Library Io.C

Require Import Effect.

The description of a computation.
Inductive t (E : Effect.t) : Type → Type :=
| Ret : ∀ (A : Type) (x : A), t E A
| Call : ∀ (command : Effect.command E), t E (Effect.answer E command)
| Let : ∀ (A B : Type), t E A → (A → t E B) → t E B
| Choose : ∀ (A : Type), t E A → t E A → t E A
| Join : ∀ (A B : Type), t E A → t E B → t E (A × B).

The implicit arguments so that the `match` command works both with Coq 8.4 and Coq 8.5.
Arguments Ret {E} _ _.
Arguments Call {E} _.
Arguments Let {E} _ _ _ _.
Arguments Choose {E} _ _ _.
Arguments Join {E} _ _ _ _.

A nicer notation for `Ret`.
Definition ret {E : Effect.t} {A : Type} (x : A) : t E A :=
  Ret A x.

A nicer notation for `Call`.
Definition call (E : Effect.t) (command : Effect.command E)
  : t E (Effect.answer E command) :=
  Call (E := E) command.

A nicer notation for `Choose`.
Definition choose {E : Effect.t} {A : Type} (x1 x2 : t E A) : t E A :=
  Choose A x1 x2.

A nicer notation for `Join`.
Definition join {E : Effect.t} {A B : Type} (x : t E A) (y : t E B)
  : t E (A × B) :=
  Join A B x y.

Some optional notations.
Module Notations.
A nicer notation for `Let`.
  Notation "´let!´ x ´:=´ X ´in´ Y" :=
    (Let _ _ X (fun x ⇒ Y))
    (at level 200, x ident, X at level 100, Y at level 200).

Let with a typed answer.
  Notation "´let!´ x : A ´:=´ X ´in´ Y" :=
    (Let _ _ X (fun (x : A) ⇒ Y))
    (at level 200, x ident, X at level 100, A at level 200, Y at level 200).

Let ignoring the unit answer.
  Notation "´do!´ X ´in´ Y" :=
    (Let _ _ X (fun (_ : unit) ⇒ Y))
    (at level 200, X at level 100, Y at level 200).
End Notations.

Module I.
The description of an infinite computation.
  CoInductive t (E : Effect.t) : Type → Type :=
  | Ret : ∀ (A : Type) (x : A), t E A
  | Call : ∀ (command : Effect.command E), t E (Effect.answer E command)
  | Let : ∀ (A B : Type), t E A → (A → t E B) → t E B
  | Choose : ∀ (A : Type), t E A → t E A → t E A
  | Join : ∀ (A B : Type), t E A → t E B → t E (A × B).

The implicit arguments so that the `match` command works both with Coq 8.4 and Coq 8.5.
  Arguments Ret {E} _ _.
  Arguments Call {E} _.
  Arguments Let {E} _ _ _ _.
  Arguments Choose {E} _ _ _.
  Arguments Join {E} _ _ _ _.

  Definition unfold {E A} (x : t E A) : t E A :=
    match x with
    | Ret _ v ⇒ Ret _ v
    | Call c ⇒ Call c
    | Let _ _ x f ⇒ Let _ _ x f
    | Choose _ x1 x2 ⇒ Choose _ x1 x2
    | Join _ _ x y ⇒ Join _ _ x y
    end.

  Definition unfold_eq {E A} (x : t E A) : x = unfold x.
    destruct x; reflexivity.
  Defined.

A lift from the finite computations.
  Fixpoint lift {E : Effect.t} {A : Type} (x : C.t E A) : t E A :=
    match x with
    | C.Ret _ v ⇒ Ret _ v
    | C.Call c ⇒ Call c
    | C.Let _ _ x f ⇒ Let _ _ (lift x) (fun v_x ⇒ lift (f v_x))
    | C.Choose _ x1 x2 ⇒ Choose _ (lift x1) (lift x2)
    | C.Join _ _ x y ⇒ Join _ _ (lift x) (lift y)
    end.

A nicer notation for `Ret`.
  Definition ret {E : Effect.t} {A : Type} (x : A) : t E A :=
    Ret A x.

A nicer notation for `Call`.
  Definition call (E : Effect.t) (command : Effect.command E)
    : t E (Effect.answer E command) :=
    Call (E := E) command.

A nicer notation for `Choose`.
  Definition choose {E : Effect.t} {A : Type} (x1 x2 : t E A) : t E A :=
    Choose A x1 x2.

A nicer notation for `Join`.
  Definition join {E : Effect.t} {A B : Type} (x : t E A) (y : t E B)
    : t E (A × B) :=
    Join A B x y.

Some optional notations.
  Module Notations.
A nicer notation for `Let`.
    Notation "´ilet!´ x ´:=´ X ´in´ Y" :=
      (Let _ _ X (fun x ⇒ Y))
      (at level 200, x ident, X at level 100, Y at level 200).

Let with a typed answer.
    Notation "´ilet!´ x : A ´:=´ X ´in´ Y" :=
      (Let _ _ X (fun (x : A) ⇒ Y))
      (at level 200, x ident, X at level 100, A at level 200, Y at level 200).

Let ignoring the unit answer.
    Notation "´ido!´ X ´in´ Y" :=
      (Let _ _ X (fun (_ : unit) ⇒ Y))
      (at level 200, X at level 100, Y at level 200).
  End Notations.
End I.