 Prime-chain lower bound for local length

A prime chain of length $s$ in a Noetherian local ring forces its maximal-ideal length function to be at least $c{t}^s$ for large $t$. Quotient by the initial prime to reduce to a domain, choose a nonzero $a$ in the next prime, and use the Artin-Rees lemma to obtain $H_R(t)\ge H_R(t-c_0)+H_{R/(a)}(t)$. Induction on chain length and summing about $t/(2c_0)$ terms prove the bound. This yields $\dim R\le d(R)$ without assuming the Krull height theorem.