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_extern_incr: forall mu12 mu12' mu23 mu23' (inc12: extern_incr mu12 mu12') (inc23: extern_incr mu23 mu23') (FRG': forall b1 b2 d1, foreign_of mu12' b1 = Some(b2,d1) -> exists b3 d2, foreign_of mu23' b2 = Some(b3,d2)) (WD12': SM_wd mu12') (WD23': SM_wd mu23'), extern_incr (compose_sm mu12 mu23) (compose_sm mu12' mu23').