mpri-funcprog-project/coq/Autosubst_Env.v

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2017-10-05 17:57:33 +02:00
Require Import List.
Require Import MyTactics. (* TEMPORARY *)
Require Import Autosubst.Autosubst.
Require Import Autosubst_EOS. (* [eos_var] *)
(* Environments are sometimes represented as finite lists. This file
provides a few notions that helps deal with this representation. *)
Section Env.
Context {A} `{Ids A, Rename A, Subst A, SubstLemmas A}.
(* -------------------------------------------------------------------------- *)
(* The function [env2subst default], where [default] is a default value,
converts an environment (a finite list) to a substitution (a total
function). *)
Definition env2subst (default : A) (e : list A) (x : var) : A :=
nth x e default.
(* -------------------------------------------------------------------------- *)
(* [env_ren_comp e xi e'] means (roughly) that the environment [e] is equal to
the composition of the renaming [xi] and the environment [e'], that is,
[e = xi >>> e']. We also explicitly require the environments [e] and [e']
to have matching lengths, up to [xi], as this is *not* a consequence of
the other premise. *)
Inductive env_ren_comp : list A -> (var -> var) -> list A -> Prop :=
| EnvRenComp:
forall e xi e',
(forall x, x < length e -> xi x < length e') ->
(forall x default, nth x e default = nth (xi x) e' default) ->
env_ren_comp e xi e'.
(* A reformulation of the second premise in the above definition. *)
Lemma env_ren_comp_eq:
forall e xi e',
(forall default, env2subst default e = xi >>> env2subst default e') <->
(forall x default, nth x e default = nth (xi x) e' default).
Proof.
unfold env2subst. split; intros h; intros.
{ change (nth x e default) with ((fun x => nth x e default) x).
rewrite h. reflexivity. }
{ f_ext; intro x. eauto. }
Qed.
(* -------------------------------------------------------------------------- *)
(* Initialization: [e = id >>> e]. *)
Lemma env_ren_comp_id:
forall e,
env_ren_comp e (fun x => x) e.
Proof.
econstructor; eauto.
Qed.
(* -------------------------------------------------------------------------- *)
(* The relation [e = xi >>> e'] can be taken under a binder, as follows. *)
Lemma env_ren_comp_up:
forall e xi e' v,
env_ren_comp e xi e' ->
env_ren_comp (v :: e) (upren xi) (v :: e').
Proof.
inversion 1; intros; subst; econstructor;
intros [|x]; intros; simpl in *; eauto with omega.
Qed.
(* -------------------------------------------------------------------------- *)
(* One element can be prepended to [e'], provided [xi] is adjusted. *)
Lemma env_ren_comp_prepend:
forall e xi e' v,
env_ren_comp e xi e' ->
env_ren_comp e (xi >>> (+1)) (v :: e').
Proof.
inversion 1; intros; subst.
econstructor; intros; simpl; eauto with omega.
Qed.
(* -------------------------------------------------------------------------- *)
(* A consequence of [env_ren_comp_id] and [env_ren_comp_prepend]. The renaming
(+1) will eat away the first entry in [v :: e]. *)
Lemma env_ren_comp_plus1:
forall e v,
env_ren_comp e (+1) (v :: e).
Proof.
econstructor; intros; simpl; eauto with omega.
Qed.
(* -------------------------------------------------------------------------- *)
(* More generally, the renaming [eos_var x], which means that [x] goes out of
scope, will eat away the entry at index [x] in [e1 ++ v :: e2]. *)
Lemma env_ren_comp_eos_var:
forall x e1 v e2,
x = length e1 ->
env_ren_comp (e1 ++ e2) (eos_var x) (e1 ++ v :: e2).
Proof.
rewrite eos_var_eq_lift_var. unfold lift_var.
econstructor; intros y; dblib_by_cases.
{ rewrite app_length in *. simpl. omega. }
{ rewrite app_length in *. simpl. omega. }
{ do 2 (rewrite app_nth2 by omega).
replace (1 + y - length e1) with (1 + (y - length e1)) by omega.
reflexivity. }
{ do 2 (rewrite app_nth1 by omega).
reflexivity. }
Qed.
(* -------------------------------------------------------------------------- *)
End Env.
Hint Resolve
env_ren_comp_id
env_ren_comp_up
env_ren_comp_prepend
env_ren_comp_plus1
env_ren_comp_eos_var
: env_ren_comp.