Require Import Zwf.
Require Import Axioms.
Require Import Coqlib.
Require Intv.
Require Import Maps.
Require Archi.
Require Import AST.
Require Import Integers.
Require Import Floats.
Require Import Values.
Require Import Values_symbolictype.
Require Import Values_symbolic.
Require Export Memtype.
Require Import IntFacts.
Require Import Normalise.
Require Import NormaliseSpec.
Require Export Memdata.
Require Import Alloc.
Require Import Permutation.

Local Unset Elimination Schemes.
Local Unset Case Analysis Schemes.

Local Notation "a # b" := (PMap.get b a) (at level 1).

Module Mem <: MEM.

Definition perm_order' (po: option permission) (p: permission) :=
  match po with
  | Some p' => perm_order p' p
  | None => False
 end.

Definition perm_order'' (po1 po2: option permission) :=
  match po1, po2 with
  | Some p1, Some p2 => perm_order p1 p2
  | _, None => True
  | None, Some _ => False
 end.

Record mem' : Type := mkmem {
  mem_contents: PMap.t (ZMap.t memval); (* block -> offset -> memval *)
  mem_access: PMap.t (Z -> perm_kind -> option permission);
  mem_blocksize: PMap.t (option (Z*Z));
  mem_mask: PMap.t (option nat); (* block -> option mask; the number of 0-trailing-bits *)
  nextblock: block;
  access_max:
    forall b ofs, perm_order'' (mem_access#b ofs Max) (mem_access#b ofs Cur);
  nextblock_noaccess:
    forall b ofs k, ~(Plt b nextblock) -> mem_access#b ofs k = None;
  contents_wt_memval:
    forall b ofs, wt_memval (ZMap.get ofs (mem_contents # b));
  access_bounds_ok: (* permissions and bounds agree *)
    forall b ofs k,
      (mem_access#b ofs k) <> None
      -> in_bound ofs (get_bounds mem_blocksize b);
  bs_valid: (* only valid blocks have non-empty bounds *)
    forall b, ~ (Plt b nextblock) -> mem_blocksize#b = None;
  msk_valid: (* only valid blocks have non-empty masks *)
    forall b, ~ (Plt b nextblock) -> mem_mask#b = None;
  wfm_alloc_ok: (* the allocation algorithm succeeds on this memory *)
    alloc_blocks
      (mk_block_list_aux (get_size ((mem_blocksize)))
                         (pred (Pos.to_nat nextblock))) <> None;
  alignment_ok: (* the alignment agrees with the bounds of blocks *)
    forall b al,
      mem_mask#b = Some al ->
      al = alignment_of_size (get_size (mem_blocksize) b);
  bounds_mask_consistency: (* bounds and masks information are present together, or not at all *)
    forall b,
      (mem_mask#b = None /\ mem_blocksize#b = None) \/
      (mem_mask#b <> None /\ mem_blocksize#b <> None);
  bounds_lo_inf0: (* the lower bound of a block is 0 *)
    forall b lo hi,
      mem_blocksize#b = Some (lo,hi) ->
      lo = 0
}.

Definition mem := mem'.


Definition size_block (m: mem) (b:block) : Z :=
  get_size (m.(mem_blocksize)) b.

Definition size_of_block (m:mem) (b:block) :=
  match PMap.get b m.(mem_blocksize) with
      | Some (lo,hi) => (Int.repr (hi-lo))
      | None => (Int.repr 0)
  end.

Definition bounds_of_block (m:mem) (b:block) : Z*Z :=
  match PMap.get b m.(mem_blocksize) with
      | Some (lo,hi) => (lo,hi)
      | None => (0,0)
  end.

Definition mask (m:mem) (b:block) :=
  match PMap.get b m.(mem_mask) with
      | Some mask => mask
      | None => O
  end.

Definition nat_mask (m:mem) (b:block) :=
  Int.not (Int.repr (two_power_nat (mask m b) - 1)).

Definition mk_block_list (m:mem) : (list (block * Z)) :=
  mk_block_list_aux (size_block m) (pred (Pos.to_nat (Mem.nextblock m))).


Definition size_mem (m:mem) : Z :=
  size_mem_aux (mk_block_list m).

Definition alloc_mem (m:mem) : option (block -> int) :=
  alloc_blocks (mk_block_list m).


Definition mem_norm (m:mem) (e: expr_sym) (r:result) : val :=
  Normalise.normalise (bounds_of_block m) (nat_mask m) e r.
  
Lemma mkmem_ext:
 forall cont1 cont2 acc1 acc2 next1 next2
        a1 a2 b1 b2 c1 c2 bs1 bs2
        mask1 mask2 Pabo1 Pabo2 comp2 comp1 vbs1 vbs2 vm1 vm2
        const1 const2 bb1 bb2
        bli1 bli2,
   cont1=cont2 -> acc1=acc2 -> next1=next2 ->
   bs1=bs2 -> mask1=mask2 ->
   mkmem cont1 acc1 bs1 mask1 next1
         a1 b1 c1 Pabo1 vbs1 vm1
         comp1 bb1 const1 bli1 =
   mkmem cont2 acc2 bs2 mask2 next2
         a2 b2 c2 Pabo2 vbs2 vm2
         comp2 bb2 const2 bli2.

Lemma alloc_compat:
  forall m,
  exists al,
    alloc_mem m = Some al /\
    compat al (bounds_of_block m) (nat_mask m).

Lemma concrete_mem:
  forall m,
  exists al,
    compat al (bounds_of_block m) (nat_mask m).

Lemma alloc_permut:
  forall l l',
    Permutation l l' ->
    (alloc_blocks l = None <-> alloc_blocks l' = None).



Lemma norm_ptr_same_eval:
  forall m l b o
         (MN : Mem.mem_norm m (Eval (Eptr l Int.zero)) Ptr = Vptr b o)
         (NIB : ~ NormaliseSpec.in_bound 0 (Mem.bounds_of_block m l)),
    False.

Validity of blocks and accesses

Definition valid_block (m: mem) (b: block) := Plt b (nextblock m).

Definition valid_block_dec (m:mem) (b:block) : {valid_block m b} + {~ valid_block m b}.

Theorem valid_not_valid_diff:
  forall m b b', valid_block m b -> ~(valid_block m b') -> b <> b'.

Hint Local Resolve valid_not_valid_diff: mem.


Definition perm (m: mem) (b: block) (ofs: Z) (k: perm_kind) (p: permission) : Prop :=
   perm_order' (m.(mem_access)#b ofs k) p.

Theorem perm_implies:
  forall m b ofs k p1 p2, perm m b ofs k p1 -> perm_order p1 p2 -> perm m b ofs k p2.

Hint Local Resolve perm_implies: mem.

Theorem perm_cur_max:
  forall m b ofs p, perm m b ofs Cur p -> perm m b ofs Max p.

Theorem perm_cur:
  forall m b ofs k p, perm m b ofs Cur p -> perm m b ofs k p.

Theorem perm_max:
  forall m b ofs k p, perm m b ofs k p -> perm m b ofs Max p.

Hint Local Resolve perm_cur perm_max: mem.

Theorem perm_valid_block:
  forall m b ofs k p, perm m b ofs k p -> valid_block m b.

Hint Local Resolve perm_valid_block: mem.

Remark perm_order_dec:
  forall p1 p2, {perm_order p1 p2} + {~perm_order p1 p2}.

Remark perm_order'_dec:
  forall op p, {perm_order' op p} + {~perm_order' op p}.

Theorem perm_dec:
  forall m b ofs k p, {perm m b ofs k p} + {~ perm m b ofs k p}.

Definition range_perm (m: mem) (b: block) (lo hi: Z) (k: perm_kind) (p: permission) : Prop :=
  forall ofs, lo <= ofs < hi -> perm m b ofs k p.

Theorem range_perm_implies:
  forall m b lo hi k p1 p2,
  range_perm m b lo hi k p1 -> perm_order p1 p2 -> range_perm m b lo hi k p2.

Theorem range_perm_cur:
  forall m b lo hi k p,
  range_perm m b lo hi Cur p -> range_perm m b lo hi k p.

