Proof with try solve [simpl; auto | congruence | inversion 1
| apply listd_pure_atomd_pureu with (s0 := y);
[ | exists x; apply join_comm; auto ]; solve [apply IHsigma; auto] ].
revert s; induction sigma... simpl; intros s [x [y [A [B C]]]].
remember a as b; destruct b... remember e as c; destruct c...
remember (next_in_dom v sigma) as d. if_tac... split...
exfalso. symmetry in Heqd. eapply nn_njoin; eauto.
Qed.

(* Specifications: *)
Lemma nl_lseg_in_sound x y sigma s1 s2 s :
space_atom_denote (Next (Var x) y) s1 ->
space_denote sigma s2 ->
join s1 s2 s ->
listd pure_atom_denote inter TT (lseg_in_dom_atoms x sigma) s.