
|
|
| M K =>
|
|
M K.[up sigma]
|
|
end.
|
|
|
|
(* [liftc i c] is the result of lifting the free names of the continuation [c]
|
|
up by [i]. The function [liftc] can be defined in terms of [substc]. *)
|
|
|
|
Notation liftc i c :=
|
|
(substc (ren (+i)) c).
|
|
|
|
(* -------------------------------------------------------------------------- *)
|
|
|
|
(* Here is the definition of [cps]. Because we must keep track of sizes and
|
|
prove that the recursive calls cause a decrease in the size, this
|
|
definition cannot be easily written as Coq terms. Instead, we switch to
|
|
proof mode and use the tactic [refine]. This allows us to write some of the
|
|
code, with holes [_] in it, and use proof mode to fill the holes. *)
|
|
|
|
(* [cps t c] is the CPS-translation of the term [t] with continuation [c]. *)
|
|
|
|
Definition cps : term -> continuation -> term.
|
|
Proof.
|
|
(* The definition is by well-founded recursion on the size of [t]. *)
|
|
refine (Fix smaller_wf_transparent (fun _ => continuation -> term) _).
|
|
(* We receive the arguments [t] and [c] as well as a function [cps_]
|
|
which we use for recursive calls. At every call to [cps_], there
|
|
is an obligation to prove that the size of the argument is less
|
|
than the size of [t]. *)
|
|
intros t cps_ c.
|
|
(* The definition is by cases on the term [t]. *)
|
|
destruct t as [ x | t | t1 t2 | t1 t2 ].
|
|
(* When [t] is a value, we return an application of the continuation [c]
|
|
to a value which will later be known as [cpsv t]. *)
|
|
(* Case: [Var x]. *)
|
|
{ refine (apply c (Var x)). }
|
|
(* Case: [Lam t]. *)
|
|
(* In informal notation, the term [\x.t] is transformed to an application of
|
|
[c] to [\x.\k.[cps t k]], where [k] is a fresh variable. Here, [k] is
|
|
represented by the de Bruijn index 0, and the term [t] must be lifted
|
|
because it is brought into the scope of [k]. *)
|
|
{ refine (apply c
|
|
(Lam (* x *) (Lam (* k *) (cps_ (lift 1 t) _ (O (Var 0)))))
|
|
); abstract size. }
|
|
(* Case: [App t1 t2]. *)
|
|
(* The idea is, roughly, to first obtain the value [v1] of [t1], then obtain
|
|
the value [v2] of [t2], then perform the application [v1 v2 c].
|
|
|
|
Two successive calls to [cps] are used to obtain [v1] and [v2]. In each
|
|
case, we avoid administrative redexes by using an [M] continuation.
|
|
Thus, [v1] and [v2] are represented by two holes, denoted by the
|
|
variables [Var 1] and [Var 0].
|
|
|
|
If [t1] is a value, then the first hole will be filled directly with (the
|
|
translation of) [t1]; otherwise, it will be filled with a fresh variable,
|
|
bound to the result of evaluating (the translation of) [t1]. The same
|
|
goes for [t2].
|
|
|
|
The application [v1 v2 c] does not directly make sense, since [c] is a
|
|
continuation, not a term. Instead of [c], we must use [reify c]. The
|
|
continuation [c] must turned into a term, so it can be used in this
|
|
term-level application.
|
|
|
|
A little de Bruijn wizardry is involved. The term [t2] must be lifted up
|
|
by 1 because it is brought into the scope of the first meta-level
|
|
continuation. Similarly, the first hole must be lifted by 1 because it is
|
|
brought into the scope of the second meta-level continuation: thus, it
|
|
becomes Var 1. Finally, the continuation [c] must be lifted up by [2]
|
|
because it is brought into the scope of both. Here, we have a choice
|
|
between [reify (liftc _ c)] and [lift _ (reify c)], which mean the same
|
|
thing. *)
|
|
{ refine (
|
|
cps_ t1 _ (M (
|
|
cps_ (lift 1 t2) _ (M (
|
|
App (App (Var 1) (Var 0)) (lift 2 (reify c))
|
|
))
|
|
))
|
|
);
|
|
abstract size.
