mpri-funcprog-project/coq/CPSSubstitution.v

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Require Import MyTactics.
Require Import LambdaCalculusSyntax.
Require Import LambdaCalculusValues.
Require Import CPSDefinition.
Require Import CPSContextSubstitution.
Require Import CPSRenaming.
(* The CPS transformation commutes with certain substitutions. More precisely,
it commutes with a substitution [sigma] of values for variables, up to a
transformation of the values in the codomain of [sigma].
In the case of [cpsv], we have the following diagram: applying [sigma]
first, followed with [cpsv], is the same as applying [cpsv] first, followed
with [sigma >>> cpsv].
cpsv v.[sigma] = (cpsv v).[sigma >>> cpsv]
This can also be written in point-free style, that is, without mentioning
the value [v]:
subst sigma >>> cpsv = cpsv >>> subst (sigma >>> cpsv)
As in the case of the renaming lemma (see CPSRenaming.v), this statement is
proved by induction on the size of terms, together with an analogous
statement about the function [cps]. *)
(* The proof depends on [CPSRenaming] via the lemmas [up_sigma_cpsv] and
[upn_sigma_cpsv], which are declared as hints for [obvious]. *)
Lemma substitution:
(
forall v sigma sigma',
sigma' = sigma >>> cpsv ->
is_value_subst sigma ->
(cpsv v).[sigma'] = cpsv v.[sigma]
) /\ (
forall t c sigma c' sigma',
sigma' = sigma >>> cpsv ->
is_value_subst sigma ->
substc sigma' c = c' ->
(cps t c).[sigma'] = cps t.[sigma] c'
).
Proof.
eapply mutual_induction.
(* [cpsv] *)
{ intros n IHcps v Hvn sigma sigma' Heq Hsigma. subst.
destruct v; asimpl; cpsv; asimpl; try reflexivity.
(* Lam *)
{ erewrite IHcps by obvious. asimpl. reflexivity. }
}
(* [cps] *)
{ intros n IHcpsv IHcps t c Htn sigma c' sigma' Heq Hsigma Hsubstc. subst.
value_or_app_or_let t; asimpl; cps.
(* Case: [t] is a value. *)
{ erewrite apply_substitution by eauto.
erewrite IHcpsv by obvious.
reflexivity. }
(* Case: [t] is an application. *)
{ eapply IHcps; obvious.
simpl. f_equal.
erewrite <- lift_up by tc.
eapply IHcps; obvious.
asimpl. do 2 f_equal.
rewrite lift_reify.
eapply reify_substitution.
rewrite substc_substc.
reflexivity. }
(* Case: [t] is a [let] construct. *)
{ eapply IHcps; obvious.
simpl.
rewrite fold_up_up.
do 2 f_equal.
erewrite IHcps by first [ eapply substc_liftc_liftc; eauto | obvious ].
autosubst. }
}
Qed.
(* The projections of the above result. *)
Definition cpsv_substitution := proj1 substitution.
Definition cps_substitution := proj2 substitution.
(* A point-free reformulation of the above result: [cpsv] commutes with an
arbitrary substitution [sigma], up to a transformation of the values in the
codomain of [sigma]. *)
Goal
forall sigma,
is_value_subst sigma ->
cpsv >>> subst (sigma >>> cpsv) =
subst sigma >>> cpsv.
Proof.
intros. f_ext; intros v. asimpl. eauto using cpsv_substitution.
Qed.
(* This technical lemma is used below. *)
Lemma cpsv_cons:
forall v,
cpsv v .: ids = (v .: ids) >>> cpsv.
Proof.
intros. f_ext; intros [|x]; autosubst.
Qed.
(* A corollary where the substitution [sigma] is [v .: ids], that is, a
substitution of the value [v] for the variable 0. This one is about
[cpsv]. *)
Lemma cpsv_substitution_0:
forall v w,
is_value v ->
(cpsv w).[cpsv v/] =
cpsv w.[v/].
Proof.
intros. rewrite cpsv_cons. erewrite cpsv_substitution by obvious. reflexivity.
Qed.
(* Another corollary where the substitution [sigma] is [v .: ids], that is, a
substitution of the value [v] for the variable 0. This one is about [cps]
and concerns the case where the continuation is of the form [liftc 1 c], so
it is unaffected. *)
Lemma cps_substitution_0:
forall t c v,
is_value v ->
(cps t (liftc 1 c)).[cpsv v/] =
cps t.[v/] c.
Proof.
intros. eapply cps_substitution.
{ autosubst. }
{ obvious. }
{ eauto using substc_liftc_single. }
Qed.
(* A corollary where the substitution [sigma] is [up (v .: ids)], that is, a
substitution of the value [v] for the variable 1, and the continuation is
the variable 0, so it is unaffected. *)
Lemma cps_substitution_1_O_Var_0:
forall t v,
is_value v ->
(cps t (O (Var 0))).[up (cpsv v .: ids)] =
cps t.[up (v .: ids)] (O (Var 0)).
Proof.
intros. eapply cps_substitution.
{ rewrite cpsv_cons. obvious. }
{ obvious. }
{ reflexivity. }
Qed.