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From Coq Require Import List String.
Require Import ExtLib.Structures.CoMonad.
Set Implicit Arguments.
Set Strict Implicit.
Inductive type : Type :=
| Self : type
| Inj : Type -> type.
Definition product := list type.
Definition variant := list product.
Section denote.
Variable M : Type.
Definition typeD (t : type) : Type :=
match t with
| Self => M
| Inj t => t
end.
Definition func (T : Type) (v : product) : Type :=
fold_right (fun x acc => typeD x -> acc) T v.
Definition data (v : product) : Type :=
fold_right (fun x acc => typeD x * acc)%type unit v.
Definition matchD (T : Type) (v : variant) : Type :=
fold_right (fun x acc => func T x -> acc)%type T v.
Definition dataD (v : variant) : Type :=
fold_right (fun x acc => data x + acc)%type Empty_set v.
Definition recD (T : Type) (c : Type -> Type) (v : variant) : Type :=
fold_right (fun x acc =>
fold_right (fun x acc =>
match x with
| Inj t => t
| Self => c T
end -> acc) (c T) x -> acc) (M -> T) v.
End denote.
Class Data (T : Type) : Type :=
{ repr : variant
; into : dataD T repr -> T
; outof : T -> forall A, matchD T A repr
; rec : forall c {_ : CoMonad c}, forall A, recD T A c repr
}.
Local Open Scope string_scope.
Global Instance Data_nat : Data nat :=
{ repr := nil :: (Self :: nil) :: nil
; outof := fun x _ z s =>
match x with
| 0 => z
| S n => s n
end
; into := fun d =>
match d with
| inl tt => 0
| inr (inl (n, tt)) => n
| inr (inr x) => match x with end
end
; rec := fun c _ A z s d =>
extract ((fix recur (d : nat) {struct d} : c A :=
match d with
| 0 => z
| S n => s (recur n)
end) d)
}.
Global Instance Data_list {A} : Data (list A) :=
{ repr := (nil) :: (Inj A :: Self :: nil) :: nil
; outof := fun x _ n c =>
match x with
| nil => n
| x :: xs => c x xs
end
; into := fun d =>
match d with
| inl tt => nil
| inr (inl (x, (xs, tt))) => x :: xs
| inr (inr x) => match x with end
end
; rec := fun c _ T n co d =>
extract ((fix recur (ds : list A) {struct ds} : c T :=
match ds with
| nil => n
| d :: ds => co d (recur ds)
end) d)
}.
(** Example of deriving Show from Data **)
Require Import ExtLib.Programming.Show.
Require Import ExtLib.Data.Monads.IdentityMonad.
Require Import ExtLib.Structures.Monads.
Global Instance Comoand_Id : CoMonad id :=
{ extract := fun _ x => x
; extend := fun _ _ x f => x f
}.
(*
Inductive AllResolve (C : Type -> Type) : list type -> Type :=
| AllResolve_nil : AllResolve C nil
| AllResolve_Self : forall ls, AllResolve C ls -> AllResolve C (Self :: ls)
| AllResolve_Inj : forall t ls, C t -> AllResolve C ls -> AllResolve C (Inj t :: ls).
Existing Class AllResolve.
*)
Definition ProductResolve (C : Type -> Type) (r : product) : Type :=
fold_right (fun t acc =>
match t with
| Inj t => C t * acc
| Self => acc
end)%type unit r.
Definition VariantResolve (C : Type -> Type) (r : variant) : Type :=
fold_right (fun p acc => ProductResolve C p * acc)%type unit r.
Existing Class VariantResolve.
Ltac all_resolve :=
simpl VariantResolve; simpl ProductResolve;
repeat match goal with
| |- unit => apply tt
| |- (unit * _)%type => constructor; [ apply tt | ]
| |- (_ * _)%type => constructor
| |- _ => solve [ eauto with typeclass_instances ]
end.
#[global]
Hint Extern 0 (ProductResolve _ _) => all_resolve : typeclass_instances.
#[global]
Hint Extern 0 (VariantResolve _ _) => all_resolve : typeclass_instances.
Definition comma_before (b : bool) (s : showM) : showM :=
if b then
cat (show_exact ",") s
else
s.
Fixpoint show_product (first : bool) (r : list type) {struct r} :
ProductResolve Show r ->
(showM -> showM) ->
(fold_right
(fun (x : type) (acc : Type) =>
match x with
| Self => showM
| Inj t => t
end -> acc) (showM) r).
refine (
match r as r
return
ProductResolve Show r ->
(showM -> showM) ->
(fold_right
(fun (x : type) (acc : Type) =>
match x with
| Self => showM
| Inj t => t
end -> acc) (showM) r)
with
| nil => fun _ f => f empty
| Self :: rs => fun a f s =>
@show_product false rs a (fun s' => f (cat s (comma_before first s')))
| Inj t :: rs => fun a f x => @show_product false rs (snd a) (fun s' => f (cat ((fst a) x) (comma_before first s')))
end); simpl in *.
Defined.
Global Instance Show_data (T : Type) (d : Data T) (AS : VariantResolve Show repr) : Show T :=
{ show :=
(fix recur (repr : variant) : VariantResolve Show repr -> recD T showM id repr -> T -> showM :=
match repr as repr return
VariantResolve Show repr -> recD T showM id repr -> T -> showM
with
| nil => fun _ x => x
| r :: rs => fun a k' =>
recur rs (snd a) (k' (show_product true _ (fst a)
(fun s' => cat (show_exact "-") (cat (show_exact "(") (cat s' (show_exact ")"))))))
end) repr AS (rec (c := id) showM)
}.
Eval compute in
to_string (M := Show_data _ _) (5 :: 6 :: 7 :: nil).
|