|
|
}
|
|
(* Case: [Let x = t1 in t2]. *)
|
|
(* The idea is to first obtain the value [v1] of [t1]. This value is
|
|
represented by the hole [Var 0] in the [M] continuation. We bind
|
|
this value via a [Let] construct to the variable [x] (represented by the
|
|
index 0 in [t2]), then execute [t2], under the original continuation [c].
|
|
All variables in [t2] except [x] must lifted up by one because they are
|
|
brought in the scope of the meta-level continuation. The continuation [c]
|
|
must be lifted up by 2 because it is brought in the scope of the
|
|
meta-level continuation and in the scope of the [Let] construct. *)
|
|
{ refine (
|
|
cps_ t1 _ (M (
|
|
Let (Var 0) (
|
|
cps_ t2.[up (ren (+1))] _ (liftc 2 c)
|
|
)
|
|
))
|
|
);
|
|
abstract size.
|
|
}
|
|
Defined.
|
|
|
|
(* -------------------------------------------------------------------------- *)
|
|
|
|
(* The above definition can be used inside Coq to compute the image of a term
|
|
through the transformation. For instance, the image under [cps] of [\x.x]
|
|
with object-level continuation [k] (a variable) is [k (\x.\k.k x)]. *)
|
|
|
|
Goal
|
|
cps (Lam (Var 0)) (O (Var 0)) =
|
|
App (Var 0) (Lam (Lam (App (Var 0) (Var 1)))).
|
|
Proof.
|
|
compute. (* optional *)
|
|
reflexivity.
|
|
Qed.
|
|
|
|
(* The image of [(\x.x) y] with continuation [k] is [(\x.\k.k x) y k]. *)
|
|
|
|
Goal
|
|
cps (App (Lam (Var 0)) (Var 0)) (O (Var 1)) =
|
|
App (App (Lam (Lam (App (Var 0) (Var 1)))) (Var 0)) (Var 1).
|
|
Proof.
|
|
compute. (* optional *)
|
|
reflexivity.
|
|
Qed.
|
|
|
|
(* -------------------------------------------------------------------------- *)
|
|
|
|
(* The initial continuation is used when invoking [cps] at the top level. *)
|
|
|
|
(* We could use [O (Lam (Var 0))] -- that is, the identity function -- as
|
|
the initial continuation. Instead, we use [M (Var 0)], that is, the
|
|
empty context. This sometimes saves one beta-redex. *)
|
|
|
|
Definition init :=
|
|
M (Var 0).
|
|
|
|
Definition cpsinit t :=
|
|
cps t init.
|
|
|
|
(* -------------------------------------------------------------------------- *)
|
|
|
|
(* Now that [cps] is defined, we can define [cpsv] in terms of it. *)
|
|
|
|
(* We explicitly check whether [v] is a value, and if it is not, we return a
|
|
dummy closed value. [cpsv] is supposed to be applied only to values,
|
|
anyway. Using a dummy value allows us to prove that [cpsv v] is always a
|
|
value, without requiring that [v] itself be a value. *)
|
|
|
|
Definition cpsv (v : term) :=
|
|
if_value v
|
|
(cpsinit v)
|
|
(Lam (Var 0)).
|
|
|
|
(* -------------------------------------------------------------------------- *)
|
|
|
|
(* The function [cps] satisfies the expected recurrence equations. *)
|
|
|
|
(* The lemmas [cps_var] and [cps_lam] are not used outside this file, as we
|
|
use [cps_value] instead, followed with [cpsv_var] or [cpsv_lam]. *)
|
|
|
|
Lemma cps_var:
|
|
forall x c,
|
|
cps (Var x) c = apply c (Var x).
