Class CCCviaNatDed (A: Type) (prod expo: A -> A -> A) {ND: NatDed A}: Prop :=
  isCCC: CartesianClosedCat.CCC A derives eq prod expo.Class CCCviaNatDed (A: Type) (prod expo: A -> A -> A) {ND: NatDed A}: Prop :=
  isCCC: CartesianClosedCat.CCC A derives eq prod expo.

(* Specifications: *)
Lemma CCC_exp_prod1: forall A prod expo {ND : NatDed A} {CCC: CCCviaNatDed A prod expo} B (P: B -> A) Q, prod (exp P) Q = exp (fun x => prod (P x) Q).
Lemma CCC_exp_prod2: forall A prod expo {ND : NatDed A} {CCC: CCCviaNatDed A prod expo} B P (Q: B -> A), prod P (exp Q) = exp (fun x => prod P (Q x)).
Lemma CCC_distrib_orp_prod: forall A prod expo {ND : NatDed A} {CCC: CCCviaNatDed A prod expo} P Q R, prod (orp P Q) R = orp (prod P R) (prod Q R).
Lemma CCC_FF_prod: forall A prod expo {ND : NatDed A} {CCC: CCCviaNatDed A prod expo} P, prod FF P = FF.
Lemma CCC_prod_FF: forall A prod expo {ND : NatDed A} {CCC: CCCviaNatDed A prod expo} P, prod P FF = FF.