Definition added (c: clause) {s s'': clauseset} (step: trans s s'') :=
(c \notin s) && (c \in s'').

(*main lemma 2' (redund_sub): if c \in s, c is not redundant in s and s -->* s', then either: (1) c \in s', or (2) c \notin s' and there exists a clauseset t such that t is a subset of s' but not a subset of the first set s  as a corollary, we have that when c is added in some step s --> s'' and s'' -->* s', then s != s'.  proof sketch: since c is added in s --> s'', c is not redundant in s''. also, we know that c \notin s and c \in s'' (since c was just added).  by lemma 2', either c \in s' (in which case s != s' since c \notin s), or there exists a clauseset t such that t is a subset of s' but not of s''. since s'' = s \cup {c}, t is not a subset of s either, so s != s'. *)
Lemma redund_sub (c: clause) (s s': clauseset) :
c \in s -> ~redund s c -> deriv s s' ->
(c \in s') \/
(c \notin s' /\ exists t: clauseset, t \subset s' /\ ~t \subset s).