Theorem range_perm_max:
  forall m b lo hi k p,
  range_perm m b lo hi k p -> range_perm m b lo hi Max p.

Hint Local Resolve range_perm_implies range_perm_cur range_perm_max: mem.

Lemma range_perm_dec:
  forall m b lo hi k p, {range_perm m b lo hi k p} + {~ range_perm m b lo hi k p}.


Definition valid_access (m: mem) (chunk: memory_chunk) (b: block) (ofs: Z) (p: permission): Prop :=
  range_perm m b ofs (ofs + size_chunk chunk) Cur p
  /\ (align_chunk chunk | ofs).

Theorem valid_access_implies:
  forall m chunk b ofs p1 p2,
  valid_access m chunk b ofs p1 -> perm_order p1 p2 ->
  valid_access m chunk b ofs p2.

Theorem valid_access_freeable_any:
  forall m chunk b ofs p,
  valid_access m chunk b ofs Freeable ->
  valid_access m chunk b ofs p.

Hint Local Resolve valid_access_implies: mem.

Theorem valid_access_valid_block:
  forall m chunk b ofs,
  valid_access m chunk b ofs Nonempty ->
  valid_block m b.

Hint Local Resolve valid_access_valid_block: mem.

Lemma valid_access_perm:
  forall m chunk b ofs k p,
  valid_access m chunk b ofs p ->
  perm m b ofs k p.

Lemma valid_access_compat:
  forall m chunk1 chunk2 b ofs p,
  size_chunk chunk1 = size_chunk chunk2 ->
  align_chunk chunk2 <= align_chunk chunk1 ->
  valid_access m chunk1 b ofs p->
  valid_access m chunk2 b ofs p.

Lemma valid_access_dec:
  forall m chunk b ofs p,
  {valid_access m chunk b ofs p} + {~ valid_access m chunk b ofs p}.

Definition valid_pointer (m: mem) (b: block) (ofs: Z): bool :=
  perm_dec m b ofs Cur Nonempty.

Theorem valid_pointer_nonempty_perm:
  forall m b ofs,
  valid_pointer m b ofs = true <-> perm m b ofs Cur Nonempty.

Theorem valid_pointer_valid_access:
  forall m b ofs,
  valid_pointer m b ofs = true <-> valid_access m Mint8unsigned b ofs Nonempty.


Definition weak_valid_pointer (m: mem) (b: block) (ofs: Z) :=
  valid_pointer m b ofs || valid_pointer m b (ofs - 1).

Lemma weak_valid_pointer_spec:
  forall m b ofs,
  weak_valid_pointer m b ofs = true <->
    valid_pointer m b ofs = true \/ valid_pointer m b (ofs - 1) = true.
Lemma valid_pointer_implies:
  forall m b ofs,
  valid_pointer m b ofs = true -> weak_valid_pointer m b ofs = true.

Operations over memory stores

Program Definition empty: mem :=
  mkmem (PMap.init (ZMap.init (Symbolic (Eval (Eint Int.zero)) O)))
        (PMap.init (fun ofs k => None))
        (PMap.init (None))
        (PMap.init None)
        2%positive _ _ _ _ _ _ _ _ _ _.


Fixpoint getN (n: nat) (p: Z) (c: ZMap.t memval) {struct n}: list memval :=
  match n with
  | O => nil
  | S n' => ZMap.get p c :: getN n' (p + 1) c
  end.

Lemma getN_Nlist : forall n p c, length (getN n p c) = n.



Fixpoint setN (vl: list memval) (p: Z) (c: ZMap.t memval) {struct vl}: ZMap.t memval :=
  match vl with
  | nil => c
  | v :: vl' => setN vl' (p + 1) (ZMap.set p v c)
  end.

Remark setN_other:
  forall vl c p q,
  (forall r, p <= r < p + Z_of_nat (length vl) -> r <> q) ->
  ZMap.get q (setN vl p c) = ZMap.get q c.

Remark setN_outside:
  forall vl c p q,
  q < p \/ q >= p + Z_of_nat (length vl) ->
  ZMap.get q (setN vl p c) = ZMap.get q c.

Remark getN_setN_same:
  forall vl p c,
  getN (length vl) p (setN vl p c) = vl.

Remark getN_exten:
  forall c1 c2 n p,
  (forall i, p <= i < p + Z_of_nat n -> ZMap.get i c1 = ZMap.get i c2) ->
  getN n p c1 = getN n p c2.

Remark getN_setN_disjoint:
  forall vl q c n p,
  Intv.disjoint (p, p + Z_of_nat n) (q, q + Z_of_nat (length vl)) ->
  getN n p (setN vl q c) = getN n p c.

Remark getN_setN_outside:
  forall vl q c n p,
  p + Z_of_nat n <= q \/ q + Z_of_nat (length vl) <= p ->
  getN n p (setN vl q c) = getN n p c.

Remark setN_default:
  forall vl q c, fst (setN vl q c) = fst c.


Fixpoint init_block' (n:nat) (b:block) (cur:int) : list memval :=
  match n with
      O => Symbolic (Eval (Eundef b cur)) O :: nil
    | S m => Symbolic (Eval (Eundef b cur)) O :: (init_block' m b (Int.add cur (Int.repr 1)))
  end.

Definition init_block (lo hi: Z) (b: block) : memval * PTree.t memval :=
  let sz := hi - lo in
  if zlt sz 0
  then (ZMap.init (Symbolic (Eval (Eint Int.zero)) O))
  else setN (init_block' (Z.to_nat sz) b (Int.repr lo)) lo (ZMap.init (Symbolic (Eval (Eint Int.zero)) O)).

Lemma init_block'_wt_memval:
  forall n b cur,
    Forall wt_memval (init_block' n b cur).

Lemma init_block_wt_memval:
  forall lo hi cur ofs,
    wt_memval (ZMap.get ofs ( (init_block lo hi cur))).

Lemma pos_ex_index:
  forall p,
    exists z,
      ZIndexed.index z = p.

Lemma init_block_wt_memval':
  forall lo hi cur ofs,
    wt_memval ((init_block lo hi cur) # ofs).

Lemma zle_zlt:
  forall a b c,
    zle a b && zlt b c = true <->
    a <= b < c.

Definition conc_mem_alloc (m: mem) (lo hi : Z) (alignment: nat) :
  option (block -> int) :=
  alloc_blocks
    (mk_block_list_aux
       (get_size
          (PMap.set (nextblock m) (Some (0, hi)) (mem_blocksize m)))
       (Pos.to_nat (nextblock m))).

Definition alloc_mem_dec:
  forall l ,
    {alloc_blocks l <> None} + {alloc_blocks l = None}.

Definition conc_mem_alloc_dec (m:mem) (lo hi: Z) (alignment: nat) :
  {conc_mem_alloc m lo hi alignment <> None} +
  {conc_mem_alloc m lo hi alignment = None}.

Program Definition low_alloc (m: mem) (lo hi: Z) : option (mem * block) :=
  let lo := 0 in
  let al := alignment_of_size (hi-lo) in
  if conc_mem_alloc_dec m lo hi al
  then
    Some (mkmem (PMap.set m.(nextblock)
                              (init_block lo hi m.(nextblock))
                              m.(mem_contents))
                (PMap.set m.(nextblock)
                              (fun ofs k => if zle lo ofs && zlt ofs hi then Some Freeable else None)
                              m.(mem_access))
                (PMap.set m.(nextblock)
                              (Some (0,hi))
                              m.(mem_blocksize))
                (PMap.set m.(nextblock)
                              (Some al)
                              m.(mem_mask))
                (Psucc m.(nextblock))
                _ _ _ _ _ _ _ _ _ _,
          m.(nextblock))
  else None.

Definition alloc (m: mem) (lo hi: Z) :=
  low_alloc m lo hi.

Lemma alloc_nat_mask:
  forall m lo hi m' b,
    Mem.alloc m lo hi = Some (m',b) ->
    Mem.nat_mask m' b = Int.not (Int.repr (two_power_nat (alignment_of_size hi) - 1)).

