Ltac despec := 
  match goal with
    | [H : comp_spec _ _ _  |- _] => destruct H
  end.

Theorem comp_spec_iso : forall (A B : Set){eqda : EqDec A}{eqdb : EqDec B}(c1 : Comp A)(c2 : Comp B)(f : A -> B)(f_inv : B -> A), (forall x : B, In x (getSupport c2) -> f (f_inv x) = x) -> (forall x : A, In x (getSupport c1) -> f_inv (f x) = x) -> (forall x : B, In x (getSupport c2) -> In (f_inv x) (getSupport c1)) -> (forall x : A, In x (getSupport c1) -> comp_spec (fun a b => a = (f x) <-> b = x) c2 c1) -> comp_spec (fun a b => f a = b) c1 c2.