Section Factorization_for_Verification.
Variable A : Set.
Variable BASE : BT.
Let b := base BASE.
Let Num := num BASE.
Let Digit := digit BASE.
Let Val_bound := val_bound BASE.
Variable R : forall n : nat, A -> inf n -> inf n -> A -> Prop.
Definition Connection := connection A Digit Digit (R b).
Notation Factorizable := (factorizable _) (only parsing).
Notation Proper := (proper _) (only parsing).

Theorem factorization_for_verification :
factorizable _ R ->
proper _ BASE R ->
forall (n : nat) (X Y : Num n) (a a' : A),
Connection n a X Y a' -> R (exp b n) a (Val_bound n X) (Val_bound n Y) a'.