Section power_decomp.
Variable (p : nat) (Hp : 2 <= p).
Let power_nzero x : power x p <> 0.

Theorem power_decomp_unique n f a b :
(forall i j, i < j < n -> f i < f j)
-> (forall i, i < n -> a i < p)
-> (forall i, i < n -> b i < p)
-> ∑ n (fun i => a i * power (f i) p)
= ∑ n (fun i => b i * power (f i) p)
-> forall i, i < n -> a i = b i.