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_dom2Lemma_strong: forall x y z sigma, Some y = lseg_in_dom2 x (Var z) sigma -> exists sigma2, Permutation sigma (Lseg (Var x) y :: sigma2) /\ (expr_eq y (Var z) =false).