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.