mpri-funcprog-project/coq/CPSContextSubstitution.v

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Require Import MyTactics.
Require Import LambdaCalculusSyntax.
Require Import LambdaCalculusValues.
Require Import CPSDefinition.
(* This file contains a few lemmas about [substc]. *)
(* Two successive applications of [substc] can be fused. *)
Lemma substc_substc:
forall sigma1 sigma2 c,
substc sigma2 (substc sigma1 c) = substc (sigma1 >> sigma2) c.
Proof.
intros. destruct c; autosubst.
Qed.
(* Two successive applications of [liftc] can be fused. *)
Lemma liftc_liftc:
forall i j c,
liftc i (liftc j c) = liftc (i + j) c.
Proof.
intros i j c. destruct c; autosubst.
Qed.
(* [apply] commutes with substitutions. *)
Lemma apply_substitution:
forall c sigma c' v,
substc sigma c = c' ->
(apply c v).[sigma] = apply c' v.[sigma].
Proof.
intros. subst. destruct c; autosubst.
Qed.
(* [reify] commutes with substitutions. *)
Lemma reify_substitution:
forall c sigma c',
substc sigma c = c' ->
(reify c).[sigma] = reify c'.
Proof.
intros. subst. destruct c; reflexivity.
Qed.
(* As a special case, [reify] commutes with lifting. *)
Lemma lift_reify:
forall i c,
lift i (reify c) = reify (liftc i c).
Proof.
intros. destruct c; reflexivity.
Qed.
(* [substc] is preserved by [liftc]. *)
Lemma substc_liftc_liftc:
forall i c sigma c',
substc sigma c = c' ->
substc (upn i sigma) (liftc i c) = liftc i c'.
Proof.
intros. subst. destruct c; simpl.
{ rewrite lift_upn by tc. reflexivity. }
{ asimpl. erewrite plus_upn by tc. autosubst. }
Qed.
Hint Resolve substc_liftc_liftc : obvious.
(* As is the case for terms, lifting [c] by 1, then applying a substitution
of the form [v .: ids], yields [c] again. *)
Lemma substc_liftc_single:
forall c v,
substc (v .: ids) (liftc 1 c) = c.
Proof.
intros. destruct c; autosubst.
Qed.