Definition mkInjections (m1 m1' m2:mem) (j k l: meminj)
                     :  meminj * meminj * block * block :=
  if plt (Mem.nextblock m1) (Mem.nextblock m1')
  then mkInjectionsN (Pos.to_nat (Mem.nextblock m1') - Pos.to_nat(Mem.nextblock m1))
                (Mem.nextblock m1)
                (Mem.nextblock m2) j k l
  else (j,k,Mem.nextblock m1,Mem.nextblock m2).

Lemma mkInjections_composememinj: forall m1 m1' m2 j k l j' k' n1' n2' (HI: mkInjections m1 m1' m2 j k l = (j',k',n1',n2')) (InjIncr: inject_incr (compose_meminj j k) l) m3 (InjSep: inject_separated (compose_meminj j k) l m1 m3) (VBj1: forall b1 b2 ofs2, j b1 = Some(b2,ofs2) -> Mem.valid_block m1 b1) (VBj2: forall b1 b2 ofs2, j b1 = Some(b2,ofs2) -> Mem.valid_block m2 b2) (VBk2: forall b1 b2 ofs2, k b1 = Some(b2,ofs2) -> Mem.valid_block m2 b1) (VBL1': forall b1 b3 ofs3, l b1 = Some (b3, ofs3) -> (b1 < n1')%positive), l = compose_meminj j' k'.