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