Journal article:

    Samuel R. Buss.
    "On Herbrand's Theorem,"
    In Logic and Computational Complexity,
    Lecture Notes in Computer Science #960, 1995, Springer-Verlag, pp. 195-209.

    Abstract: We firstly survey several forms of Herbrand's theorem. What is commonly called ``Herbrand's theorem'' in many textbooks is actually a very simple form of Herbrand's theorem which applies only to $\forall\exists$-formulas; but the original statement of Herbrand's theorem applied to arbitrary first-order formulas. We give a direct proof, based on cut-elimination, of what is essentially Herbrand's original theorem. The ``nocounterexample theorems'' recently used in bounded and Peano arithmetic are immediate corollaries of this form of Herbrand's theorem.
    Secondly, we discuss the results proved in Herbrand's 1930 dissertation.

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