mpri-funcprog-project/coq/Autosubst_IsRen.v
2017-09-28 10:36:07 +02:00

87 lines
2.4 KiB
Coq

Require Import Coq.Logic.ClassicalUniqueChoice.
Require Import Autosubst.Autosubst.
Require Import AutosubstExtra.
Section Lemmas.
Context {A} `{Ids A, Rename A, Subst A, SubstLemmas A}.
(* The predicate [is_ren sigma] means that the substitution [sigma] is in fact
a renaming [ren xi]. *)
(* When stating a lemma that involves a renaming, it is preferable to use a
substitution [sigma], together with a hypothesis [is_ren sigma], rather
than request that [sigma] be of the form [ren xi]. This allows us to use
[obvious] to check that [sigma] is a renaming, whereas we would otherwise
have to manually rewrite [sigma] to the form [ren xi]. *)
Definition is_ren sigma :=
exists xi, sigma = ren xi.
(* One way of proving that [sigma] is a renaming is to prove that [sigma] maps
every variable [x] to a variable [y]. *)
Lemma prove_is_ren:
forall sigma,
(forall x y, ids x = ids y -> x = y) ->
(forall x, exists y, sigma x = ids y) ->
is_ren sigma.
Proof.
(* This proof uses the axiom of unique choice. If one is willing to use
the stronger axiom of choice, then one can remove the hypothesis that
[ids] is injective. *)
intros ? Hinj Hxy.
assert (Hxi: exists xi : var -> var, forall x, sigma x = ids (xi x)).
{ eapply unique_choice with (R := fun x y => sigma x = ids y).
intros x. destruct (Hxy x) as [ y Heqy ]. exists y.
split.
{ assumption. }
{ intros x' Heqx'. eapply Hinj. congruence. }
}
destruct Hxi as [ xi ? ].
exists xi.
f_ext; intros x. eauto.
Qed.
(* Applying [up] or [upn i] to a renaming produces a renaming. *)
Lemma up_is_ren:
forall sigma,
is_ren sigma ->
is_ren (up sigma).
Proof.
intros ? [ xi ? ]. subst. exists (upren xi).
erewrite <- up_ren by eauto with typeclass_instances. reflexivity.
Qed.
Lemma upn_is_ren:
forall sigma i,
is_ren sigma ->
is_ren (upn i sigma).
Proof.
intros ? ? [ xi ? ]. subst. exists (iterate upren i xi).
erewrite <- upn_ren by eauto with typeclass_instances. reflexivity.
Qed.
(* Composing two renamings yields a renaming. *)
Lemma comp_is_ren:
forall sigma1 sigma2,
is_ren sigma1 ->
is_ren sigma2 ->
is_ren (sigma1 >> sigma2).
Proof.
intros ? ? [ xi1 ? ] [ xi2 ? ]. subst. exists (xi1 >>> xi2). autosubst.
Qed.
Lemma is_ren_ids:
is_ren ids.
Proof.
exists id. autosubst.
Qed.
End Lemmas.
Hint Unfold is_ren : is_ren obvious.
Hint Resolve up_is_ren upn_is_ren comp_is_ren is_ren_ids : is_ren obvious.