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_valid: forall mu1 mu2 m1 m2 m2' m3 (SMV1: sm_valid mu1 m1 m2) (SMV2: sm_valid mu2 m2' m3), sm_valid (compose_sm mu1 mu2) m1 m3.