Fixpoint mkInjectionsN (N:nat)(n1 n2:block)(j k l: meminj)
                     :  meminj * meminj * block * block :=
   match N with O => (j,k,n1,n2)
    | S M => mkInjectionsN M (Pos.succ n1) (Pos.succ n2)
                             (fun b => if eq_block b n1
                                       then Some (n2,0)
                                       else j b)
                             (fun b => if eq_block b n2 then l n1 else k b)
                             l
   end.

Lemma mkInjectionsN_0: forall N n1 n2 j k l j' k' n1' n2' (HI: mkInjectionsN N n1 n2 j k l = (j',k',n1',n2')), match N with O => (n1 = n1' /\ n2 = n2' /\ j' = j /\ k' = k) | S n => (n1 + (Pos.of_nat N) = n1')%positive /\ (n2 + (Pos.of_nat N) = n2')%positive end.