All presented properties are showed by automated theorem-prover Prover9.
(There is no special order in the presentation of the theorems.)

ID	Name	Lemma	Duals
OAT1	iteration of tests	$p^\omega = p\cdot 1^\omega$	n/a

In general, we write $x,y...$ and $a,b...$ for arbitrary semiring elements and $p,q...$ for tests. If other letters are involved we adapted the original notation of the authors.