From Undecidability.L Require Export Util.L_facts.
(* **** Closure calculus *)
Inductive Comp : Type :=
| CompVar (x:nat)
| CompApp (s : Comp) (t : Comp) : Comp
| CompClos (s : term) (A : list Comp) : Comp.
Coercion CompApp : Comp >-> Funclass.
Inductive lamComp : Comp -> Prop := lambdaComp s A: lamComp (CompClos (lam s) A).
Inductive validComp : Comp -> Prop :=
| validCompApp s t : validComp s -> validComp t -> validComp (s t)
| validCompClos (s : term) (A : list Comp) :
(forall a, a el A -> validComp a) -> (forall a, a el A -> lamComp a) -> bound (length A) s -> validComp (CompClos s A).
Hint Constructors Comp lamComp validComp : core.
Definition validEnv A := forall a, a el A -> validComp a (*/\ lamComp a)*).
Definition validEnv' A := forall a, a el A -> closed a.
Hint Unfold validEnv validEnv' : core.
Ltac inv_validComp :=
match goal with
| H : validComp (CompApp _ _) |- _ => inv H
| H : validComp (CompClos _ _)   |- _ => inv H
end.
Definition Comp_ind_deep'
(P : Comp -> Prop)
(Pl : list Comp -> Prop)
(IHVar : forall x : nat, P (CompVar x))
(IHApp : forall s : Comp, P s -> forall t : Comp, P t -> P (s t))
(IHClos : forall (s : term) (A : list Comp),
Pl A -> P (CompClos s A))
(IHNil : Pl nil)
(IHCons : forall (a:Comp) (A : list Comp),
P a -> Pl A -> Pl (a::A))
(x:Comp) : P x :=
(fix f c : P c:=
match c with
| CompVar x => IHVar x
| CompApp s t => IHApp s (f s) t (f t)
| CompClos s A => IHClos s A
((fix g A : Pl A :=
match A with
[] => IHNil
| a::A => IHCons a A (f a) (g A)
end) A)
end) x
.
Definition Comp_ind_deep
(P : Comp -> Prop)
(IHVar : forall x : nat, P (CompVar x))
(IHApp : forall s : Comp, P s -> forall t : Comp, P t -> P (s t))
(IHClos : forall (s : term) (A : list Comp),
(forall a, a el A -> P a) -> P (CompClos s A)) : forall x, P x.
(* Lemma subst_comm s x1 u1 x2 u2 : closed u1 -> closed u2 -> x1 <> x2 -> subst (subst s x1 u1) x2 u2 = subst (subst s x2 u2) x1 u1. Proof with try (congruence||auto). intros cl1 cl2 neq. revert x1 x2 neq;induction s;simpl;intros. -decide (n=x1); decide (n=x2); try rewrite cl1;try rewrite cl2;subst;simpl... +decide (x1=x1)... +decide (x2=x2)... +decide (n=x2);decide (n=x1)... -rewrite IHs1,IHs2... -rewrite IHs... Qed. *) (* Lemma subst_twice s x u1 u2 : closed u1 -> subst (subst s x u1) x u2 = subst s x u1. Proof with try (congruence||auto). intros cl. revert x;induction s;simpl;intros. -decide (n=x);subst. now rewrite cl. simpl. decide (n=x);subst;congruence. -rewrite IHs1,IHs2... -rewrite IHs... Qed.*)
(* Lemma subst_free a s k u y: closed a -> subst s k u = s -> subst (subst s y a) k u = subst s y a. (* Lemma bound_ge k s m: bound k s -> m >= k -> bound m s. *)
(* Lemma bound_subst' x s a y:  bound x s -> closed a -> bound x (subst s y a). *)
(* Fixpoint substList' (s:term) (x:nat) (A: list term): term := match A with | nil => s | a::A => substList' (subst s x a) (S x) A end.*)
Fixpoint substList (s:term) (x:nat) (A: list term): term :=
match s with
| var n => if Dec (x>n) then var n else nth (n-x) A (var n)
| app s t => app (substList s x A) (substList t x A)
| lam s => lam (substList s (S x) A)
end.
