mpri-funcprog-project/coq/Autosubst_FreeVars.v

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2017-10-05 17:57:33 +02:00
Require Import Omega.
Require Import Autosubst.Autosubst.
Require Import AutosubstExtra.
Require Import Autosubst_EOS.
(* -------------------------------------------------------------------------- *)
Class IdsLemmas (term : Type) {Ids_term : Ids term} := {
(* The identity substitution is injective. *)
ids_inj:
forall x y,
ids x = ids y ->
x = y
}.
(* -------------------------------------------------------------------------- *)
Section FreeVars.
Context
{A : Type}
{Ids_A : Ids A}
{Rename_A : Rename A}
{Subst_A : Subst A}
{IdsLemmas_A : IdsLemmas A}
{SubstLemmas_A : SubstLemmas A}.
(* -------------------------------------------------------------------------- *)
(* A reformulation of [ids_inj]. *)
Lemma ids_inj_False:
forall x y,
ids x = ids y ->
x <> y ->
False.
Proof.
intros.
assert (x = y). { eauto using ids_inj. }
unfold var in *.
omega.
Qed.
(* -------------------------------------------------------------------------- *)
(* The predicate [fv k t] means that the free variables of the term [t] are
contained in the semi-open interval [0..k). *)
Definition fv k t :=
t.[upn k (ren (+1))] = t.
(* -------------------------------------------------------------------------- *)
(* The predicate [closed t] means that the term [t] is closed, that is, [t]
has no free variables. *)
Definition closed :=
fv 0.
(* -------------------------------------------------------------------------- *)
(* This technical lemma states that the renaming [+1] is injective. *)
Lemma lift_inj_ids:
forall t x,
t.[ren (+1)] = ids (S x) <-> t = ids x.
Proof.
split; intros.
{ eapply lift_inj. autosubst. }
{ subst. autosubst. }
Qed.
(* -------------------------------------------------------------------------- *)
(* This lemma characterizes the meaning of [fv k] when applied to a variable. *)
Lemma fv_ids_eq:
forall k x,
fv k (ids x) <-> x < k.
Proof.
unfold fv. induction k; intros.
(* Base case. *)
{ asimpl. split; intros; elimtype False.
{ eauto using ids_inj_False. }
{ omega. }
}
(* Step. *)
{ destruct x; asimpl.
{ split; intros. { omega. } { reflexivity. } }
{ rewrite lift_inj_ids.
rewrite <- id_subst.
rewrite IHk. omega. }
}
Qed.
(* -------------------------------------------------------------------------- *)
(* A simplification lemma. *)
Lemma fv_lift:
forall k i t,
fv (k + i) t.[ren (+i)] <-> fv k t.
Proof.
unfold fv. intros. asimpl.
rewrite Nat.add_comm.
rewrite <- upn_upn.
erewrite plus_upn by eauto.
rewrite <- subst_comp.
split; intros.
{ eauto using lift_injn. }
{ f_equal. eauto. }
Qed.
(* -------------------------------------------------------------------------- *)
(* If [t] has at most [n - 1] free variables,
and if [x] is inserted among them,
then we get [eos x t],
which has at most [n] free variables. *)
Lemma fv_eos:
forall x n t,
x < n ->
fv (n - 1) t ->
fv n (eos x t).
Proof.
unfold fv. intros x n t ? ht.
rewrite eos_eq in ht.
rewrite eos_eq.
rewrite eos_eos_reversed by omega. (* nice! *)
rewrite ht.
reflexivity.
Qed.
Lemma fv_eos_eq:
forall x n t,
x < n ->
fv n (eos x t) <->
fv (n - 1) t.
Proof.
unfold fv. intros x n t ?.
rewrite eos_eq.
rewrite eos_eq.
rewrite eos_eos_reversed by omega. (* nice! *)
split; intros h.
{ eauto using eos_injective. }
{ rewrite h. reflexivity. }
Qed.
(* -------------------------------------------------------------------------- *)
(* A substitution [sigma] is regular if and only if, for some [j], for
sufficiently large [x], [sigma x] is [x + j]. *)
Definition regular (sigma : var -> A) :=
exists i j,
ren (+i) >> sigma = ren (+j).
Lemma regular_ids:
regular ids.
Proof.
exists 0. exists 0. autosubst.
Qed.
Lemma regular_plus:
forall i,
regular (ren (+i)).
Proof.
intros. exists 0. exists i. autosubst.
Qed.
Lemma regular_upn:
forall n sigma,
regular sigma ->
regular (upn n sigma).
Proof.
intros ? ? (i&j&hsigma).
exists (n + i). eexists (n + j).
replace (ren (+(n + i))) with (ren (+i) >> ren (+n)) by autosubst.
rewrite <- scompA.
rewrite up_liftn.
rewrite scompA.
rewrite hsigma.
autosubst.
