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.