Lemma alloc_contents:
  forall m lo hi m' b,
    alloc m lo hi = Some (m',b) ->
    forall b', b <> b' ->
               (mem_contents m') !! b' = (mem_contents m) !! b'.

Opaque zeq.
Program Definition unchecked_free (m: mem) (b: block) (lo hi: Z): mem :=
  mkmem m.(mem_contents)
            (PMap.set b
                      (fun ofs k => if zle lo ofs && zlt ofs hi then None else m.(mem_access)#b ofs k)
                      m.(mem_access))
            (PMap.set b (match m.(mem_blocksize)#b with
                             Some (l,h) => if zeq l lo && zeq h hi then
                                             None
                                           else m.(mem_blocksize)#b
                           | None => None
                         end) m.(mem_blocksize))
            (PMap.set b (match m.(mem_blocksize)#b with
                             Some (l,h) => if zeq l lo && zeq h hi
                                           then None
                                           else m.(mem_mask)#b
                           | None => None
                         end) m.(mem_mask))
            m.(nextblock) _ _ _ _ _ _ _ _ _ _.

Lemma unchecked_free_mask:
  forall m b lo hi b',
    mask (unchecked_free m b lo hi) b' =
    if eq_block b' b
    then match bounds_of_block m b with
             (lo',hi') =>
                            (if zeq lo' lo && zeq hi' hi
                             then O
                             else mask m b')
         end
    else mask m b'.

Lemma unchecked_free_nat_mask:
  forall m b lo hi b',
    nat_mask (unchecked_free m b lo hi) b' =
    if eq_block b' b
    then match bounds_of_block m b with
             (lo',hi') => if zeq lo' lo && zeq hi' hi
                          then Int.mone
                          else nat_mask m b'
         end
    else nat_mask m b'.

Definition free (m: mem) (b: block) (lo hi: Z): option mem :=
  if range_perm_dec m b lo hi Cur Freeable
  then Some(unchecked_free m b lo hi)
  else None.

Fixpoint free_list (m: mem) (l: list (block * Z * Z)) {struct l}: option mem :=
  match l with
  | nil => Some m
  | (b, lo, hi) :: l' =>
      match free m b lo hi with
      | None => None
      | Some m' => free_list m' l'
      end
  end.

Lemma bounds_of_block_valid:
  forall m b lo hi,
    bounds_of_block m b = (lo,hi) ->
    lo <> hi ->
    valid_block m b.


Lemma size_mem_aux_sup1:
  forall m,
    Alloc.size_mem_aux m >= 8.

Lemma size_mem_sup1:
  forall m,
    Mem.size_mem m >= 8.




Definition load (chunk: memory_chunk) (m: mem) (b: block) (ofs: Z): option expr_sym :=
  if valid_access_dec m chunk b ofs Readable
  then Some (decode_val chunk (getN (size_chunk_nat chunk) ofs (m.(mem_contents)#b)))
  else None.


Definition loadv (chunk: memory_chunk) (m: mem) (addr: expr_sym) : option expr_sym :=
  let v := mem_norm m addr Values_symbolictype.Ptr
  in match v with
         Vptr b ofs => load chunk m b (Int.unsigned ofs)
       | _ => None
     end.


Definition loadbytes (m: mem) (b: block) (ofs n: Z): option (list memval) :=
  if range_perm_dec m b ofs (ofs + n) Cur Readable
  then Some (getN (nat_of_Z n) ofs (m.(mem_contents)#b))
  else None.




Definition optForall {A} P (l: option (list A)) :=
  match l with
      Some ll => Forall P ll
    | None => True
  end.

Lemma encode_val_wt':
  forall c v,
    optForall wt_memval (encode_val c v).


Lemma wt_memval_setN:
  forall l t ofs ofs0,
  Forall wt_memval l ->
  (forall o, wt_memval (ZMap.get o t)) ->
  wt_memval (ZMap.get ofs0 (setN l ofs t)).

Definition is_encodable chunk v :=
  match encode_val chunk v with
      Some enc => true
    | None => false
  end.

Definition encode_val_tot chunk v :=
  match encode_val chunk v with
      Some enc => enc
    | _ => nil
  end.

Program Definition store (chunk: memory_chunk) (m: mem) (b: block) (ofs: Z) (v: expr_sym): option mem :=
  if valid_access_dec m chunk b ofs Writable then
    if is_encodable chunk v then
      Some (mkmem (PMap.set b
                            (setN (encode_val_tot chunk v) ofs (m.(mem_contents)#b))
                            m.(mem_contents))
                  m.(mem_access)
                      m.(mem_blocksize)
                          m.(mem_mask)
                              m.(nextblock)
                                  _ _ _ _ _ _ _ _ _ _)
    else None
  else
    None.


Definition storev (chunk: memory_chunk) (m: mem) (addr v: expr_sym) : option mem :=
  match mem_norm m addr Values_symbolictype.Ptr with
      Vptr b ofs => store chunk m b (Int.unsigned ofs) v
    | _ => None
  end.

Lemma check_bytes_correct:
  forall l,
    check_bytes l = true ->
    Forall wt_memval l.


Program Definition storebytes (m: mem) (b: block) (ofs: Z) (bytes: list memval) : option mem :=
  if bool_dec (check_bytes bytes) true then
    if range_perm_dec m b ofs (ofs + Z_of_nat (length bytes)) Cur Writable then
      Some (mkmem
              (PMap.set b (setN bytes ofs (m.(mem_contents)#b)) m.(mem_contents))
              m.(mem_access)
                  m.(mem_blocksize)
                      m.(mem_mask)
                          m.(nextblock)
                              _ _ _ _ _ _ _ _ _ _)
    else None
  else
    None.


Program Definition drop_perm (m: mem) (b: block) (lo hi: Z) (p: permission): option mem :=
  if range_perm_dec m b lo hi Cur Freeable then
    Some (mkmem m.(mem_contents)
                (PMap.set b
                        (fun ofs k => if zle lo ofs && zlt ofs hi then Some p else m.(mem_access)#b ofs k)
                        m.(mem_access))
                m.(mem_blocksize)
                m.(mem_mask)
                m.(nextblock) _ _ _ _ _ _ _ _ _ _)
  else None.

Properties of the memory operations

Theorem nextblock_empty: nextblock empty = 2%positive.

Theorem perm_empty: forall b ofs k p, ~perm empty b ofs k p.

Theorem valid_access_empty: forall chunk b ofs p, ~valid_access empty chunk b ofs p.

Properties related to load

Lemma Forall_rib:
  forall {A} (P : A -> Prop) l,
    Forall P l ->
    Forall P (rev_if_be l).

Lemma wt_memval_getN:
  forall n b ofs,
    (forall o, wt_memval (ZMap.get o b) ) ->
    Forall wt_memval (getN n ofs b).

Theorem valid_access_load:
  forall m chunk b ofs,
  valid_access m chunk b ofs Readable ->
  exists v, load chunk m b ofs = Some v.

Theorem load_valid_access:
  forall m chunk b ofs v,
  load chunk m b ofs = Some v ->
  valid_access m chunk b ofs Readable.

Lemma load_result:
  forall chunk m b ofs v,
  load chunk m b ofs = Some v ->
  v = decode_val chunk (getN (size_chunk_nat chunk) ofs (m.(mem_contents)#b)).

Hint Local Resolve load_valid_access valid_access_load: mem.

Theorem load_type:
  forall m chunk b ofs v,
  load chunk m b ofs = Some v ->
  Val.has_type v ((type_of_chunk chunk)).

Theorem load_cast:
  forall m chunk b ofs v,
  load chunk m b ofs = Some v ->
  match chunk with
  | Mint8signed => v == Val.sign_ext 8 v
  | Mint8unsigned => v == Val.zero_ext 8 v
  | Mint16signed => v == Val.sign_ext 16 v
  | Mint16unsigned => v == Val.zero_ext 16 v
  | _ => True
  end.

Lemma ps_int8_signed_unsigned:
  forall l,
    proj_symbolic_le l Mint8signed =
    proj_symbolic_le l Mint8unsigned.

