Definition Rule_U3 (nc pc : clause) :=
match nc , pc with
PosSpaceClause Gamma Delta ((Lseg x y)::Sigma) ,
NegSpaceClause Gamma' (Lseg x' Nil::Sigma') Delta' =>
match expr_cmp x x' with
Eq => Some (NegSpaceClause Gamma' (Lseg x y::Lseg y Nil::Sigma') Delta')
|_ => None
end
| _ , _ => None
end.

Lemma next_in_dom_true: forall v sigma s,
true = next_in_dom v sigma ->
space_denote sigma s ->
exists w, exists sig, space_denote (Next (Var v) w::sig) s /\
Permutation (Next (Var v) w::sig) sigma.