Definition reestablish (mu0 mu: SM_Injection): SM_Injection :=
  match mu0, mu with
    Build_SM_Injection locBSrc0 locBTgt0 pSrc0 pTgt0 local0
                       extBSrc0 extBTgt0 fSrc0 fTgt0 extern0,
    Build_SM_Injection locBSrc locBTgt pSrc pTgt local
                       extBSrc extBTgt fSrc fTgt extern =>
    Build_SM_Injection locBSrc0 locBTgt0 pSrc0 pTgt0 local0
                       (fun b => if locBSrc0 b then false else locBSrc b || extBSrc b)
                       (fun b => if locBTgt0 b then false else locBTgt b || extBTgt b)
                       fSrc0 fTgt0 (*We have not YET learned about additional blocks - that's
                          done by replace_externs in AfterEtxernal!*)
                       (fun b => if locBSrc0 b then None else join extern local b)
  end.

Lemma reestablish_sm_injsep: forall mu0 mu (WD0:SM_wd mu0) (WD:SM_wd mu) (REST: restrict (as_inj mu) (DomSrc mu0) = as_inj mu0) (LocSrc: forall b, locBlocksSrc mu0 b = true -> DomSrc mu b = true) (LocTgt: forall b, locBlocksTgt mu0 b = true -> DomTgt mu b = true) m1 m2 (SEP: sm_inject_separated mu0 mu m1 m2), sm_inject_separated mu0 (reestablish mu0 mu) m1 m2.