Section sinc_decidable.
Variable (P : nat -> Prop)
(f : nat -> nat)
(Hf : forall n, f n < f (S n))
(HP : forall n, P n <-> exists k, n = f k).
Let f_mono x y : x <= y -> f x <= f y.
Let f_smono x y : x < y -> f x < f y.
Let f_ge_n n : n <= f n.
Let unbounded n : exists k, n <= k /\ P k.

Theorem sinc_decidable : (forall n, exists k, n <= k /\ P k)
* (forall n, { P n } + { ~ P n }).