|
|
Proof.
|
|
reflexivity.
|
|
Qed.
|
|
|
|
Lemma cps_lam:
|
|
forall t c,
|
|
cps (Lam t) c =
|
|
apply c (Lam (* x *) (Lam (* k *) (cps (lift 1 t) (O (Var 0))))).
|
|
Proof.
|
|
intros. erewrite Fix_eq_simplified with (f := cps) by reflexivity.
|
|
reflexivity.
|
|
Qed.
|
|
|
|
Lemma cps_app:
|
|
forall t1 t2 c,
|
|
cps (App t1 t2) c =
|
|
cps t1 (M (
|
|
cps (lift 1 t2) (M (
|
|
App (App (Var 1) (Var 0)) (lift 2 (reify c))
|
|
))
|
|
)).
|
|
Proof.
|
|
intros. erewrite Fix_eq_simplified with (f := cps) by reflexivity.
|
|
reflexivity.
|
|
Qed.
|
|
|
|
Lemma cps_let:
|
|
forall t1 t2 c,
|
|
cps (Let t1 t2) c =
|
|
cps t1 (M (
|
|
Let (Var 0) (
|
|
cps t2.[up (ren (+1))] (liftc 2 c)
|
|
)
|
|
)).
|
|
Proof.
|
|
intros. erewrite Fix_eq_simplified with (f := cps) by reflexivity.
|
|
reflexivity.
|
|
Qed.
|
|
|
|
(* -------------------------------------------------------------------------- *)
|
|
|
|
(* The translation of values is uniform: we also have the following equation. *)
|
|
|
|
Lemma cps_value:
|
|
forall v c,
|
|
is_value v ->
|
|
cps v c = apply c (cpsv v).
|
|
Proof.
|
|
destruct v; simpl; intros; try not_a_value; unfold cpsv, cpsinit.
|
|
{ rewrite !cps_var. reflexivity. }
|
|
{ rewrite !cps_lam. reflexivity. }
|
|
Qed.
|
|
|
|
(* -------------------------------------------------------------------------- *)
|
|
|
|
(* The function [cpsv] satisfies the expected recurrence equations. *)
|
|
|
|
Lemma cpsv_var:
|
|
forall x,
|
|
cpsv (Var x) = Var x.
|
|
Proof.
|
|
reflexivity.
|
|
Qed.
|
|
|
|
Lemma cpsv_lam:
|
|
forall t,
|
|
cpsv (Lam t) = Lam (Lam (cps (lift 1 t) (O (Var 0)))).
|
|
Proof.
|
|
intros. unfold cpsv, cpsinit. rewrite cps_lam. reflexivity.
|
|
Qed.
|
|
|
|
(* If [theta] is a renaming, then [theta x] is a variable, so [cpsv (theta x)]
|
|
is [theta x], which can also be written [(Var x).[theta]]. *)
|
|
|
|
Lemma cpsv_var_theta:
|
|
forall theta x,
|
|
is_ren theta ->
|
|
cpsv (theta x) = (Var x).[theta].
|
|
Proof.
|
|
inversion 1. subst. asimpl. reflexivity.
|
|
Qed.
|
|
|
|
(* -------------------------------------------------------------------------- *)
|
|
|
|
(* The tactic [cps] applies the rewriting rules [cps_value] and [cps_app] as
|
|
many times as possible, therefore expanding applications of the function
|
|
[cps] to values and to applications. *)
|
|
|
|
Ltac cps :=
|
|
repeat first [ rewrite cps_value by obvious
|
|
| rewrite cps_app | rewrite cps_let ].
|
|
|
|
(* -------------------------------------------------------------------------- *)
|
|
|
|
(* The translation of a value is a value. *)
|
|
|
|
(* In fact, thanks to the manner in which we have defined [cpsv], the image of
|
|
every term through [cpsv] is a value. This turns out to be quite pleasant,
|
|
as it allows removing nasty side conditions in several lemmas. *)
|
|
|
|
Lemma is_value_cpsv:
|
|
forall v,
|
|
(* is_value v -> *)
|
|
is_value (cpsv v).
