From semantics.ts.systemf_mu Require Export lang.
Set Default Proof Using "Type".


(** Coercions to make programs easier to type. *)
Coercion of_val : val >-> expr.
Coercion LitInt : Z >-> base_lit.
Coercion LitBool : bool >-> base_lit.
Coercion App : expr >-> Funclass.
Coercion Var : string >-> expr.

(** Define some derived forms. *)
Notation Let x e1 e2 := (App (Lam x e2) e1) (only parsing).
Notation Seq e1 e2 := (Let BAnon e1 e2) (only parsing).
Notation Match e0 x1 e1 x2 e2 := (Case e0 (Lam x1 e1) (Lam x2 e2)) (only parsing).

(* No scope for the values, does not conflict and scope is often not inferred
properly. *)
Notation "# l" := (LitV l%Z%V%stdpp) (at level 8, format "# l").
Notation "# l" := (Lit l%Z%E%stdpp) (at level 8, format "# l") : expr_scope.

(** Syntax inspired by Coq/Ocaml. Constructions with higher precedence come
    first. *)
Notation "( e1 , e2 , .. , en )" := (Pair .. (Pair e1 e2) .. en) : expr_scope.
Notation "( e1 , e2 , .. , en )" := (PairV .. (PairV e1 e2) .. en) : val_scope.

Notation "'match:' e0 'with' 'InjL' x1 => e1 | 'InjR' x2 => e2 'end'" :=
  (Match e0 x1%binder e1 x2%binder e2)
  (e0, x1, e1, x2, e2 at level 200,
   format "'[hv' 'match:'  e0  'with'  '/  ' '[' 'InjL'  x1  =>  '/  ' e1 ']'  '/' '[' |  'InjR'  x2  =>  '/  ' e2 ']'  '/' 'end' ']'") : expr_scope.
Notation "'match:' e0 'with' 'InjR' x1 => e1 | 'InjL' x2 => e2 'end'" :=
  (Match e0 x2%binder e2 x1%binder e1)
  (e0, x1, e1, x2, e2 at level 200, only parsing) : expr_scope.

Notation "()" := LitUnit : val_scope.

Notation "e1 + e2" := (BinOp PlusOp e1%E e2%E) : expr_scope.
Notation "e1 - e2" := (BinOp MinusOp e1%E e2%E) : expr_scope.
Notation "e1 * e2" := (BinOp MultOp e1%E e2%E) : expr_scope.
Notation "e1 ≤ e2" := (BinOp LeOp e1%E e2%E) : expr_scope.
Notation "e1 = e2" := (BinOp EqOp e1%E e2%E) : expr_scope.
Notation "e1 < e2" := (BinOp LtOp e1%E e2%E) : expr_scope.

(*Notation "~ e" := (UnOp NegOp e%E) (at level 75, right associativity) : expr_scope.*)

Notation "'if:' e1 'then' e2 'else' e3" := (If e1%E e2%E e3%E)
  (at level 200, e1, e2, e3 at level 200) : expr_scope.

Notation "λ: x , e" := (Lam x%binder e%E)
  (at level 200, x at level 1, e at level 200,
   format "'[' 'λ:'  x ,  '/  ' e ']'") : expr_scope.
Notation "λ: x y .. z , e" := (Lam x%binder (Lam y%binder .. (Lam z%binder e%E) ..))
  (at level 200, x, y, z at level 1, e at level 200,
   format "'[' 'λ:'  x  y  ..  z ,  '/  ' e ']'") : expr_scope.

Notation "λ: x , e" := (LamV x%binder e%E)
  (at level 200, x at level 1, e at level 200,
   format "'[' 'λ:'  x ,  '/  ' e ']'") : val_scope.
Notation "λ: x y .. z , e" := (LamV x%binder (Lam y%binder .. (Lam z%binder e%E) .. ))
  (at level 200, x, y, z at level 1, e at level 200,
   format "'[' 'λ:'  x  y  ..  z ,  '/  ' e ']'") : val_scope.

Notation "'let:' x := e1 'in' e2" := (Lam x%binder e2%E e1%E)
  (at level 200, x at level 1, e1, e2 at level 200,
   format "'[' 'let:'  x  :=  '[' e1 ']'  'in'  '/' e2 ']'") : expr_scope.
Notation "e1 ;; e2" := (Lam BAnon e2%E e1%E)
  (at level 100, e2 at level 200,
   format "'[' '[hv' '[' e1 ']' ;;  ']' '/' e2 ']'") : expr_scope.


Notation "'Λ' , e" := (TLam e%E)
  (at level 200, e at level 200,
    format "'[' 'Λ' , '/  ' e ']'") : expr_scope.
Notation "'Λ' , e" := (TLamV e%E)
  (at level 200, e at level 200,
    format "'[' 'Λ' , '/  ' e ']'") : val_scope.

(* the [e] always needs to be paranthesized, due to the level
   (chosen to make this cooperate with the [App] coercion) *)
Notation "e '<>'" := (TApp e%E)
  (at level 10) : expr_scope.
(*Check ((Λ, #4) <>)%E.*)
(*Check (((λ: "x", "x") #5) <>)%E.*)

Notation "'pack' e" := (Pack e%E)
  (at level 200, e at level 200) : expr_scope.
Notation "'pack' v" := (PackV v%V)
  (at level 200, v at level 200) : val_scope.
Notation "'unpack'  e1  'as'  x  'in'  e2" := (Unpack x%binder e1%E e2%E)
  (at level 200, e1, e2 at level 200, x at level 1) : expr_scope.

Notation "'roll'  e" := (Roll e%E)
  (at level 200, e at level 200) : expr_scope.
Notation "'roll'  v" := (RollV v%E)
  (at level 200, v at level 200) : val_scope.
Notation "'unroll'  e" := (Unroll e%E)
  (at level 200, e at level 200) : expr_scope.