We present three finitary cut-free sequent calculi for the modal [my]-calculus.
Two of these derive annotated sequents in the style of Stirling’s ‘tableau proof
system with names’ (4236) and feature special inferences that discharge open
assumptions. The third system is a variant of Kozen’s axiomatisation in which
cut is replaced by a strengthening of the v-induction inference rule. Soundness
and completeness for the three systems is proved by establishing a sequence
of embeddings between the calculi, starting at Stirling’s tableau-proofs and
ending at the original axiomatisation of the [my]-calculus due to Kozen. As a
corollary we obtain a completeness proof for Kozen’s axiomatisation which
avoids the usual detour through automata or games