Fixpoint deClos (s:Comp) : term :=
match s with
| CompVar x => var x
| CompApp s t => app (deClos s) (deClos t)
| CompClos s A => substList s 0 (map deClos A)
end.
(* Reduction *)
Reserved Notation "s '>[(' l ')]' t" (at level 50, format "s  '>[(' l ')]'  t").
Declare Scope LClos.
Inductive CPow : nat -> Comp -> Comp -> Prop :=
| CPowRefl (s:Comp) : s >[(0)] s
| CPowTrans (s t u:Comp) i j : s >[(i)] t -> t >[(j)] u -> s >[(i+j)] u
| CPowAppL (s s' t :Comp) l: s >[(l)] s' -> (s t) >[(l)] (s' t)
| CPowAppR (s t t':Comp) l: t >[(l)] t' -> (s t) >[(l)] (s t')
| CPowApp  (s t:term) (A:list Comp) :
CompClos (app s t) A >[(0)] (CompClos s A) (CompClos t A)
| CPowVar (x:nat) (A:list Comp):
CompClos (var x) A >[(0)] nth x A (CompVar x)
| CPowVal (s t:term) (A B:list Comp):
lambda t -> (CompClos (lam s) A) (CompClos t B) >[(1)] (CompClos s ((CompClos t B)::A))
where "s  '>[(' l ')]'  t" := (CPow l s t)  : LClos.
Open Scope LClos.
Ltac inv_CompStep :=
match goal with
| H : (CompApp _ _) >(_) CompClos _ _ |- _ => inv H
| H : (CompClos _ _) >(_) CompApp _ _ |- _ => inv H
end.
Hint Constructors CPow : core.
Instance CPow'_App_properR n:
Proper (eq ==> (CPow n) ==> (CPow n)) CompApp.
(* Definition CStar s t:= exists k , CPow k s t .  Notation "s '>[]*' t" := (CStar s t) (at level 50)  : L.los.  Instance rStar'_PreOrder : PreOrder CStar.  Lemma rStar'_trans_l s s' t : s >[]* s' ->  s t >[]* s' t.  Lemma rStar'_trans_r (s t t' : Comp): t >[]* t' -> s t >[]* s t'.  Instance rStar'_App_proper : Proper ((star CStep) ==> (star CStep) ==> (star CStep)) CompApp.  Instance CStep_star_subrelation : subrelation CStep (star CStep).  *)

(* Properties of step-indexed version *)
(* Notation "x '>[]^' n y" := (ARS.pow CStep n x y) (at level 50) : L.cope.  Lemma CStep_Lam  n: forall (s t u:Comp), lamComp u -> (ARS.pow CStep n (s t) u) -> exists m1 m2 (s' t':Comp),(m1 < n /\ ARS.pow CStep m1 s s' /\ lamComp s') /\ (m2 < n /\ ARS.pow CStep m2 t t' /\ lamComp t'). Proof with repeat intuition;try now reflexivity. induction n;intros ? ? ? lu R. -inv R. inv lu. -destruct R as [u' [R R']]. inv R. +apply IHn in R'... decompose [ex and] R'. exists (S x), x0, x1, x2... change (S x) with (1+x). apply pow_add;simpl. exists s';intuition. eexists;simpl... +apply IHn in R'... decompose [ex and] R'. exists x, (S x0), x1, x2... change (S x0) with (1+x0). apply pow_add;simpl. exists t';intuition. eexists;simpl... +inv H2. eexists 0,0,_,_... Qed.  Lemma CStep_Lam' (s t u:Comp) : lamComp u -> (s t) >[]* u -> exists (s' t':Comp),( s >[]* s' /\ lamComp s') /\ (t >[]* t' /\ lamComp t'). Proof with repeat intuition;try now reflexivity. intros lu R. apply star_pow in R. destruct R as [n R]. revert s t u lu R. induction n;intros. -inv R. inv lu. -destruct R as [u' [R R']]. inv R. +apply IHn in R'... decompose [ex and] R'. exists x, x0... eauto using star. +apply IHn in R'... decompose [ex and] R'. exists x, x0... eauto using star. +inv H2. eexists _,_... Qed.    *)
Lemma substList_bound x s A: bound x s -> substList s x A = s.