  We study the complexity of a range of propositional proof systems which allow
inference rules of the form: from a set of clauses $\Gamma$ derive the set of
clauses $\Gamma \cup \{ C \}$ where, due to some syntactic condition, $\Gamma
\cup \{ C \}$ is satisfiable if $\Gamma$ is, but where $\Gamma$ does not
necessarily imply $C$. These inference rules include BC, RAT, SPR and PR
(respectively short for blocked clauses, resolution asymmetric tautologies,
subset propagation redundancy and propagation redundancy), which arose from
work in satisfiability (SAT) solving. We introduce a new, more general rule SR
(substitution redundancy).
  If the new clause $C$ is allowed to include new variables then the systems
based on these rules are all equivalent to extended resolution. We focus on
restricted systems that do not allow new variables. The systems with deletion,
where we can delete a clause from our set at any time, are denoted DBC${}^-$,
DRAT${}^-$, DSPR${}^-$, DPR${}^-$ and DSR${}^-$. The systems without deletion
are BC${}^-$, RAT${}^-$, SPR${}^-$, PR${}^-$ and SR${}^-$.
  With deletion, we show that DRAT${}^-$, DSPR${}^-$ and DPR${}^-$ are
equivalent. By earlier work of Kiesl, Rebola-Pardo and Heule, they are also
equivalent to DBC${}^-$. Without deletion, we show that SPR${}^-$ can simulate
PR${}^-$ provided only short clauses are inferred by SPR inferences. We also
show that many of the well-known "hard" principles have small SPR${}^-$
refutations. These include the pigeonhole principle, bit pigeonhole principle,
parity principle, Tseitin tautologies and clique-coloring tautologies.
SPR${}^-$ can also handle or-fication and xor-ification, and lifting with an
index gadget. Our final result is an exponential size lower bound for RAT${}^-$
refutations, giving exponential separations between RAT${}^-$ and both
DRAT${}^-$ and SPR${}^-$.
