Unwinding a Non-effective Cut Elimination Proof

Non-effective cut elimination proof uses Koenig’s lemma to obtain a non-closed branch of a proof-search tree τ (without cut) for a first order formula A, if A is not cut free provable. A partial model (semi-valuation) corresponding to this branch and verifying ¬A is extended to a total model for ¬A using arithmetical comprehension. This contradicts soundness, if A is derivable with cut. Hence τ is a cut free proof of A. The same argument works for Herbrand Theorem. We discuss algorithms of obtaining cut free proofs corresponding to this schema and quite different from complete search through all possible proofs.