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