Module MemObsEq.
(* A compcert injection would not work because it allows permissions to go up *)
(* Moreover, we require that undefined values are matched by the target memory, unlike compcert injections. Although the latter is not neccessary and in retrospect may be a limiting factor. For example we would be able to reuse this development for our final erasure if we allowed undefined values to become more defined. *)
(** Weak injection between memories *)
Record weak_mem_obs_eq (f : memren) (mc mf : mem) :=
{
domain_invalid: forall b, ~(Mem.valid_block mc b) -> f b = None;
domain_valid: forall b, Mem.valid_block mc b -> exists b', f b = Some b';
codomain_valid: forall b1 b2, f b1 = Some b2 -> Mem.valid_block mf b2;
injective: forall b1 b1' b2, f b1 = Some b2 ->
f b1' = Some b2 ->
b1 = b1';
perm_obs_weak :
forall b1 b2 ofs (Hrenaming: f b1 = Some b2),
Mem.perm_order''
(permission_at mc b1 ofs Cur)
(permission_at mf b2 ofs Cur)}.
(** Strong injection between memories *)
Record strong_mem_obs_eq (f : memren) (mc mf : mem) :=
{ perm_obs_strong :
forall b1 b2 ofs (Hrenaming: f b1 = Some b2),
permission_at mf b2 ofs Cur =
(permission_at mc b1 ofs Cur);
val_obs_eq :
forall b1 b2 ofs (Hrenaming: f b1 = Some b2)
(Hperm: Mem.perm mc b1 ofs Cur Readable),
memval_obs_eq f (Maps.ZMap.get ofs mc.(Mem.mem_contents)#b1)
(Maps.ZMap.get ofs mf.(Mem.mem_contents)#b2)}.
(** Renaming between memories *)
Record mem_obs_eq (f : memren) (mc mf : mem) :=
{ weak_obs_eq : weak_mem_obs_eq f mc mf;
strong_obs_eq : strong_mem_obs_eq f mc mf }.

Lemma weak_obs_eq_domain_ren:
forall f m m',
weak_mem_obs_eq f m m' ->
domain_memren f m.