Definition mkinitial_SM (mu: SM_Injection) frgnS frgnT :=
  match mu with
  Build_SM_Injection locBSrc locBTgt pSrc pTgt local extBSrc extBTgt fSrc fTgt extern =>
  Build_SM_Injection (fun b => false) (fun b => false) (fun b => false) (fun b => false) (fun b => None)
                     (DomSrc mu) (DomTgt mu) frgnS frgnT (as_inj mu)
  end.

Lemma forall_vals_inject_extern_incr mu mu' vals1 vals2 (Inj : Forall2 (val_inject (as_inj mu)) vals1 vals2) (Incr : extern_incr mu mu') (WD : SM_wd mu') : Forall2 (val_inject (as_inj mu')) vals1 vals2.