Lemma ps_int16_signed_unsigned:
  forall l,
    proj_symbolic_le l Mint16signed =
    proj_symbolic_le l Mint16unsigned.

Theorem load_int8_signed_unsigned:
  forall m b ofs v1 v2,
  load Mint8signed m b ofs = Some v1 ->
  option_map (Val.sign_ext 8) (load Mint8unsigned m b ofs) = Some v2 ->
  v1 == v2.

Theorem load_int8_signed_unsigned_none:
  forall m b ofs,
    load Mint8signed m b ofs = None ->
    load Mint8unsigned m b ofs = None.

Theorem load_int16_signed_unsigned:
  forall m b ofs v1 v2,
  load Mint16signed m b ofs = Some v1 ->
  option_map (Val.sign_ext 16) (load Mint16unsigned m b ofs) = Some v2 ->
  v1 == v2.

Theorem load_int16_signed_unsigned_none:
  forall m b ofs,
    load Mint16signed m b ofs = None ->
    load Mint16unsigned m b ofs = None.

Properties related to loadbytes

Theorem range_perm_loadbytes:
  forall m b ofs len,
  range_perm m b ofs (ofs + len) Cur Readable ->
  exists bytes, loadbytes m b ofs len = Some bytes.

Theorem loadbytes_range_perm:
  forall m b ofs len bytes,
  loadbytes m b ofs len = Some bytes ->
  range_perm m b ofs (ofs + len) Cur Readable.

Theorem loadbytes_load:
  forall chunk m b ofs bytes,
  loadbytes m b ofs (size_chunk chunk) = Some bytes ->
  (align_chunk chunk | ofs) ->
  load chunk m b ofs = Some (decode_val chunk bytes).

Theorem load_loadbytes:
  forall chunk m b ofs v,
  load chunk m b ofs = Some v ->
  exists bytes, loadbytes m b ofs (size_chunk chunk) = Some bytes
             /\ v = decode_val chunk bytes.

Lemma getN_length:
  forall c n p, length (getN n p c) = n.

Theorem loadbytes_length:
  forall m b ofs n bytes,
  loadbytes m b ofs n = Some bytes ->
  length bytes = nat_of_Z n.

Theorem loadbytes_empty:
  forall m b ofs n,
  n <= 0 -> loadbytes m b ofs n = Some nil.
  
Lemma getN_concat:
  forall c n1 n2 p,
  getN (n1 + n2)%nat p c = getN n1 p c ++ getN n2 (p + Z_of_nat n1) c.

Theorem loadbytes_concat:
  forall m b ofs n1 n2 bytes1 bytes2,
  loadbytes m b ofs n1 = Some bytes1 ->
  loadbytes m b (ofs + n1) n2 = Some bytes2 ->
  n1 >= 0 -> n2 >= 0 ->
  loadbytes m b ofs (n1 + n2) = Some(bytes1 ++ bytes2).

Theorem loadbytes_split:
  forall m b ofs n1 n2 bytes,
  loadbytes m b ofs (n1 + n2) = Some bytes ->
  n1 >= 0 -> n2 >= 0 ->
  exists bytes1, exists bytes2,
     loadbytes m b ofs n1 = Some bytes1
  /\ loadbytes m b (ofs + n1) n2 = Some bytes2
  /\ bytes = bytes1 ++ bytes2.

Lemma Forall_dec:
  forall {A} (P: A -> Prop)
         (Pdec: forall a, P a \/ ~ P a)
         l,
    Forall P l \/ ~ Forall P l.

Lemma wt_memval_dec:
  forall a,
    wt_memval a \/ ~ wt_memval a.

Lemma Forall_app_l:
  forall {A} (P: A -> Prop) l1 l2,
    Forall P (l1 ++ l2) ->
    Forall P l1.

Lemma Forall_rev':
  forall {A} (P: A -> Prop) l,
    Forall P l ->
    Forall P (rev l).

Lemma Forall_app_r:
  forall {A} (P: A -> Prop) l1 l2,
    Forall P (l1 ++ l2) ->
    Forall P l2.

Lemma loadbytes_wt_memval:
  forall m b ofs n bytes,
    loadbytes m b ofs n = Some bytes ->
    Forall wt_memval bytes.

Theorem load_int64_split:
  forall m b ofs v,
  load Mint64 m b ofs = Some v ->
  exists v1 v2,
     load Mint32 m b ofs = Some (if Archi.big_endian then v1 else v2)
  /\ load Mint32 m b (ofs + 4) = Some (if Archi.big_endian then v2 else v1)
  /\ v == Val.longofwords v1 v2.

Properties related to store

 
Theorem valid_access_store:
  forall m1 chunk b ofs v,
  valid_access m1 chunk b ofs Writable ->
  tcheck_expr v = Some ((type_of_chunk chunk)) ->
  { m2: mem | store chunk m1 b ofs v = Some m2 }.

Hint Local Resolve valid_access_store: mem.
 
Section STORE.
Variable chunk: memory_chunk.
Variable m1: mem.
Variable b: block.
Variable ofs: Z.
Variable v: expr_sym.
Variable m2: mem.
Hypothesis STORE: store chunk m1 b ofs v = Some m2.
 
Lemma store_access: mem_access m2 = mem_access m1.

Lemma store_mem_contents:
  mem_contents m2 = PMap.set b (setN (encode_val_tot chunk v) ofs m1.(mem_contents)#b) m1.(mem_contents).


Lemma store_encode_some:
  encode_val chunk v <> None.


Lemma encode_val_tot_length:
  length (encode_val_tot chunk v) = size_chunk_nat chunk.

Theorem perm_store_1:
  forall b' ofs' k p, perm m1 b' ofs' k p -> perm m2 b' ofs' k p.

Theorem perm_store_2:
  forall b' ofs' k p, perm m2 b' ofs' k p -> perm m1 b' ofs' k p.

Local Hint Resolve perm_store_1 perm_store_2: mem.

Theorem size_of_block_store:
  forall b',
    size_of_block m1 b' = size_of_block m2 b'.

Theorem mask_store:
  forall b',
    mask m1 b' = mask m2 b'.

Theorem bounds_of_block_store:
  forall b',
    bounds_of_block m1 b' = bounds_of_block m2 b'.

Theorem nat_mask_store:
  forall b',
    nat_mask m1 b' = nat_mask m2 b'.


Theorem nextblock_store:
  nextblock m2 = nextblock m1.

Theorem store_valid_block_1:
  forall b', valid_block m1 b' -> valid_block m2 b'.

Theorem store_valid_block_2:
  forall b', valid_block m2 b' -> valid_block m1 b'.

Local Hint Resolve store_valid_block_1 store_valid_block_2: mem.

Theorem store_valid_access_1:
  forall chunk' b' ofs' p,
  valid_access m1 chunk' b' ofs' p -> valid_access m2 chunk' b' ofs' p.

Theorem store_valid_access_2:
  forall chunk' b' ofs' p,
  valid_access m2 chunk' b' ofs' p -> valid_access m1 chunk' b' ofs' p.

Theorem store_valid_access_3:
  valid_access m1 chunk b ofs Writable.

Local Hint Resolve store_valid_access_1 store_valid_access_2 store_valid_access_3: mem.

Lemma tcheck_expr_store_chunk:
  tcheck_expr v = Some ((type_of_chunk chunk)).

Theorem load_store_similar:
  forall chunk',
  size_chunk chunk' = size_chunk chunk ->
  align_chunk chunk' <= align_chunk chunk ->
  exists v', load chunk' m2 b ofs = Some v' /\ decode_encode_expr_sym v chunk chunk' v'.

Lemma store_some_type:
  tcheck_expr v = Some (type_of_chunk chunk).

Theorem load_store_similar_2:
  forall chunk' v1,
  size_chunk chunk' = size_chunk chunk ->
  align_chunk chunk' <= align_chunk chunk ->
  type_of_chunk chunk' = type_of_chunk chunk ->
  load chunk' m2 b ofs = Some v1 ->
  chunk' <> Many32 /\ chunk' <> Many64 ->
  v1 == Val.load_result_expr chunk' v.

