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_unique: forall x y s (L:lseg x y s) s' (L': lseg x y s') t r (J: join s t r) t' (J': join s' t' r), s'=s /\ t'=t.