Completeness of atomic initial sequents

In sequent calculus, the completeness of atomic initial sequents states that initial sequents $A ⊢ A$ (where $A$ is an arbitrary formula) can be derived from only atomic initial sequents $p ⊢ p$ (where $p$ is an atomic formula). This theorem plays a role analogous to eta expansion in lambda calculus, and dual to cut-elimination and beta reduction. Typically it can be established by induction on the structure of $A$, much more easily than cut-elimination.