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).