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_dom_atoms_space_denote_perm_strong:
forall sig p v s,
(space_denote sig) s -> In p (lseg_in_dom_atoms v sig) ->
exists y, (p=Eqv (Var v) y \/ p=Eqv y (Var v)) /\
(pure_atom_denote p s \/
exists sig1,
space_denote (Lseg (Var v) y::sig1) s /\
Permutation sig (Lseg (Var v) y :: sig1)).