From Undecidability.L Require Export Util.L_facts.
From Undecidability.L.Tactics Require Import Lproc Lbeta Lrewrite Reflection mixedTactics.
Local Ltac wLsimpl' _n := intros;try reflexivity';try standardizeGoal _n ; try reflexivity'.
Local Ltac wLsimpl := wLsimpl' 100.
(* Lsimpl' uses correctnes lemmas and wLsimpl*)
Ltac Lsimpl' :=
match goal with
| |- eval ?s _ => assert (lambda s) by Lproc;split;[ (exact (starR _ _);fail 1)|Lproc]
| |- eval ?s _ => (progress (eapply eval_helper;[Lsimpl';reflexivity|]))
| _ => try Lrewrite;try wLsimpl' 100
end.
Ltac Lreduce :=
repeat progress (try Lrewrite;try Lbeta).
(*Lsimpl that uses correctnes lemmas*)
Ltac Lsimpl :=intros(*;repeat foldLocalInts*);
once lazymatch goal with
| |- _ >(<= _ ) _ => Lreduce;try Lreflexivity
| |- _ ⇓(_ ) _ => repeat progress Lbeta;try Lreflexivity
| |- _ ⇓(<= _ ) _ => Lreduce;try Lreflexivity
| |- _ >(_) _ => repeat progress Lbeta;try Lreflexivity
| |- _ >* _ => Lreduce;try Lreflexivity (* test *)
| |- eval _ _ => Lreduce;try Lreflexivity (* test *)
(*| |- _ >* _  => repeat Lsimpl';try reflexivity' | |- eval _ _  => repeat Lsimpl';try reflexivity'*)
| |- _ == _  => repeat Lsimpl';try reflexivity'
end.
Ltac LsimplHypo := standardizeHypo 100.
Tactic Notation "closedRewrite" :=
match goal with
| [ |- context[subst ?s _ _] ] =>
let cl := fresh "cl" in assert (cl:closed s);[Lproc|rewrite !cl;clear cl]
end.
Tactic Notation "closedRewrite" "in" hyp(h):=
match type of h with
| context[subst ?s _ _] =>
let cl := fresh "cl" in assert (cl:closed s);[Lproc|rewrite !cl in h;clear cl]
end.
Tactic Notation "redStep" "at" integer(pos) := rewrite step_Lproc at pos;[simpl;try closedRewrite|Lproc].
Tactic Notation "redStep" "in" hyp(h) "at" integer(pos) := rewrite step_Lproc in h at pos;[simpl in h;try closedRewrite in h|Lproc].
(* Tactic Notation "redStep" := redStep at 1. *)
Tactic Notation "redStep" "in" hyp(h) := redStep in h at 1.
(* register needed lemmas:*)
Hint Resolve rho_correct : Lrewrite.
Hint Resolve rho_lambda rho_cls : LProc.
Tactic Notation "recStep" constr(P) "at" integer(i):=
match eval lazy [P] in P with
| rho ?rP => unfold P;rewrite rho_correct at i;[|Lproc..];fold P;try unfold rP
end.
Tactic Notation "recStep" constr(P) :=
intros;recStep P at 1.
(* Lemma rClosed_closed s: recProc s -> proc s. intros [? [? ?]]. subst. split; auto with LProc. Qed.  Hint Resolve rClosed_closed : LProc cbv. *)
fLproc.
Qed.
fLproc.
Qed.
fLproc.
Qed.

Lemma Omega_closed : closed Omega.