Qed.
(* -------------------------------------------------------------------------- *)
(* If the free variables of the term [t] are below [k], then [t] is unaffected
by a substitution of the form [upn k sigma]. *)
(* Unfortunately, in this file, where the definition of type [A] is unknown, I
am unable to establish this result for arbitrary substitutions [sigma]. I
am able to establish it for *regular* substitutions, where The proof is somewhat interesting, so it is given here, even
though, once the definition of the type [A] is known, a more direct proof,
without a regularity hypothesis, can usually be given. *)
(* An intermediate result states that, since [upn k (ren (+1))] does not
affect [t], then (by iteration) neither does [upn k (ren (+j))]. *)
Lemma fv_unaffected_lift:
forall j t k,
fv k t ->
t.[upn k (ren (+j))] = t.
Proof.
induction j as [| j ]; intros t k ht.
{ asimpl. rewrite up_id_n. autosubst. }
{ replace (ren (+S j)) with (ren (+1) >> ren (+j)) by autosubst.
rewrite <- up_comp_n.
replace (t.[upn k (ren (+1)) >> upn k (ren (+j))])
with (t.[upn k (ren (+1))].[upn k (ren (+j))]) by autosubst.
rewrite ht.
rewrite IHj by eauto.
eauto. }
Qed.
(* There follows that a substitution of the form [upn k sigma], where [sigma]
is regular, does not affect [t]. The proof is slightly subtle but very
short. The previous lemma is used twice. *)
Lemma fv_unaffected_regular:
forall k t sigma,
fv k t ->
regular sigma ->
t.[upn k sigma] = t.
Proof.
intros k t sigma ? (i&j&hsigma).
rewrite <- (fv_unaffected_lift i t k) at 1 by eauto.
asimpl. rewrite up_comp_n.
rewrite hsigma.
rewrite fv_unaffected_lift by eauto.
reflexivity.
Qed.
(* A corollary. *)
Lemma closed_unaffected_regular:
forall t sigma,
closed t ->
regular sigma ->
t.[sigma] = t.
Proof.
unfold closed. intros.
rewrite <- (upn0 sigma).
eauto using fv_unaffected_regular.
Qed.
(*One might also wish to prove a result along the following lines:
Goal
forall t k sigma1 sigma2,
fv k t ->
(forall x, x < k -> sigma1 x = sigma2 x) ->
t.[sigma1] = t.[sigma2].
I have not yet investigated how this could be proved. *)
(* -------------------------------------------------------------------------- *)
(* If some term [t] has free variables under [j], then it also has free
variables under [k], where [j <= k]. *)
Lemma fv_monotonic:
forall j k t,
fv j t ->
j <= k ->
fv k t.
Proof.
intros. unfold fv.
replace k with (j + (k - j)) by omega.
rewrite <- upn_upn.
eauto using fv_unaffected_regular, regular_upn, regular_plus.
Qed.
(* -------------------------------------------------------------------------- *)
(* These little lemmas may be occasionally useful. *)
Lemma use_fv_length_cons:
forall A (x : A) (xs : list A) n t,
(forall x, fv (length (x :: xs)) t) ->
n = length xs ->
fv (n + 1) t.
Proof.
intros. subst.
replace (length xs + 1) with (length (x :: xs)) by (simpl; omega).
eauto.
Qed.
Lemma prove_fv_length_cons:
forall A (x : A) (xs : list A) n t,
n = length xs ->
fv (n + 1) t ->
fv (length (x :: xs)) t.
Proof.
intros. subst.
replace (length (x :: xs)) with (length xs + 1) by (simpl; omega).
eauto.
Qed.
(* -------------------------------------------------------------------------- *)
(* [closed t] is equivalent to [t.[ren (+1)] = t]. *)
Lemma closed_eq:
forall t,
closed t <-> t.[ren (+1)] = t.
Proof.
unfold closed, fv. asimpl. tauto.
Qed.
(* -------------------------------------------------------------------------- *)
(* A variable is not closed. *)
Lemma closed_ids:
forall x,
~ closed (ids x).
Proof.
unfold closed, fv. intros. asimpl. intro.
eauto using ids_inj_False.
Qed.
End FreeVars.
(* -------------------------------------------------------------------------- *)
(* The tactic [fv] is intended to use a number of lemmas as rewriting rules.
The hint database [fv] can be extended with language-specific lemmas. *)
Hint Rewrite @fv_ids_eq @fv_lift @fv_eos_eq : fv.
Ltac fv :=
autorewrite with fv in *;
eauto with typeclass_instances.
(* -------------------------------------------------------------------------- *)
(* A hint database to prove goals of the form [~ (closed _)] or [closed _]. *)
Hint Resolve closed_ids : closed.
(* -------------------------------------------------------------------------- *)
Hint Resolve regular_ids regular_plus regular_upn : regular.