Definition Rule_U2 (nc pc : clause) :=
match nc , pc with
PosSpaceClause Gamma Delta ((Next x y)::Sigma) ,
NegSpaceClause Gamma' (Lseg x' z::Sigma') Delta' =>
match expr_cmp x x' with
Eq => Some (NegSpaceClause Gamma' (Next x y::Lseg y z::Sigma')
(Eqv x z :: Delta'))
|_ => None
end
| _ , _ => None
end.

(*The axiom schema for U2 from Figure 2, used as an auxiliary lemma for the proof of U2*)
Lemma U2_axiomSchema: forall x y z Sigma s,
space_denote (Next x y::Lseg y z::Sigma) s ->
pure_atom_denote (Eqv x z) s \/ space_denote (Lseg x z::Sigma) s.