Cut-elimination for ω1

Annals of Pure and Applied Logic 169 (12):1246-1269 (2018)
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A new system of proof-theoretic ordinal functions.W. Buchholz - 1986 - Annals of Pure and Applied Logic 32:195-207.
Proof theory for theories of ordinals—I: recursively Mahlo ordinals.Toshiyasu Arai - 2003 - Annals of Pure and Applied Logic 122 (1-3):1-85.
Proof theory of weak compactness.Toshiyasu Arai - 2013 - Journal of Mathematical Logic 13 (1):1350003.
Intuitionistic fixed point theories over set theories.Toshiyasu Arai - 2015 - Archive for Mathematical Logic 54 (5-6):531-553.

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