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.
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_0: forall m1 m1' m2 j k l j' k' n1' n2'
(HI: mkInjections m1 m1' m2 j k l = (j',k',n1',n2')),
( Pos.le (Mem.nextblock m1') (Mem.nextblock m1)  /\
Mem.nextblock  m1 = n1' /\ Mem.nextblock  m2 = n2' /\
j' = j /\ k' = k) \/
(exists N, (N > 0)%nat /\
(Mem.nextblock  m1 + (Pos.of_nat N) = n1')%positive /\
(n1' = Mem.nextblock  m1') /\
(Mem.nextblock  m2 + (Pos.of_nat N) = n2')%positive).