Section diophantine_necessity.
Variables (a b c : nat).

(* Specifications: *)
Theorem alpha_necessity : 3 < b /\ a = alpha_nat b c -> exists u t r s v w x y, alpha_conditions a b c u t r s v w x y.
End diophantine_necessity.
Theorem alpha_diophantine a b c : 3 < b /\ a = alpha_nat b c
<-> exists u t r s v w x y, alpha_conditions a b c u t r s v w x y.