mpri-funcprog-project/coq/CPSRenaming.v
2017-10-11 15:28:20 +02:00

92 lines
2.6 KiB
Coq

Require Import MyTactics.
Require Import LambdaCalculusSyntax.
Require Import LambdaCalculusValues.
Require Import CPSDefinition.
Require Import CPSContextSubstitution.
(* The CPS transformation commutes with renamings, where a renaming [sigma] is
a substitution that maps variables to variables. (Note that [sigma] is not
necessarily injective.) *)
Lemma renaming:
(
forall v sigma,
is_ren sigma ->
(cpsv v).[sigma] = cpsv v.[sigma]
) /\ (
forall t c sigma c',
is_ren sigma ->
substc sigma c = c' ->
(cps t c).[sigma] = cps t.[sigma] c'
).
Proof.
eapply mutual_induction.
(* [cpsv] *)
{ intros n IHcps v Hvn sigma Hsigma.
destruct v; asimpl; cpsv; asimpl; try reflexivity.
(* [Var] *)
(* The CPS transformation maps variables to variables. *)
{ destruct Hsigma as [ xi ? ]. subst sigma. reflexivity. }
(* [Lam] *)
{ erewrite IHcps by obvious. asimpl. reflexivity. }
}
(* [cps] *)
{ intros n IHcpsv IHcps t c Htn sigma c' Hsigma Hsubstc.
(* Perform case analysis on [t]. The first two cases, [Var] and [Lam],
can be shared by treating the case where [t] is a value. *)
value_or_app_or_let t; asimpl; cps.
(* Case: [t] is a value. *)
{ erewrite apply_substitution by eauto.
rewrite IHcpsv by obvious.
reflexivity. }
(* Case: [t] is an application. *)
{ eapply IHcps; obvious.
erewrite <- lift_upn by tc.
simpl. f_equal.
eapply IHcps; obvious.
simpl.
rewrite fold_up_upn, lift_upn by tc.
do 3 f_equal.
eauto using reify_substitution. }
(* Case: [t] is a [let] construct. *)
{ eapply IHcps; obvious.
simpl. do 2 f_equal.
rewrite fold_up_up.
erewrite IHcps by first [ eapply substc_liftc_liftc; eauto | obvious ].
autosubst. }
}
Qed.
(* The projections of the above result. *)
Definition cpsv_renaming := proj1 renaming.
Definition cps_renaming := proj2 renaming.
(* A point-free reformulation of the above result: [cpsv] commutes with
an arbitrary renaming [xi]. *)
Goal
forall sigma,
is_ren sigma ->
cpsv >>> subst sigma = subst sigma >>> cpsv.
Proof.
intros. f_ext; intros t. asimpl. eauto using cpsv_renaming.
Qed.
(* Corollaries. *)
Lemma up_sigma_cpsv:
forall sigma,
up (sigma >>> cpsv) = up sigma >>> cpsv.
Proof.
eauto using up_sigma_f, cpsv_renaming with is_ren typeclass_instances.
Qed.
Lemma upn_sigma_cpsv:
forall i sigma,
upn i (sigma >>> cpsv) = upn i sigma >>> cpsv.
Proof.
eauto using upn_sigma_f, cpsv_renaming with is_ren typeclass_instances.
Qed.
Hint Resolve up_sigma_cpsv upn_sigma_cpsv : obvious.