Theorem load_store_same:
  exists v1, load chunk m2 b ofs = Some v1 /\
             v1 == (Val.load_result chunk v).

Theorem load_store_other:
  forall chunk' b' ofs',
  b' <> b
  \/ ofs' + size_chunk chunk' <= ofs
  \/ ofs + size_chunk chunk <= ofs' ->
  load chunk' m2 b' ofs' = load chunk' m1 b' ofs'.

Theorem loadbytes_store_same:
  loadbytes m2 b ofs (size_chunk chunk) = encode_val chunk v.

Theorem loadbytes_store_other:
  forall b' ofs' n,
  b' <> b
  \/ n <= 0
  \/ ofs' + n <= ofs
  \/ ofs + size_chunk chunk <= ofs' ->
  loadbytes m2 b' ofs' n = loadbytes m1 b' ofs' n.

Lemma setN_property:
  forall (P: memval -> Prop) vl p q c,
  (forall v, In v vl -> P v) ->
  p <= q < p + Z_of_nat (length vl) ->
  P(ZMap.get q (setN vl p c)).

Lemma getN_in:
  forall c q n p,
  p <= q < p + Z_of_nat n ->
  In (ZMap.get q c) (getN n p c).

Theorem load_pointer_store:
  forall chunk' b' ofs' v_b v_o,
    load chunk' m2 b' ofs' = Some(Eval (Eptr v_b v_o)) ->
    (chunk = Mint32 /\ v = Eval (Eptr v_b v_o) /\ chunk' = Mint32 /\ b' = b /\ ofs' = ofs)
    \/ (b' <> b \/ ofs' + size_chunk chunk' <= ofs \/ ofs + size_chunk chunk <= ofs').

End STORE.

Local Hint Resolve perm_store_1 perm_store_2: mem.
Local Hint Resolve store_valid_block_1 store_valid_block_2: mem.
Local Hint Resolve store_valid_access_1 store_valid_access_2
             store_valid_access_3: mem.



Lemma is_encodable_eq:
  forall v1 v2 chunk1 chunk2 ,
    encode_val chunk1 v1 = encode_val chunk2 v2 ->
    align_chunk chunk1 = align_chunk chunk2 ->
    size_chunk chunk1 = size_chunk chunk2 ->
    is_encodable chunk1 v1 = is_encodable chunk2 v2.

Lemma store_similar_chunks:
  forall chunk1 chunk2 v1 v2 m b ofs,
  encode_val chunk1 v1 = encode_val chunk2 v2 ->
  align_chunk chunk1 = align_chunk chunk2 ->
  store chunk1 m b ofs v1 = store chunk2 m b ofs v2.

Theorem store_signed_unsigned_8:
  forall m b ofs v,
  store Mint8signed m b ofs v = store Mint8unsigned m b ofs v.

Theorem store_signed_unsigned_16:
  forall m b ofs v,
  store Mint16signed m b ofs v = store Mint16unsigned m b ofs v.

Definition memval_equiv_typ (e1 e2: memval) :=
  match tcheck_memval e1, tcheck_memval e2 with
      Some t1, Some t2 =>
      if styp_eq t1 t2
      then forall alloc em,
             eSexpr alloc em eb eu t1 (expr_of_memval e1 (ltyp_of_typ t1)) =
             eSexpr alloc em eb eu t2 (expr_of_memval e2 (ltyp_of_typ t2))
      else False
    | _,_ => False
  end.

Record mem_equiv (m1 m2: mem) : Prop :=
  mk_mem_equiv {
      mem_contents_eq:
        forall b ofs,
          memval_equiv_typ (ZMap.get ofs ((mem_contents m1) # b))
                           (ZMap.get ofs ((mem_contents m2) # b));
      mem_access_eq:
        (mem_access m1) = (mem_access m2);
      mem_blocksize_eq:
        (mem_blocksize m1) = (mem_blocksize m2);
      mem_mask_eq:
        (mem_mask m1) = (mem_mask m2);
      nextblock_eq:
        (nextblock m1) = (nextblock m2) }.

Definition opt_mem_equiv (m1 m2: option mem) : Prop :=
  match m1, m2 with
      None, None => True
    | Some mm1 , Some mm2 => mem_equiv mm1 mm2
    | _ , _ => False
  end.

Lemma memval_equiv_typ_refl:
  forall ofs0 m b,
    memval_equiv_typ (ZMap.get ofs0 (mem_contents m) # b)
                     (ZMap.get ofs0 (mem_contents m) # b).

Lemma get_setN_not_last:
  forall al ofs0 ofs o a t,
    ofs0 <> ofs ->
    (ZMap.get ofs0 (setN al o (ZMap.set ofs a t))) = ZMap.get ofs0 (setN al o t).

Lemma met_eq:
  forall i1 i2,
    list_forall2 memval_equiv_typ i1 i2 ->
    forall m b ofs b0 ofs0,
    memval_equiv_typ
      (ZMap.get ofs0 ((PMap.set b (setN i1 ofs (mem_contents m) # b) (mem_contents m)) # b0))
      (ZMap.get ofs0 ((PMap.set b (setN i2 ofs (mem_contents m) # b) (mem_contents m)) # b0)).

Lemma memval_equiv_tint_typ:
  forall i1 i2,
    Forall (fun x => tcheck_memval x = Some Tint) i1 ->
    Forall (fun x => tcheck_memval x = Some Tint) i2 ->
    (list_forall2 memval_equiv_tint i1 i2 <->
    list_forall2 memval_equiv_typ i1 i2).

Theorem store_int8_zero_ext:
  forall m b ofs n,
    opt_mem_equiv
      (store Mint8unsigned m b ofs (Eval (Eint (Int.zero_ext 8 n))))
      (store Mint8unsigned m b ofs (Eval (Eint n))).

Theorem store_int8_sign_ext:
  forall m b ofs n,
    opt_mem_equiv
      (store Mint8signed m b ofs (Eval (Eint (Int.sign_ext 8 n))))
      (store Mint8signed m b ofs (Eval (Eint n))).

Theorem store_int16_zero_ext:
  forall m b ofs n,
    opt_mem_equiv
      (store Mint16unsigned m b ofs (Eval (Eint (Int.zero_ext 16 n))))
      (store Mint16unsigned m b ofs (Eval (Eint n))).

Theorem store_int16_sign_ext:
  forall m b ofs n,
    opt_mem_equiv
      (store Mint16signed m b ofs (Eval (Eint (Int.sign_ext 16 n))))
      (store Mint16signed m b ofs (Eval (Eint n))).

Properties related to storebytes.

Theorem range_perm_storebytes:
  forall m1 b ofs bytes,
  range_perm m1 b ofs (ofs + Z_of_nat (length bytes)) Cur Writable ->
  check_bytes bytes = true ->
  { m2 : mem | storebytes m1 b ofs bytes = Some m2 }.

Lemma check_bytes_complete:
  forall l : list memval, Forall wt_memval l -> check_bytes l = true.

Theorem storebytes_store:
  forall m1 b ofs chunk v m2 enc,
    encode_val chunk v = Some enc ->
  storebytes m1 b ofs enc = Some m2 ->
  (align_chunk chunk | ofs) ->
  store chunk m1 b ofs v = Some m2.

Lemma check_bytes_encode:
  forall chunk v l,
  encode_val chunk v = Some l ->
  check_bytes l = true.

Theorem store_storebytes:
  forall m1 b ofs chunk v m2 enc,
  store chunk m1 b ofs v = Some m2 ->
  encode_val chunk v = Some enc ->
  storebytes m1 b ofs enc = Some m2.
  
Section STOREBYTES.
Variable m1: mem.
Variable b: block.
Variable ofs: Z.
Variable bytes: list memval.
Variable m2: mem.
Hypothesis STORE: storebytes m1 b ofs bytes = Some m2.

Lemma storebytes_access: mem_access m2 = mem_access m1.

Lemma storebytes_mem_contents:
   mem_contents m2 = PMap.set b (setN bytes ofs m1.(mem_contents)#b) m1.(mem_contents).

