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In this paper, we study the provability logic of intuitionistic theories of arithmetic that prove their own completeness. We prove a completeness theorem for theories equipped with two provability predicates β‘ and β³ that prove the schemes Aββ³A and β‘β³Sββ‘S for SβΞ£1β. Using this theorem, we determine the logic of fast provability for a number of intuitionistic theories. Furthermore, we reprove a theorem previously obtained by M. Ardeshir and S. Mojtaba Mojtahedi determining the Ξ£1β-provability logic of Heyting Arithmetic
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