Definition AccessMap_II_Property  (j12 j23 j12' :meminj) (m1 m1' m2 : mem)
           (AM:ZMap.t (Z -> perm_kind -> option permission)):Prop :=
  forall b2,
    (Mem.valid_block m2 b2 -> forall k ofs2,
       match j23 b2 with
         None => PMap.get b2 AM ofs2 k  = PMap.get b2 m2.(Mem.mem_access) ofs2 k
       | Some (b3,d3) =>
         match source j12 m1 b2 ofs2 with
             Some(b1,ofs1) =>  PMap.get b2 AM ofs2 k =
                               PMap.get b1 m1'.(Mem.mem_access) ofs1 k
           | None =>  PMap.get b2 AM ofs2 k =
                      PMap.get b2 m2.(Mem.mem_access) ofs2 k
           end
        end)
     /\ (~ Mem.valid_block m2 b2 -> forall k ofs2,
           match source j12' m1' b2 ofs2 with
              Some(b1,ofs1) => PMap.get b2 AM ofs2 k =
                               PMap.get b1 m1'.(Mem.mem_access) ofs1 k
            | None =>  PMap.get b2 AM ofs2 k = None
          end).

Lemma II_ok: forall m1 m2 j12 (MInj12 : Mem.inject j12 m1 m2) m1' (Fwd1: mem_forward m1 m1') j23 m3 (MInj23 : Mem.inject j23 m2 m3) m3' (Fwd3: mem_forward m3 m3') j' (MInj13': Mem.inject j' m1' m3') (InjIncr: inject_incr (compose_meminj j12 j23) j') (InjSep: inject_separated (compose_meminj j12 j23) j' m1 m3) (Unch11': Mem.unchanged_on (loc_unmapped (compose_meminj j12 j23)) m1 m1') (Unch33': Mem.unchanged_on (loc_out_of_reach (compose_meminj j12 j23) m1) m3 m3') prej12' j23' n1' n2' (HeqMKI: mkInjections m1 m1' m2 j12 j23 j' = (prej12', j23', n1', n2')) j12' (Hj12': j12'= removeUndefs j12 j' prej12') m2' (NB: m2'.(Mem.nextblock)=n2') (CONT: Content_II_Property j12 j12' j23' m1 m1' m2 (m2'.(Mem.mem_contents))) (ACCESS: AccessMap_II_Property j12 j23 j12' m1 m1' m2 (m2'.(Mem.mem_access))), j'=compose_meminj j12' j23' /\ inject_incr j12 j12' /\ inject_incr j23 j23' /\ Mem.inject j12' m1' m2' /\ mem_forward m2 m2' /\ Mem.inject j23' m2' m3' /\ Mem.unchanged_on (loc_out_of_reach j12 m1) m2 m2' /\ inject_separated j12 j12' m1 m2 /\ inject_separated j23 j23' m2 m3 /\ Mem.unchanged_on (loc_unmapped j23) m2 m2' /\ Mem.unchanged_on (loc_out_of_reach j23 m2) m3 m3'.