Section Factorization_for_Synthesis.
Variable A : Set.
Variable BASE : BT.
Let b := base BASE.
Let Num := num BASE.
Let Digit := digit BASE.
Let Val_bound := val_bound BASE.
Definition Tl := tl Digit.
Variable FR : forall n : nat, A -> inf n -> inf n -> A.
Let R (n : nat) (a : A) (x y : inf n) (a' : A) : Prop := a' = FR n a x y.
Notation Factorizable := (factorizable _) (only parsing).
Notation Proper := (proper _) (only parsing).

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