|
|
Proof.
|
|
intros. destruct v; simpl; tauto.
|
|
Qed.
|
|
|
|
Hint Resolve is_value_cpsv : is_value obvious.
|
|
|
|
Hint Rewrite cpsv_var cpsv_lam : cpsv.
|
|
Ltac cpsv := autorewrite with cpsv.
|
|
|
|
(* -------------------------------------------------------------------------- *)
|
|
|
|
(* As a final step, we prove an induction principle that helps work with the
|
|
functions [cpsv] and [cps]. When trying to establish a property of the
|
|
function [cps], we often need to prove this property by induction on the
|
|
size of terms. Furthermore, we usually need to state an auxiliary property
|
|
of the function [cpsv] and to prove the two statements simultaneously, by
|
|
induction on the size of terms. The following lemma is tailored for this
|
|
purpose. It proves the properties [Pcpsv] and [Pcps] simultaneously. The
|
|
manner in which the index [n] is used reflects precisely the manner in
|
|
which each function depends on the other, with or without a decrease in
|
|
[n]. The dependencies are as follows:
|
|
|
|
[cpsv] invokes [cps] with a size decrease.
|
|
|
|
[cps] invokes [cpsv] without a size decrease and
|
|
[cps] with a size decrease.
|
|
|
|
It is worth noting that this proof has nothing to do with the definitions
|
|
of [cpsv] and [cps]. It happens to reflect just the right dependencies
|
|
between them. *)
|
|
|
|
Lemma mutual_induction:
|
|
forall
|
|
(Pcpsv : term -> Prop)
|
|
(Pcps : term -> continuation -> Prop),
|
|
(forall n,
|
|
(forall t c, size t < n -> Pcps t c) ->
|
|
(forall v, size v < S n -> Pcpsv v)
|
|
) ->
|
|
(forall n,
|
|
(forall v, size v < S n -> Pcpsv v) ->
|
|
(forall t c, size t < n -> Pcps t c) ->
|
|
(forall t c, size t < S n -> Pcps t c)
|
|
) ->
|
|
(
|
|
(forall v, Pcpsv v) /\
|
|
(forall t c, Pcps t c)
|
|
).
|
|
Proof.
|
|
intros Pcpsv Pcps IHcpsv IHcps.
|
|
assert (fact:
|
|
forall n,
|
|
(forall v, size v < n -> Pcpsv v) /\
|
|
(forall t k, size t < n -> Pcps t k)
|
|
).
|
|
{ induction n; intros; split; intros;
|
|
try solve [ elimtype False; omega ];
|
|
destruct IHn as (?&?); eauto. }
|
|
split; intros; eapply fact; eauto.
|
|
Qed.
|
|
|
|
(* -------------------------------------------------------------------------- *)
|
|
|
|
(* In the proofs that follow, we never expand the definition of [cpsv] or
|
|
[cps]: we use the tactics [cpsv] and [cps] instead. We actually forbid
|
|
unfolding [cpsv] and [cps], so our proofs cannot depend on the details of
|
|
the above definitions.
|
|
|
|
In general, when working with complex objects, it is good practice anyway
|
|
to characterize an object and forget how it was defined. Here, the
|
|
functions [cpsv] and [cps] are characterized by the equations that they
|
|
satisfy; the rest does not matter.
|
|
|
|
Attempting to work with transparent [cpsv] and [cps] would be problematic
|
|
for several reasons. The tactics [simpl] and [asimpl] would sometimes
|
|
expand these functions too far. Furthermore, because we have used the term
|
|
[smaller_wf_transparent] inside the definition of [cps], expanding [cps]
|
|
definition would often give rise to uncontrollably large terms. *)
|
|
|
|
Global Opaque cps cpsv.
|