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).
Definition Content_II_Property (j12 j12' j23':meminj) (m1 m1' m2:Mem.mem)
(CM:ZMap.t (ZMap.t memval)):=
forall b2,
(Mem.valid_block m2 b2 -> forall ofs2,
match source j12 m1 b2 ofs2 with
Some(b1,ofs1) =>
match j23' b2 with
None => ZMap.get ofs2 (PMap.get b2 CM) =
ZMap.get ofs2 (PMap.get b2 m2.(Mem.mem_contents))
| Some(b3,ofs3) =>
ZMap.get ofs2 (PMap.get b2 CM) =
inject_memval j12'
(ZMap.get ofs1 (PMap.get b1 m1'.(Mem.mem_contents)))
end
| None => ZMap.get ofs2 (PMap.get b2 CM) =
ZMap.get ofs2 (PMap.get b2 m2.(Mem.mem_contents))
end)
/\ (~ Mem.valid_block m2 b2 -> forall ofs2,
match source j12' m1' b2 ofs2 with
None => ZMap.get ofs2 (PMap.get b2 CM) =Undef
| Some(b1,ofs1) =>
ZMap.get ofs2 (PMap.get b2 CM) =
inject_memval j12'
(ZMap.get ofs1 (PMap.get b1 m1'.(Mem.mem_contents)))
end)
/\ fst CM !! b2 = Undef.

Lemma add_no_neutral2: forall (p q:positive), (p <> p + q)%positive.