Definition Rule_U2 (nc pc : clause) :=
 match nc , pc with
   PosSpaceClause Gamma Delta ((Next x y)::Sigma) ,
   NegSpaceClause Gamma' (Lseg x' z::Sigma') Delta' =>
     match expr_cmp x x' with
      Eq => Some (NegSpaceClause Gamma' (Next x y::Lseg y z::Sigma')
                   (Eqv x z :: Delta'))
     |_ => None
     end
  | _ , _ => None
 end.

Lemma unfolding2'_sound: forall sigma2 sigma0 sigma1 p sig2' s, In (p,sig2') (unfolding2' sigma0 sigma1 sigma2) -> space_denote sig2' s -> (pure_atom_denote p s \/ space_denote (sigma0++sigma2) s).