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 Perm_aux1: forall {A} (a:A) l0 l1 l2, Permutation l0 (l1 ++ a :: l2) -> exists l3 : list A, exists l4 : list A, l0 = l3 ++ a :: l4.