Theorem perm_storebytes_1:
  forall b' ofs' k p, perm m1 b' ofs' k p -> perm m2 b' ofs' k p.

Theorem perm_storebytes_2:
  forall b' ofs' k p, perm m2 b' ofs' k p -> perm m1 b' ofs' k p.

Local Hint Resolve perm_storebytes_1 perm_storebytes_2: mem.

Theorem storebytes_valid_access_1:
  forall chunk' b' ofs' p,
  valid_access m1 chunk' b' ofs' p -> valid_access m2 chunk' b' ofs' p.

Theorem storebytes_valid_access_2:
  forall chunk' b' ofs' p,
  valid_access m2 chunk' b' ofs' p -> valid_access m1 chunk' b' ofs' p.

Local Hint Resolve storebytes_valid_access_1 storebytes_valid_access_2: mem.

Theorem size_of_block_storebytes:
  forall b',
    size_of_block m1 b' = size_of_block m2 b'.

Theorem mask_storebytes:
  forall b',
    mask m1 b' = mask m2 b'.

Theorem bounds_of_block_storebytes:
  forall b',
    bounds_of_block m1 b' = bounds_of_block m2 b'.

Theorem nat_mask_storebytes:
  forall b',
    nat_mask m1 b' = nat_mask m2 b'.

Theorem nextblock_storebytes:
  nextblock m2 = nextblock m1.

Theorem storebytes_valid_block_1:
  forall b', valid_block m1 b' -> valid_block m2 b'.

Theorem storebytes_valid_block_2:
  forall b', valid_block m2 b' -> valid_block m1 b'.

Local Hint Resolve storebytes_valid_block_1 storebytes_valid_block_2: mem.

Theorem storebytes_range_perm:
  range_perm m1 b ofs (ofs + Z_of_nat (length bytes)) Cur Writable.

Theorem loadbytes_storebytes_same:
  loadbytes m2 b ofs (Z_of_nat (length bytes)) = Some bytes.

Theorem loadbytes_storebytes_disjoint:
  forall b' ofs' len,
  len >= 0 ->
  b' <> b \/ Intv.disjoint (ofs', ofs' + len) (ofs, ofs + Z_of_nat (length bytes)) ->
  loadbytes m2 b' ofs' len = loadbytes m1 b' ofs' len.

Theorem loadbytes_storebytes_other:
  forall b' ofs' len,
  len >= 0 ->
  b' <> b
  \/ ofs' + len <= ofs
  \/ ofs + Z_of_nat (length bytes) <= ofs' ->
  loadbytes m2 b' ofs' len = loadbytes m1 b' ofs' len.

Theorem load_storebytes_other:
  forall chunk b' ofs',
  b' <> b
  \/ ofs' + size_chunk chunk <= ofs
  \/ ofs + Z_of_nat (length bytes) <= ofs' ->
  load chunk m2 b' ofs' = load chunk m1 b' ofs'.

End STOREBYTES.

Lemma setN_concat:
  forall bytes1 bytes2 ofs c,
  setN (bytes1 ++ bytes2) ofs c = setN bytes2 (ofs + Z_of_nat (length bytes1)) (setN bytes1 ofs c).

Lemma check_bytes_Forall:
  forall b,
  check_bytes b = true <-> Forall wt_memval b.

Lemma check_bytes_false_Forall:
  forall b,
  check_bytes b = false <-> ~ Forall wt_memval b.

Lemma Forall_app_not:
  forall {A} P (a b:list A),
  (~ Forall P a) \/ (~ Forall P b) ->
  ~ Forall P (a++b).

Lemma check_bytes_append:
  forall a b,
    check_bytes (a ++ b) = check_bytes a && check_bytes b.

Theorem storebytes_concat:
  forall m b ofs bytes1 m1 bytes2 m2,
  storebytes m b ofs bytes1 = Some m1 ->
  storebytes m1 b (ofs + Z_of_nat(length bytes1)) bytes2 = Some m2 ->
  storebytes m b ofs (bytes1 ++ bytes2) = Some m2.

Theorem storebytes_check_bytes:
  forall m b ofs bytes m2,
    storebytes m b ofs bytes = Some m2 ->
    check_bytes bytes = true.

Theorem storebytes_split:
  forall m b ofs bytes1 bytes2 m2,
  storebytes m b ofs (bytes1 ++ bytes2) = Some m2 ->
  exists m1,
     storebytes m b ofs bytes1 = Some m1
  /\ storebytes m1 b (ofs + Z_of_nat(length bytes1)) bytes2 = Some m2.

Definition mem_eq_modulo_normalise (m1 m2:mem) :=
  forall chunk b ofs v1 v2,
    load chunk m1 b ofs = Some v1 ->
    load chunk m2 b ofs = Some v2 ->
    v1 == v2.

Notation "a ==m b" := (mem_eq_modulo_normalise a b) (at level 80).

Lemma load_align_chunk:
  forall m b ofs chunk v,
    load chunk m b ofs = Some v ->
    (align_chunk chunk | ofs).

Lemma loadbytes_one:
  forall m b o n bytes,
    loadbytes m b o n = Some bytes ->
    n > 0 ->
    exists b1 b2,
      loadbytes m b o 1 = Some b1 /\
      loadbytes m b (o+1) (n-1) = Some b2 /\
      bytes = b1 ++ b2.

Lemma expr_of_memval_zero_ext_eq:
  forall a1 a2 alloc0 i i0 em,
    wt_memval a1 -> wt_memval a2 ->
    tcheck_expr (expr_of_memval a2 Lint) = Some Tint ->
    tcheck_expr (expr_of_memval a1 Lint) = Some Tint ->
    eSexpr alloc0 em eb eu Tint (expr_of_memval a1 Lint) = Vi i ->
    eSexpr alloc0 em eb eu Tint (expr_of_memval a2 Lint) = Vi i0 ->
    Int.zero_ext 8 i = Int.zero_ext 8 i0 ->
    i = i0.

Lemma decode_val_spec':
  forall l l',
    Forall wt_memval l ->
    Forall wt_memval l' ->
    list_forall2 memval_equiv_tint l l' ->
      forall c alloc em,
        eSexpr alloc em eb eu ((type_of_chunk c)) (decode_val c l) =
        eSexpr alloc em eb eu ((type_of_chunk c)) (decode_val c l').

Transparent decode_val.
Lemma mem_normalise_byte :
  forall m1 m2,
    (forall b ofs v1 v2,
       load Mint8unsigned m1 b ofs = Some v1 ->
       load Mint8unsigned m2 b ofs = Some v2 ->
       forall alloc em t,
         eSexpr alloc em eb eu t v1 =
         eSexpr alloc em eb eu t v2) ->
     (forall chunk b ofs v1 v2,
       load chunk m1 b ofs = Some v1 ->
       load chunk m2 b ofs = Some v2 ->
       forall alloc em,
         eSexpr alloc em eb eu ((type_of_chunk chunk)) v1 =
         eSexpr alloc em eb eu ((type_of_chunk chunk)) v2).

Lemma bytewise_eq_all :
  forall m1 m2
         ,
    (forall b ofs v1 v2,
       load Mint8unsigned m1 b ofs = Some v1 ->
       load Mint8unsigned m2 b ofs = Some v2 ->
       forall alloc em t,
         eSexpr alloc em eb eu t v1 =
         eSexpr alloc em eb eu t v2) ->
    m1 ==m m2.

Lemma eSexpr_eq_int_eq_all:
  forall x1 x2 alloc em,
    memval_equiv x1 x2 ->
    eSexpr alloc em eb eu Tint (decode_val Mint8unsigned (x1::nil)) =
    eSexpr alloc em eb eu Tint (decode_val Mint8unsigned (x2::nil)) ->
    forall t,
    eSexpr alloc em eb eu t (decode_val Mint8unsigned (x1::nil)) = eSexpr alloc em eb eu t (decode_val Mint8unsigned (x2::nil)).

Require Import Fappli_IEEE_bits.
Require Import Psatz.


Lemma int_range_int64:
  forall i,
  0 <= Int.unsigned i <= Int64.max_unsigned.

