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.

References

• Gaisi Takeuti. Proof theory. Volume 81 of Studies in Logic and the Foundation of Mathematics. North-Holland, Amsterdam, 1975.
• Anne Sjerp Troelstra and Helmut Schwichtenberg. Basic Proof Theory. Edition: 2, illustrated, revised. Published by Cambridge University Press, 2000.