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_intern_incr: forall mu12 mu12' mu23 mu23' (inc12: intern_incr mu12 mu12') (inc23: intern_incr mu23 mu23'), intern_incr (compose_sm mu12 mu23) (compose_sm mu12' mu23').