Eliminating definitions and Skolem functions in first-order logic

In any classical first-order theory that proves the existence of at least two elements, one can eliminate definitions with a polynomial bound on the increase in proof length. The author considers how in any classical first-order theory strong enough to code finite functions, including sequential theories, one can also eliminate Skolem functions with a polynomial bound on the increase in proof length.

[1]  Helmut Schwichtenberg.  Logic and the Axiom of Choice , 1979 .

[2]  Jeremy Avigad,et al.  Algebraic proofs of cut elimination , 2001, J. Log. Algebraic Methods Program..

[3]  Jeremy Avigad,et al.  Interpreting classical theories in constructive ones , 2000, Journal of Symbolic Logic.

[4]  Jeremy Avigad.  Weak Theories of Nonstandard Arithmetic and Analysis , 2000 .

[5]  Joseph R. Shoenfield,et al.  Mathematical logic , 1967 .

[6]  P. Bernays,et al.  Grundlagen der Mathematik , 1934 .

[7]  A. Wilkie,et al.  Counting problems in bounded arithmetic , 1985 .

[8]  Jeremy Avigad,et al.  A Realizability Interpretation for Classical Arithmetic , 2002 .

[9]  Jan Krajícek,et al.  Bounded arithmetic, propositional logic, and complexity theory , 1995, Encyclopedia of mathematics and its applications.

[10]  Jeremy Avigad.  Formalizing Forcing Arguments in Subsystems of Second-Order Arithmetic , 1996, Ann. Pure Appl. Log..

[11]  Petr Hájek,et al.  Metamathematics of First-Order Arithmetic , 1993, Perspectives in mathematical logic.

[12]  P. Pudlák.  Chapter VIII - The Lengths of Proofs , 1998 .

[13]  Miklós Ajtai,et al.  The complexity of the Pigeonhole Principle , 1988, [Proceedings 1988] 29th Annual Symposium on Foundations of Computer Science.