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_emp_at: forall h x y h1 (L:lseg x y h1) h2 (L2: lseg x y h2) h1' (J1:join h1 h1' h) h2' (J2: join h2 h2' h) l (E: emp_at l h2), emp_at l h1.