Lemma int_inf_two_p_32:
  forall i,
    0 <= Int.unsigned i < two_power_nat 32.

Lemma decode_val_eq_memval_equiv:
  forall l l'
         (FWMl: Forall wt_memval l)
         (FWMl': Forall wt_memval l')
         (EVAL: forall alloc em ,
                  eSexpr alloc em eb eu Tlong (decode_val Mint64 l) =
                  eSexpr alloc em eb eu Tlong (decode_val Mint64 l')),
    length l = length l' ->
    (length l <= 8)%nat ->
    list_forall2 memval_equiv_tint l l'.

Lemma memval_equiv_tint_sym:
  forall a b,
    memval_equiv_tint a b ->
    memval_equiv_tint b a.

Lemma eSexpr_eq_int_eq_all_int:
  forall v1 v2 x1 x2 alloc em,
    wt_memval x1 -> wt_memval x2 ->
    decode_val Mint8unsigned (x1::nil) = v1 ->
    decode_val Mint8unsigned (x2::nil) = v2 ->
    memval_equiv_tint x1 x2 ->
    eSexpr alloc em eb eu Tint v1 = eSexpr alloc em eb eu Tint v2 ->
    forall t,
    eSexpr alloc em eb eu t v1 = eSexpr alloc em eb eu t v2.

Require Import Setoid.

Theorem storebytes_int64_split:
  forall m b ofs v m' x x0 enc64 encLoword encHiword
         (Hwt: tcheck_expr v = Some Tlong)
         (H : store Mint64 m b ofs v = Some m')
         (Enc64: encode_val Mint64 v = Some enc64)
         (EncLoword: encode_val Mint32 (if big_endian then Val.hiword v else Val.loword v) = Some encLoword)
         (EncHiword: encode_val Mint32 (if big_endian then Val.loword v else Val.hiword v) = Some encHiword)
         (SB : storebytes m b ofs enc64 = Some m')
         (e : storebytes m b ofs encLoword = Some x)
         (e0 : storebytes x b (ofs + 4) encHiword = Some x0),
    m' ==m x0.

Lemma encode_val_wt'':
  forall c v,
    tcheck_expr v = Some ((type_of_chunk c)) ->
    encode_val c v <> None.

 Lemma ex_enc32_64:
   forall (b:bool) v enc64,
     encode_val Mint64 v = Some enc64 ->
     (exists enc32_1, encode_val Mint32 (if b then Val.hiword v else Val.loword v) = Some enc32_1).

Theorem store_int64_split:
  forall m b ofs v m',
  store Mint64 m b ofs v = Some m' ->
  exists m1 m2,
     store Mint32 m b ofs (if Archi.big_endian then Val.hiword v else Val.loword v) = Some m1
     /\ store Mint32 m1 b (ofs + 4) (if Archi.big_endian then Val.loword v else Val.hiword v) = Some m2
     /\ m' ==m m2.

Lemma store_int64_eq_add:
  forall m b i v m',
    store Mint64 m b (Int.unsigned i) v = Some m' ->
    Int.unsigned (Int.add i (Int.repr 4)) = Int.unsigned i + 4.

Properties related to alloc.

Section ALLOC.

Variable m1: Mem.mem.
Variables lo hi: Z.
Variable m2: Mem.mem.
Variable b: block.
Hypothesis ALLOC: alloc m1 lo hi = Some (m2, b).

Theorem mask_alloc_other:
  forall b',
    b <> b' ->
    mask m1 b' = mask m2 b'.

Theorem size_of_block_alloc_other:
  forall b',
    b <> b' ->
    size_of_block m1 b' = size_of_block m2 b'.

Theorem bounds_of_block_alloc_other:
  forall b',
    b <> b' ->
    bounds_of_block m1 b' = bounds_of_block m2 b'.

Theorem nat_mask_alloc_other:
  forall b',
    b <> b' ->
    nat_mask m1 b' = nat_mask m2 b'.

Theorem nextblock_alloc:
  nextblock m2 = Psucc (nextblock m1).

Theorem alloc_result:
  b = nextblock m1.

Theorem valid_block_alloc:
  forall b', valid_block m1 b' -> valid_block m2 b'.

Theorem fresh_block_alloc:
  ~(valid_block m1 b).

Theorem valid_new_block:
  valid_block m2 b.

Local Hint Resolve valid_block_alloc fresh_block_alloc valid_new_block: mem.

Theorem valid_block_alloc_inv:
  forall b', valid_block m2 b' -> b' = b \/ valid_block m1 b'.

Theorem perm_alloc_1:
  forall b' ofs k p, perm m1 b' ofs k p -> perm m2 b' ofs k p.

Theorem perm_alloc_2:
  forall ofs k, 0 <= ofs < hi -> perm m2 b ofs k Freeable.

Theorem perm_alloc_inv:
  forall b' ofs k p,
  perm m2 b' ofs k p ->
  if eq_block b' b then 0 <= ofs < hi else perm m1 b' ofs k p.

Theorem perm_alloc_3:
  forall ofs k p, perm m2 b ofs k p -> 0 <= ofs < hi.

Theorem perm_alloc_4:
  forall b' ofs k p, perm m2 b' ofs k p -> b' <> b -> perm m1 b' ofs k p.

Local Hint Resolve perm_alloc_1 perm_alloc_2 perm_alloc_3 perm_alloc_4: mem.

Theorem valid_access_alloc_other:
  forall chunk b' ofs p,
  valid_access m1 chunk b' ofs p ->
  valid_access m2 chunk b' ofs p.

Theorem valid_access_alloc_same:
  forall chunk ofs,
  0 <= ofs -> ofs + size_chunk chunk <= hi -> (align_chunk chunk | ofs) ->
  valid_access m2 chunk b ofs Freeable.

Local Hint Resolve valid_access_alloc_other valid_access_alloc_same: mem.

Theorem valid_access_alloc_inv:
  forall chunk b' ofs p,
  valid_access m2 chunk b' ofs p ->
  if eq_block b' b
  then 0 <= ofs /\ ofs + size_chunk chunk <= hi /\ (align_chunk chunk | ofs)
  else valid_access m1 chunk b' ofs p.

Theorem load_alloc_unchanged:
  forall chunk b' ofs,
  valid_block m1 b' ->
  load chunk m2 b' ofs = load chunk m1 b' ofs.

Theorem load_alloc_other:
  forall chunk b' ofs v,
  load chunk m1 b' ofs = Some v ->
  load chunk m2 b' ofs = Some v.

Require Import Classical.

Lemma init_block'_eundef:
  forall n b cur o,
    (0 <= o <= n)%nat ->
    nth_error (init_block' n b cur) o =
    Some (Symbolic
            (Eval (Eundef b (Int.add cur (Int.repr (Z.of_nat o)))))
            O).

Lemma in_nth_error:
  forall (A: Type) (vl: list A) (x:A),
    In x vl ->
    exists n, nth_error vl n = Some x.

Lemma nth_error_length:
  forall (A:Type) (l:list A) x v,
    nth_error l x = Some v ->
    (length l > x)%nat.

Lemma init_block'_length:
  forall n b c,
    length (init_block' n b c) = S n.

Lemma init_block_eundef:
  forall n o lo hi c,
    lo <= o -> o + Z.of_nat n <= hi ->
    Forall (fun x => exists y z, x = Symbolic (Eval (Eundef y z)) O) (getN n o (init_block lo hi c)).

Lemma decode_list_undef_is_undef:
  forall l chunk,
    length l = size_chunk_nat chunk ->
    Forall
      (fun x => exists y z, x = Symbolic (Eval (Eundef y z)) 0) l ->
    is_undef (decode_val chunk (rev_if_be l)).

Theorem load_alloc_same:
  forall chunk ofs v,
  load chunk m2 b ofs = Some v ->
  is_undef v.

Theorem load_alloc_same':
  forall chunk ofs,
    0 <= ofs -> ofs + size_chunk chunk <= hi -> (align_chunk chunk | ofs) ->
  exists v, load chunk m2 b ofs = Some v /\
            is_undef v.

