intros ? ? ge mu mu' mu'' smwd incr gsep12 gsep23 b1 b3 d3 map1 map3.
assert (INCR: Values.inject_incr (as_inj mu') (as_inj mu'')) by (apply intern_incr_as_inj; auto).
eapply gsep_trans; eauto.
Qed.

(* Specifications: *)
(* Lemma gsep_compose: forall {F V} (ge:  Genv.t F V) mu12 mu23 mu12' mu23', SM_wd mu12 -> SM_wd mu23 -> SM_wd mu12' -> SM_wd mu23' -> Values.inject_incr (as_inj mu12) (as_inj mu12') -> (locBlocksTgt mu12 = locBlocksSrc mu23 /\ extBlocksTgt mu12 = extBlocksSrc mu23) -> globals_separate ge mu12 mu12' -> globals_separate ge mu23 mu23' -> globals_separate ge (compose_sm mu12 mu23) (compose_sm mu12' mu23'). intros ? ? ge mu12 mu23 mu12' mu23' WD12 WD23 WD12' WD23' INCR GLUE gsep12 gsep23 b1 b3 d3 map13 map13'. destruct (compose_sm_as_injD _ _ _ _ _ map13' WD12' WD23') as [b2 [d1 [d2 [map1 [map2 extra]]]]]. destruct (compose_sm_as_injD_None _ _ _ WD12 WD23 GLUE map13) as [map12| [b2' [d [map12 map23]]]]. - assert (isGlobalBlock ge b2 = false). eapply gsep12; eauto. destruct (isGlobalBlock ge b3) eqn:isglob; [ | reflexivity]. assert (meminj_preserves_globals ge (extern_of mu12') /\ (forall b, isGlobalBlock ge b = true -> frgnBlocksSrc mu12' b = true)). ad_it. destruct H0 as [A B]. apply B in isglob. destruct A as [A1 [A2 A3]].  ad_it. - assert (HH:as_inj mu12' b1 = Some (b2', d)) by (eapply INCR; auto). rewrite HH in map1; inversion map1; subst. eapply gsep23; eauto. Qed. *)
Lemma meminj_preserves_globals_extern_incr_separate {F V:Type} (ge: Genv.t F V) mu nu:
forall (INC: extern_incr mu nu)
(PG: meminj_preserves_globals ge (as_inj mu))
(WDnu: SM_wd nu)
(GSep: globals_separate ge mu nu),
meminj_preserves_globals ge (as_inj nu).
Lemma meminj_preserves_globals_intern_incr_separate {F V:Type} (ge': Genv.t F V) mu nu:
forall (INC: intern_incr mu nu)
(PG: meminj_preserves_globals ge' (as_inj mu))
(WDnu: SM_wd nu)
(GSep: globals_separate ge' mu nu),
meminj_preserves_globals ge' (as_inj nu).
Lemma intern_incr_globals_separate
{F V:Type} (ge: Genv.t F V) mu nu:
forall (INC: intern_incr mu nu)
(PG: meminj_preserves_globals ge (as_inj mu))
(GF: forall b, isGlobalBlock ge b = true -> frgnBlocksSrc mu b = true)
(WD: SM_wd mu) (WDnu: SM_wd nu),
globals_separate ge mu nu.
Lemma exter_incr_globals_separate
{F V:Type} (ge: Genv.t F V) mu nu:
forall (EE: extern_of mu = extern_of nu)
(PG: meminj_preserves_globals ge (as_inj mu))
(GF: forall b, isGlobalBlock ge b = true -> frgnBlocksSrc mu b = true)
(WD: SM_wd mu) (WDnu: SM_wd nu),
globals_separate ge mu nu.