(* Class wrapper around "decidable" *)
(* begin decidable_class *)
    Class Dec (P : Prop) : Type := { dec : decidable P }.(* Class wrapper around "decidable" *)
(* begin decidable_class *)
    Class Dec (P : Prop) : Type := { dec : decidable P }.(* end decidable_class *)

Class DecOpt (P : Prop) := { decOpt : nat -> option bool }.(* end decidable_class *)

Class DecOpt (P : Prop) := { decOpt : nat -> option bool }.Axiom checkable_size_limit : nat.(* Discard tests that run further than the limit *)
(* For proofs, the size parameter will need to be taken into account
   to prove limit results. We just add it to the large, practical constant.
 *)
Global Instance decOpt__checkable {P} `{DecOpt P} : Checkable P :=
  {| checker _ :=
       sized (fun s =>
                match decOpt (checkable_size_limit + s) with
                | Some b => checker b
                | None => checker tt
                end
             )
  |}.Global Instance dec_decOpt {P} `{Dec P} : DecOpt P :=
  {| decOpt := fun _ => match @dec P _ with
                        | left  _ => Some true
                        | right _ => Some false
                        end |}.(* Note: maybe this should become thunked? *)
Definition checker_backtrack (l : list (unit -> option bool)) : option bool :=
  let fix aux l b :=
      match l with
      | t :: ts =>
        match t tt with
        | Some true  => Some true
        | Some false => aux ts b
        | None => aux ts true
        end
      | nil => if b then None else Some false
      end
  in aux l false.(* BCP: If I understand correctly, removing "Global" everywhere would
   change nothing... Or? *)

(* Additional Checkable instance *)
Global Instance testDec {P} `{H : Dec P} : Checkable P.Global Instance Dec_neg {P} {H : Dec P} : Dec (~ P).Global Instance Dec_conj {P Q} {H : Dec P} {I : Dec Q} : Dec (P /\ Q).Global Instance Dec_disj {P Q} {H : Dec P} {I : Dec Q} : Dec (P \/ Q).(* BCP: Not clear this is a good idea, but... *)
(* Leo: Should be ok with really low priority *)
Global Instance Dec_impl {P Q} {H : Dec P} {I : Dec Q} : Dec (P -> Q) | 100.Global Instance Dec_In {A} (Eq: forall (x y : A), Dec (x = y))
         (x : A) (l : list A) : Dec (In x l) :=
  {| dec := in_dec (fun x' y' => @dec _ (Eq x' y')) x l |}.Class Dec_Eq (A : Type) :=
  {
    dec_eq : forall (x y : A), decidable (x = y)
  }.Ltac dec_eq :=
  repeat match goal with
         | [ |- _ ] => solve [auto with eq_dec]
         | [ |- Dec _ ] => constructor
         | [ |- context[decidable _] ] => unfold decidable

         | [ H: context[decidable _] |- _ ] => unfold decidable in H

         | [ |- {?x = ?y} + {?x <> ?y} ] =>
           multimatch goal with
             | [ H: forall x y, Dec _ |- _ ] => apply H
             | [ H: Dec_Eq _ |- _ ] => apply H
             | [ |- _ ] => decide equality
           end
         end.

Theorem dec_if_dec_eq {A} (x y: A): Dec (x = y) -> {x = y} + {x <> y}.