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 â–¡ and â–³
that prove the schemes A→△A and □△S→□S for
S∈Σ1​. 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
Σ1​-provability logic of Heyting Arithmetic