(*********************************************************************************************************************************)
(* HaskStrongToWeak: convert HaskStrong to HaskWeak                                                                              *)
(*********************************************************************************************************************************)

Generalizable All Variables.
Require Import Preamble.
Require Import General.
Require Import NaturalDeduction.
Require Import Coq.Strings.String.
Require Import Coq.Lists.List.
Require Import Coq.Init.Specif.
Require Import HaskKinds.
Require Import HaskCoreLiterals.
Require Import HaskWeakTypes.
Require Import HaskWeakVars.
Require Import HaskWeak.
Require Import HaskStrongTypes.
Require Import HaskStrong.

CoInductive Fresh A T :=
{ next  : forall a:A, (T a * Fresh A T)
; split : Fresh A T * Fresh A T
}.

Fixpoint rawHaskTypeToWeakType (f:Fresh Kind (fun _ => WeakTypeVar)){κ}(rht:RawHaskType (fun _ => WeakTypeVar) κ)
 {struct rht} : WeakType :=
match rht with
  | TVar  _  v                 => WTyVarTy v
  | TCon      tc              => WTyCon tc
  | TCoerc _ t1 t2 t3          => WCoFunTy (rawHaskTypeToWeakType f t1) (rawHaskTypeToWeakType f t2) (rawHaskTypeToWeakType f t3)
  | TArrow                    => WFunTyCon
  | TApp  _ _  t1 t2             => WAppTy (rawHaskTypeToWeakType f t1) (rawHaskTypeToWeakType f t2)
  | TAll    k rht'            => let (tv,f') := next _ _ f k in WForAllTy tv (rawHaskTypeToWeakType f' (rht' tv))
  | TCode   t1 t2             => 
    match (rawHaskTypeToWeakType f t1) with
      | WTyVarTy ec => WCodeTy ec (rawHaskTypeToWeakType f t2)
      | impossible  => impossible  (* TODO: find a way to convince Coq this can't happen *)
    end
  | TyFunApp    tfc tls         => WTyFunApp tfc (rawHaskTypeListToWeakType f tls)
end
with rawHaskTypeListToWeakType (f:Fresh Kind (fun _ => WeakTypeVar)){κ}(rht:RawHaskTypeList κ) :=
match rht with
  | TyFunApp_nil   => nil
  | TyFunApp_cons  κ kl rht' rhtl' => (rawHaskTypeToWeakType f rht')::(rawHaskTypeListToWeakType f rhtl')
end.

Definition typeToWeakType (f:Fresh Kind (fun _ => WeakTypeVar))
  {Γ}{κ}(τ:HaskType Γ κ)(φ:InstantiatedTypeEnv (fun _ => WeakTypeVar) Γ) : WeakType :=
  rawHaskTypeToWeakType f (τ _ φ).

Variable unsafeCoerce           : WeakCoercion.    Extract Inlined Constant unsafeCoerce => "Coercion.unsafeCoerce".

Definition strongCoercionToWeakCoercion {Γ}{Δ}{ck}(τ:HaskCoercion Γ Δ ck)(φ:InstantiatedTypeEnv (fun _ => WeakTypeVar) Γ)
  : WeakCoercion.
  unfold HaskCoercion in τ.
  admit.
  Defined.

(* This can be used to turn HaskStrong's with arbitrary expression variables into HaskStrong's which use WeakExprVar's
 * for their variables; those can subsequently be passed to the following function (strongExprToWeakExpr) *)
(*
Definition reindexStrongExpr {VV}{HH}{eqVV:EqDecidable VV}{eqHH:EqDecidable HH}
  {Γ}{Δ}{ξ}{lev}(exp:@Expr VV eqVV Γ Δ ξ lev) : { ξ' : HH -> LeveledHaskType Γ & @Expr HH eqHH Γ Δ ξ' lev }.
  Defined.
  *)

Definition updateITE {Γ:TypeEnv}{TV:Kind->Type}{κ}(tv:TV κ)(ite:InstantiatedTypeEnv TV Γ) : InstantiatedTypeEnv TV (κ::Γ)
  := tv::::ite.

