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_as_inj: forall mu12 mu23 (WD1: SM_wd mu12) (WD2: SM_wd mu23) (SrcTgtLoc: locBlocksTgt mu12 = locBlocksSrc mu23) (SrcTgtExt: extBlocksTgt mu12 = extBlocksSrc mu23), as_inj (compose_sm mu12 mu23) = compose_meminj (as_inj mu12) (as_inj mu23).