Geodesic
Simplification of Gentzen's reductions in the classical arithmetic
Zapiski Nauchnykh Seminarov POMI, Theoretical application of methods of mathematical logic. Part III, Tome 105 (1981), pp. 45-52

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The reduction of the classical arithmetic derivations used by Gentzen in [1] has not been till now put in practive. In the article a reduction which presents a considerable simplification of the reduction in [1] is suggested. Applying the methods of [3] this reduction can be used for normalization of the derivations. The derivation ordinal as in [1] is a strictly monotonous one unline that used by Gentzen in [2]. 
@article{ZNSL_1981_105_a5,
     author = {L. M. Kogan-Bernstein},
     title = {Simplification of {Gentzen's} reductions in the classical arithmetic},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {45--52},
     publisher = {mathdoc},
     volume = {105},
     year = {1981},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1981_105_a5/}
}
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L. M. Kogan-Bernstein. Simplification of Gentzen's reductions in the classical arithmetic. Zapiski Nauchnykh Seminarov POMI, Theoretical application of methods of mathematical logic. Part III, Tome 105 (1981), pp. 45-52. http://geodesic.mathdoc.fr/item/ZNSL_1981_105_a5/