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 lseg_in_dom_atoms_space_denote_perm_strong: forall sig p v s, (space_denote sig) s -> In p (lseg_in_dom_atoms v sig) -> exists y, (p=Eqv (Var v) y \/ p=Eqv y (Var v)) /\ (pure_atom_denote p s \/ exists sig1, space_denote (Lseg (Var v) y::sig1) s /\ Permutation sig (Lseg (Var v) y :: sig1)).