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