mpri-funcprog-project/coq/AutosubstExtra.v

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Require Import Omega.
Require Import Autosubst.Autosubst.
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Require Import MyTactics. (* TEMPORARY *)
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(* -------------------------------------------------------------------------- *)
(* A more recognizable notation for lifting. *)
Notation lift i t := (t.[ren(+i)]).
(* -------------------------------------------------------------------------- *)
Section Extras.
Context A `{Ids A, Rename A, Subst A, SubstLemmas A}.
Lemma up_ren:
forall xi,
ren (upren xi) = up (ren xi).
Proof.
intros. autosubst.
Qed.
Lemma upn_ren:
forall i xi,
ren (iterate upren i xi) = upn i (ren xi).
Proof.
induction i; intros.
{ reflexivity. }
{ rewrite <- fold_up_upn. rewrite <- IHi. asimpl. reflexivity. }
Qed.
Lemma plus_upn: (* close to [up_liftn] *)
forall i sigma,
(+i) >>> upn i sigma = sigma >> ren (+i).
Proof.
induction i; intros.
{ rewrite iterate_0. autosubst. }
{ rewrite iterate_S. asimpl. rewrite IHi. autosubst. }
Qed.
Lemma up_sigma_up_ren:
forall t i sigma,
t.[up sigma].[up (ren (+i))] = t.[up (ren (+i))].[upn (1 + i) sigma].
Proof.
intros. asimpl. rewrite plus_upn. asimpl. reflexivity.
Qed.
Lemma upn_k_sigma_x:
forall k sigma x,
x < k ->
upn k sigma x = ids x.
Proof.
induction k; intros; asimpl.
{ omega. }
{ destruct x; asimpl.
{ eauto. }
{ rewrite IHk by omega. autosubst. }
}
Qed.
Lemma push_substitution_last:
forall t v i,
t.[v .: ren (+i)] = t.[up (ren (+i))].[v/].
Proof.
intros. autosubst.
Qed.
Lemma push_substitution_last_up_hoist:
forall t v i j,
t.[up (v .: ren (+i))].[up (ren (+j))] =
t.[up (up (ren (+j + i)))].[up (lift j v .: ids)].
Proof.
intros. autosubst.
Qed.
Lemma lift_lift:
forall i j t,
lift i (lift j t) = lift (i + j) t.
Proof.
intros. autosubst.
Qed.
Lemma lift_upn:
forall t i sigma,
(lift i t).[upn i sigma] = lift i t.[sigma].
Proof.
intros. asimpl.
erewrite plus_upn.
reflexivity.
Qed.
Lemma lift_up:
forall t sigma,
(lift 1 t).[up sigma] = lift 1 t.[sigma].
Proof.
intros. change up with (upn 1). eapply lift_upn.
Qed.
Lemma up_sigma_f:
forall (sigma : var -> A) (f : A -> A),
f (ids 0) = ids 0 ->
(forall i t, lift i (f t) = f (lift i t)) ->
up (sigma >>> f) = up sigma >>> f.
Proof.
intros. f_ext. intros [|x]; asimpl; eauto.
Qed.
Lemma upn_sigma_f:
forall (sigma : var -> A) (f : A -> A),
f (ids 0) = ids 0 ->
(forall i t, lift i (f t) = f (lift i t)) ->
forall i,
upn i (sigma >>> f) = upn i sigma >>> f.
Proof.
induction i; intros.
{ reflexivity. }
{ do 2 rewrite <- fold_up_upn. rewrite IHi. erewrite up_sigma_f by eauto. reflexivity. }
Qed.
Lemma upn_theta_sigma_ids:
forall theta sigma i,
theta >> sigma = ids ->
upn i theta >> upn i sigma = ids.
Proof.
intros theta sigma i Hid.
rewrite up_comp_n.
rewrite Hid.
rewrite up_id_n.
reflexivity.
Qed.
Lemma up_theta_sigma_ids:
forall theta sigma,
theta >> sigma = ids ->
up theta >> up sigma = ids.
Proof.
change up with (upn 1). eauto using upn_theta_sigma_ids.
Qed.
Lemma scons_scomp:
forall (T : A) Gamma theta,
T.[theta] .: (Gamma >> theta) = (T .: Gamma) >> theta.
Proof.
intros. autosubst.
Qed.
(* BUG: the two sides of this equation are distinct, yet they are
printed identically. *)
Goal
forall v f,
v .: (ids >>> f) = (v .: ids) >>> f.
Proof.
intros.
Fail reflexivity.
Abort.
End Extras.
(* This incantation means that [eauto with autosubst] can use the tactic
[autosubst] to prove an equality. *)
Hint Extern 1 (_ = _) => autosubst : autosubst.
(* This incantation means that [eauto with autosubst] can use the lemmas
whose names are listed here. This is useful when an equality involves
metavariables, so the tactic [autosubst] fails. *)
Hint Resolve scons_scomp : autosubst.
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(* -------------------------------------------------------------------------- *)