  We study the complexity of proof systems augmenting resolution with inference
rules that allow, given a formula $\Gamma$ in conjunctive normal form, deriving
clauses that are not necessarily logically implied by $\Gamma$ but whose
addition to $\Gamma$ preserves satisfiability. When the derived clauses are
allowed to introduce variables not occurring in $\Gamma$, the systems we
consider become equivalent to extended resolution. We are concerned with the
versions of these systems without new variables. They are called BC${}^-$,
RAT${}^-$, SBC${}^-$, and GER${}^-$, denoting respectively blocked clauses,
resolution asymmetric tautologies, set-blocked clauses, and generalized
extended resolution. Each of these systems formalizes some restricted version
of the ability to make assumptions that hold "without loss of generality,"
which is commonly used informally to simplify or shorten proofs.
  Except for SBC${}^-$, these systems are known to be exponentially weaker than
extended resolution. They are, however, all equivalent to it under a relaxed
notion of simulation that allows the translation of the formula along with the
proof when moving between proof systems. By taking advantage of this fact, we
construct formulas that separate RAT${}^-$ from GER${}^-$ and vice versa. With
the same strategy, we also separate SBC${}^-$ from RAT${}^-$. Additionally, we
give polynomial-size SBC${}^-$ proofs of the pigeonhole principle, which
separates SBC${}^-$ from GER${}^-$ by a previously known lower bound. These
results also separate the three systems from BC${}^-$ since they all simulate
it. We thus give an almost complete picture of their relative strengths.
