  We study propositional proof systems with inference rules that formalize
restricted versions of the ability to make assumptions that hold without loss
of generality, commonly used informally to shorten proofs. Each system we study
is built on resolution. They are called BC${}^-$, RAT${}^-$, SBC${}^-$, and
GER${}^-$, denoting respectively blocked clauses, resolution asymmetric
tautologies, set-blocked clauses, and generalized extended resolution - all
"without new variables." They may be viewed as weak versions of extended
resolution (ER) since they are defined by first generalizing the extension rule
and then taking away the ability to introduce new variables. Except for
SBC${}^-$, they are known to be strictly between resolution and extended
resolution.
  Several separations between these systems were proved earlier by exploiting
the fact that they effectively simulate ER. We answer the questions left open:
We prove exponential lower bounds for SBC${}^-$ proofs of a binary encoding of
the pigeonhole principle, which separates ER from SBC${}^-$. Using this new
separation, we prove that both RAT${}^-$ and GER${}^-$ are exponentially
separated from SBC${}^-$. This completes the picture of their relative
strengths.
