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

Theorem alpha_diophantine a b c : 3 < b /\ a = alpha_nat b c
<-> exists a' b' c', a' = a /\ b' = b /\ c' = c
/\ exists u t r s v w x y,
alpha_conditions a' b' c' u t r s v w x y.