Definition MATCH (d:CMinSel_core) mu c1 m1 c2 m2:Prop :=
match_states (restrict (as_inj mu) (vis mu)) c1 m1 c2 m2 /\
REACH_closed m1 (vis mu) /\
meminj_preserves_globals ge (as_inj mu) /\
globalfunction_ptr_inject (as_inj mu) /\
(forall b, isGlobalBlock ge b = true -> frgnBlocksSrc mu b = true) /\
sm_valid mu m1 m2 /\ SM_wd mu /\ Mem.inject (as_inj mu) m1 m2.

Lemma MATCH_afterExternal: forall
(GDE : genvs_domain_eq ge tge)
mu st1 st2 m1 e vals1 m2 ef_sig vals2 e' ef_sig'
(MemInjMu : Mem.inject (as_inj mu) m1 m2)
(MatchMu: MATCH st1 mu st1 m1 st2 m2)
(AtExtSrc : at_external cminsel_eff_sem st1 = Some (e, ef_sig, vals1))
(AtExtTgt : at_external rtl_eff_sem st2 = Some (e', ef_sig', vals2))
(ValInjMu : Forall2 (val_inject (restrict (as_inj mu) (vis mu))) vals1 vals2)
(pubSrc' : block -> bool)
(pubSrcHyp : pubSrc' =
(fun b : block =>
locBlocksSrc mu b && REACH m1 (exportedSrc mu vals1) b))
(pubTgt' : block -> bool)
(pubTgtHyp: pubTgt' =
(fun b : block =>
locBlocksTgt mu b && REACH m2 (exportedTgt mu vals2) b))
nu (NuHyp: nu = replace_locals mu pubSrc' pubTgt')
nu' ret1 m1' ret2 m2'
(INC: extern_incr nu nu')
(SEP: sm_inject_separated nu nu' m1 m2)
(WDnu': SM_wd nu')
(SMvalNu': sm_valid nu' m1' m2')
(MemInjNu': Mem.inject (as_inj nu') m1' m2')
(RValInjNu': val_inject (as_inj nu') ret1 ret2)
(FwdSrc: mem_forward m1 m1')
(FwdTgt: mem_forward m2 m2')
(frgnSrc' : block -> bool)
(frgnSrcHyp: frgnSrc' =
(fun b : block => DomSrc nu' b &&
(negb (locBlocksSrc nu' b) && REACH m1' (exportedSrc nu' (ret1 :: nil)) b)))
(frgnTgt' : block -> bool)
(frgnTgtHyp: frgnTgt' =
(fun b : block => DomTgt nu' b &&
(negb (locBlocksTgt nu' b) && REACH m2' (exportedTgt nu' (ret2 :: nil)) b)))
mu' (Mu'Hyp: mu' = replace_externs nu' frgnSrc' frgnTgt')
(UnchPrivSrc: Mem.unchanged_on
(fun b z => locBlocksSrc nu b = true /\ pubBlocksSrc nu b = false) m1 m1')
(UnchLOOR: Mem.unchanged_on (local_out_of_reach nu m1) m2 m2'),
exists st1' st2',
after_external cminsel_eff_sem (Some ret1) st1 =Some st1' /\
after_external rtl_eff_sem (Some ret2) st2 = Some st2' /\
MATCH st1' mu' st1' m1' st2' m2'.