  We prove three switching lemmas, for random restrictions for which variables
are set independently; for random restrictions where variables are set in
blocks (both due to Hastad [Hastad 86]); and for a distribution appropriate for
the bijective pigeonhole principle [Beame et al. 94, Krajicek et al. 95]. The
proofs are based on Beame's version [Beame 94] of Razborov's proof of the
switching lemma in [Razborov 93], except using families of weighted
restrictions rather than families of restrictions which are all the same size.
This follows a suggestion of Beame in [Beame 94]. The result is something
between Hastad's and Razborov's methods of proof. We use probabilistic
arguments rather than counting ones, in a similar way to Hastad, but rather
than doing induction on the terms in our formula with an inductive hypothesis
involving conditional probability, as Hastad does, we explicitly build one
function to bound the probabilities for the whole formula.