Fixpoint strongExprToWeakExpr {Γ}{Δ}{ξ}{lev}
  (ftv:Fresh Kind                (fun _ => WeakTypeVar))
  (fcv:Fresh (WeakType*WeakType) (fun _ => WeakCoerVar))
  (exp:@Expr WeakExprVar _ Γ Δ ξ lev)
  : InstantiatedTypeEnv (fun _ => WeakTypeVar) Γ -> WeakExpr :=
match exp as E in Expr G D X L return InstantiatedTypeEnv (fun _ => WeakTypeVar) G -> WeakExpr with
| EVar  _ _ _ ev                => fun ite => WEVar ev
| ELit  _ _ _ lit _             => fun ite => WELit lit
| EApp  Γ' _ _ _ _ _ e1 e2      => fun ite => WEApp (strongExprToWeakExpr ftv fcv e1 ite) (strongExprToWeakExpr ftv fcv e2 ite)
| ELam  Γ'   _ _ _ _ _ cv e     => fun ite => WELam cv (strongExprToWeakExpr ftv fcv e ite)
| ELet  Γ' _ _ _ _ _ ev e1 e2   => fun ite => WELet ev (strongExprToWeakExpr ftv fcv e1 ite) (strongExprToWeakExpr ftv fcv e2 ite)
| EEsc  Γ' _ _ ec t _ e         => fun ite => WEEsc  (ec _ ite) (strongExprToWeakExpr ftv fcv e ite) (typeToWeakType ftv t ite)
| EBrak Γ' _ _ ec t _ e         => fun ite => WEBrak (ec _ ite) (strongExprToWeakExpr ftv fcv e ite) (typeToWeakType ftv t ite)
| ECast Γ Δ ξ t1 t2 γ l e       => fun ite => WECast (strongExprToWeakExpr ftv fcv e ite) (strongCoercionToWeakCoercion γ ite)
| ENote _ _ _ _ n e             => fun ite => WENote n (strongExprToWeakExpr ftv fcv e ite)
| ETyApp  Γ Δ κ σ τ ξ l       e => fun ite => WETyApp (strongExprToWeakExpr ftv fcv e ite) (typeToWeakType ftv τ ite)
| ELetRec _ _ _ _ _ vars elrb e =>fun ite=>WELetRec (exprLetRec2WeakExprLetRec ftv fcv elrb ite)(strongExprToWeakExpr ftv fcv e ite)
| ECoApp Γ Δ κ σ₁ σ₂ γ σ ξ l e  => fun ite => WECoApp (strongExprToWeakExpr ftv fcv e ite) (strongCoercionToWeakCoercion γ ite)
| ECase Γ Δ ξ l tc tbranches sac atypes e alts =>
  fun ite => WECase
    (strongExprToWeakExpr ftv fcv e ite)
    (@typeToWeakType ftv Γ _ tbranches ite)
    tc
    (ilist_to_list (@ilmap _ _ (fun _ => WeakType) _ (fun _ t => typeToWeakType ftv t ite) atypes))
    ((fix caseBranches
      (tree:Tree ??{ scb : StrongCaseBranchWithVVs _ _ tc sac atypes
                            & Expr (sac_Γ scb Γ)
                                   (sac_Δ scb Γ atypes (weakCK'' Δ))
                                   (update_ξ (weakLT'○ξ) (vec2list (vec_map
                                     (fun x => ((fst x),(snd x @@ weakL' l))) (scbwv_varstypes scb))))
                                   (weakLT' (tbranches@@l)) })
      : Tree ??(AltCon * list WeakTypeVar * list WeakCoerVar * list WeakExprVar * WeakExpr) :=
      match tree with
        | T_Leaf None     => []
        | T_Leaf (Some x) => let (scb,e) := x in
          (*[(sac_altcon scb,
            nil, (* FIXME *)
            nil, (* FIXME *)
            (*map (fun t => typeToWeakType ftv t ite') (vec2list (sac_types scb))*)nil, (* FIXME *)
            strongExprToWeakExpr ftv fcv e (weakITE' ite))]*) []
        | T_Branch b1 b2  => (caseBranches b1),,(caseBranches b2)
      end
    ) alts)
| ETyLam _ _ _ k _ _ e          => fun ite =>
  let (tv,ftv') := next _ _ ftv k in WETyLam tv (strongExprToWeakExpr ftv' fcv e (updateITE tv ite))
| ECoLam Γ Δ κ σ σ₁ σ₂ ξ l e => fun ite =>
  let t1' := typeToWeakType ftv σ₁ ite in 
  let t2' := typeToWeakType ftv σ₂ ite in
  let (cv,fcv') := next _ _ fcv (t1',t2') in WECoLam cv (strongExprToWeakExpr ftv fcv' e ite)
end
with exprLetRec2WeakExprLetRec {Γ}{Δ}{ξ}{lev}{vars}
  (ftv:Fresh Kind                (fun _ => WeakTypeVar))
  (fcv:Fresh (WeakType*WeakType) (fun _ => WeakCoerVar))
  (elrb:@ELetRecBindings WeakExprVar _ Γ Δ ξ lev vars)
  : InstantiatedTypeEnv (fun _ => WeakTypeVar) Γ -> Tree ??(WeakExprVar * WeakExpr) :=
match elrb as E in ELetRecBindings G D X L V
   return InstantiatedTypeEnv (fun _ => WeakTypeVar) G -> Tree ??(WeakExprVar * WeakExpr) with
| ELR_nil    _ _ _ _           => fun ite => []
| ELR_leaf   _ _ _ _ cv v e    => fun ite => [(v,(strongExprToWeakExpr ftv fcv e ite))]
| ELR_branch _ _ _ _ _ _ b1 b2 => fun ite => (exprLetRec2WeakExprLetRec ftv fcv b1 ite),, (exprLetRec2WeakExprLetRec ftv fcv b2 ite)
end.