  For a given intuitionistic propositional formula A and a propositional
variable x occurring in it, define the infinite sequence of formulae { A \_i |
i$\ge$1} by letting A\_1 be A and A\_{i+1} be A(A\_i/x). Ruitenburg's Theorem
[8] says that the sequence { A \_i } (modulo logical equivalence) is ultimately
periodic with period 2, i.e. there is N $\ge$ 0 such that A N+2
$\leftrightarrow$ A N is provable in intuitionistic propositional calculus. We
give a semantic proof of this theorem, using duality techniques and bounded
bisimulations ranks.
