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.