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 extern_incr_inject_incr: forall nu12 nu23 nu' (WDnu' : SM_wd nu') (EXT: extern_incr (compose_sm nu12 nu23) nu') (GlueInvNu: SM_wd nu12 /\ SM_wd nu23 /\ locBlocksTgt nu12 = locBlocksSrc nu23 /\ extBlocksTgt nu12 = extBlocksSrc nu23 /\ (forall b, pubBlocksTgt nu12 b = true -> pubBlocksSrc nu23 b = true) /\ (forall b, frgnBlocksTgt nu12 b = true -> frgnBlocksSrc nu23 b = true)), inject_incr (compose_meminj (as_inj nu12) (as_inj nu23)) (as_inj nu').