Definition compose_sm (mu1 mu2 : SM_Injection) : SM_Injection :=
 Build_SM_Injection
   (locBlocksSrc mu1) (locBlocksTgt mu2)
   (pubBlocksSrc mu1) (pubBlocksTgt mu2)
   (compose_meminj (local_of mu1) (local_of mu2))
   (extBlocksSrc mu1) (extBlocksTgt mu2)
   (frgnBlocksSrc mu1) (frgnBlocksTgt mu2)
   (compose_meminj (extern_of mu1) (extern_of mu2)).

Lemma compose_sm_wd: forall mu1 mu2 (WD1: SM_wd mu1) (WD2:SM_wd mu2) (HypPub: forall b, pubBlocksTgt mu1 b = true -> pubBlocksSrc mu2 b = true) (HypFrg: forall b, frgnBlocksTgt mu1 b = true -> frgnBlocksSrc mu2 b = true), SM_wd (compose_sm mu1 mu2).