Section LOCATIONS.
Variable ge: genv.
Variable sp: block.
Inductive nlive: nmem -> block -> Z -> Prop :=
| nlive_intro: forall stk gl b ofs
(STK: b = sp -> ~ISet.In ofs stk)
(GL: forall id iv,
Genv.find_symbol ge id = Some b ->
gl!id = Some iv ->
~ISet.In ofs iv),
nlive (NMem stk gl) b ofs.
(** All locations are live *)
Definition nmem_all := NMem ISet.empty (PTree.empty _).
(** Add a range of live locations to [nm].  The range starts at the abstract pointer [p] and has length [sz]. *)
Definition nmem_add (nm: nmem) (p: aptr) (sz: Z) : nmem :=
match nm with
| NMemDead => nmem_all       (**r very conservative, should never happen *)
| NMem stk gl =>
match p with
| Gl id ofs =>
match gl!id with
| Some iv => NMem stk (PTree.set id (ISet.remove (Ptrofs.unsigned ofs) (Ptrofs.unsigned ofs + sz) iv) gl)
| None => nm
end
| Glo id =>
NMem stk (PTree.remove id gl)
| Stk ofs =>
NMem (ISet.remove (Ptrofs.unsigned ofs) (Ptrofs.unsigned ofs + sz) stk) gl
| Stack =>
NMem ISet.empty gl
| _ => nmem_all
end
end.
(** Remove a range of locations from [nm], marking these locations as dead. The range starts at the abstract pointer [p] and has length [sz]. *)
Definition nmem_remove (nm: nmem) (p: aptr) (sz: Z) : nmem :=
match nm with
| NMemDead => NMemDead
| NMem stk gl =>
match p with
| Gl id ofs =>
let iv' :=
match gl!id with
| Some iv => ISet.add (Ptrofs.unsigned ofs) (Ptrofs.unsigned ofs + sz) iv
| None    => ISet.interval (Ptrofs.unsigned ofs) (Ptrofs.unsigned ofs + sz)
end in
NMem stk (PTree.set id iv' gl)
| Stk ofs =>
NMem (ISet.add (Ptrofs.unsigned ofs) (Ptrofs.unsigned ofs + sz) stk) gl
| _ => nm
end
end.
(** Test (conservatively) whether some locations in the range delimited by [p] and [sz] can be live in [nm]. *)
Definition nmem_contains (nm: nmem) (p: aptr) (sz: Z) :=
match nm with
| NMemDead => false
| NMem stk gl =>
match p with
| Gl id ofs =>
match gl!id with
| Some iv => negb (ISet.contains (Ptrofs.unsigned ofs) (Ptrofs.unsigned ofs + sz) iv)
| None => true
end
| Stk ofs =>
negb (ISet.contains (Ptrofs.unsigned ofs) (Ptrofs.unsigned ofs + sz) stk)
| _ => true  (**r conservative answer *)
end
end.
(** Kill all stack locations between 0 and [sz], and mark everything else as live.  This reflects the effect of freeing the stack block at a [Ireturn] or [Itailcall] instruction. *)
Definition nmem_dead_stack (sz: Z) :=
NMem (ISet.interval 0 sz) (PTree.empty _).

Lemma nlive_dead_stack:
forall sz b' i, b' <> sp \/ ~(0 <= i < sz) -> nlive (nmem_dead_stack sz) b' i.