Theorem loadbytes_alloc_unchanged:
  forall b' ofs n,
  valid_block m1 b' ->
  loadbytes m2 b' ofs n = loadbytes m1 b' ofs n.

Theorem loadbytes_alloc_same:
  forall n ofs bytes byte,
  loadbytes m2 b ofs n = Some bytes ->
  In byte bytes -> exists y z, byte = Symbolic (Eval (Eundef y z)) O.

End ALLOC.


Lemma bounds_of_block_alloc:
  forall m1 lo hi m1' b1,
    alloc m1 lo hi = Some (m1', b1) ->
    bounds_of_block m1' b1 = (0,hi).


Lemma bounds_of_block_alloc_old:
  forall m1 lo hi m1' b1,
    alloc m1 lo hi = Some (m1', b1) ->
    bounds_of_block m1 b1 = (0,0).


Lemma alloc_mask:
  forall m1 lo hi m1' b1,
    alloc m1 lo hi = Some (m1',b1) ->
    mask m1' b1 = alignment_of_size (hi).

Lemma pos2nat_id:
  forall p,
    Pos.of_nat (S (pred (Pos.to_nat p))) = p.

Lemma size_mem_al_aligned:
  forall l z,
    z = align z 8 ->
    size_mem_al 8 l z = align (size_mem_al 8 l z) 8.

Lemma size_mem_aux_aligned:
  forall l,
    size_mem_aux l = align (size_mem_aux l) 8.

Opaque Pos.of_nat.
Lemma alloc_size_mem':
  forall m1 lo hi m1' b1
         (ALLOC: alloc m1 lo hi = Some (m1', b1)),
    size_mem m1' = size_mem m1 + align (Z.max 0 hi) 8.

Lemma size_mem_alloc_0:
  forall tm1 tm2 sp
         (ALLOC: alloc tm1 0 0 = Some (tm2, sp)),
    size_mem tm1 = size_mem tm2.

Lemma size_mem_aligned:
  forall m z,
    Mem.size_mem m = z ->
    align z 8 = z.

Lemma alloc_lo_0:
  forall m lo hi,
    alloc m lo hi = alloc m 0 hi.

Local Hint Resolve valid_block_alloc fresh_block_alloc valid_new_block: mem.
Local Hint Resolve valid_access_alloc_other valid_access_alloc_same: mem.

Properties related to free.

Theorem range_perm_free:
  forall m1 b lo hi,
  range_perm m1 b lo hi Cur Freeable ->
  { m2: mem | free m1 b lo hi = Some m2 }.

Section FREE.

Variable m1: mem.
Variable bf: block.
Variables lo hi: Z.
Variable m2: mem.
Hypothesis FREE: free m1 bf lo hi = Some m2.

Theorem free_range_perm:
  range_perm m1 bf lo hi Cur Freeable.

Lemma free_result:
  m2 = unchecked_free m1 bf lo hi.

Theorem mask_free:
  forall b',
    bf <> b' ->
    mask m1 b' = mask m2 b'.

Theorem size_of_block_free:
  forall b',
    bf <> b' ->
    size_of_block m1 b' = size_of_block m2 b'.

Theorem nextblock_free:
  nextblock m2 = nextblock m1.

Theorem valid_block_free_1:
  forall b, valid_block m1 b -> valid_block m2 b.

Theorem valid_block_free_2:
  forall b, valid_block m2 b -> valid_block m1 b.

Local Hint Resolve valid_block_free_1 valid_block_free_2: mem.

Theorem perm_free_1:
  forall b ofs k p,
  b <> bf \/ ofs < lo \/ hi <= ofs ->
  perm m1 b ofs k p ->
  perm m2 b ofs k p.

Theorem perm_free_2:
  forall ofs k p, lo <= ofs < hi -> ~ perm m2 bf ofs k p.

Theorem perm_free_3:
  forall b ofs k p,
  perm m2 b ofs k p -> perm m1 b ofs k p.

Theorem perm_free_inv:
  forall b ofs k p,
  perm m1 b ofs k p ->
  (b = bf /\ lo <= ofs < hi) \/ perm m2 b ofs k p.

Theorem valid_access_free_1:
  forall chunk b ofs p,
  valid_access m1 chunk b ofs p ->
  b <> bf \/ lo >= hi \/ ofs + size_chunk chunk <= lo \/ hi <= ofs ->
  valid_access m2 chunk b ofs p.

Theorem valid_access_free_2:
  forall chunk ofs p,
  lo < hi -> ofs + size_chunk chunk > lo -> ofs < hi ->
  ~(valid_access m2 chunk bf ofs p).

Theorem valid_access_free_inv_1:
  forall chunk b ofs p,
  valid_access m2 chunk b ofs p ->
  valid_access m1 chunk b ofs p.

Theorem valid_access_free_inv_2:
  forall chunk ofs p,
  valid_access m2 chunk bf ofs p ->
  lo >= hi \/ ofs + size_chunk chunk <= lo \/ hi <= ofs.

Theorem load_free:
  forall chunk b ofs,
  b <> bf \/ lo >= hi \/ ofs + size_chunk chunk <= lo \/ hi <= ofs ->
  load chunk m2 b ofs = load chunk m1 b ofs.

Theorem load_free_2:
  forall chunk b ofs v,
  load chunk m2 b ofs = Some v -> load chunk m1 b ofs = Some v.

Theorem loadbytes_free:
  forall b ofs n,
  b <> bf \/ lo >= hi \/ ofs + n <= lo \/ hi <= ofs ->
  loadbytes m2 b ofs n = loadbytes m1 b ofs n.

Theorem loadbytes_free_2:
  forall b ofs n bytes,
  loadbytes m2 b ofs n = Some bytes -> loadbytes m1 b ofs n = Some bytes.

End FREE.



Lemma unchecked_free_bounds_of_block:
  forall m b lo hi,
    bounds_of_block (unchecked_free m b lo hi) b =
    match (bounds_of_block m b) with
        (lo',hi') => if zeq lo' lo && zeq hi' hi then (0,0) else (lo',hi')
    end.

Lemma unchecked_free_bounds_of_block_other:
  forall m b lo hi b', b <> b' ->
    bounds_of_block (unchecked_free m b lo hi) b' =
    bounds_of_block m b'.

Lemma unchecked_free_bounds:
  forall m b lo hi b',
    bounds_of_block (unchecked_free m b lo hi) b' =
    if eq_block b' b
    then match bounds_of_block m b with
             (lo',hi') => (if zeq lo' lo && zeq hi' hi
                           then (0,0)
                           else (lo',hi'))
         end
    else bounds_of_block m b'.

Lemma bounds_free:
  forall m m' b b' lo hi ,
  Mem.free m b lo hi = Some m' ->
  b <> b' ->
  Mem.bounds_of_block m' b' = Mem.bounds_of_block m b'.


Lemma bounds_freelist:
  forall fbl m m' b ,
  Mem.free_list m fbl = Some m' ->
  ~ In b (map (fun a => fst (fst a)) fbl) ->
  Mem.bounds_of_block m' b = Mem.bounds_of_block m b.

Lemma range_perm_freelist:
  forall fbl m m' b,
    Mem.free_list m fbl = Some m' ->
    ~ In b (map (fun a => fst (fst a)) fbl) ->
    forall lo hi k p,
      Mem.range_perm m' b lo hi k p ->
      Mem.range_perm m b lo hi k p.


Lemma mask_freelist:
  forall fbl m m' b ,
  Mem.free_list m fbl = Some m' ->
  ~ In b (map (fun a => fst (fst a)) fbl) ->
  Mem.mask m' b = Mem.mask m b.



Lemma contents_free:
  forall m b lo hi m' b',
    Mem.free m b lo hi = Some m' ->
    b <> b' ->
    (Mem.mem_contents m) !! b' = (Mem.mem_contents m') !! b'.


Lemma contents_freelist:
  forall fbl m m' b',
    Mem.free_list m fbl = Some m' ->
    ~ In b' (map (fun a => fst (fst a)) fbl) ->
    (Mem.mem_contents m) !! b' = (Mem.mem_contents m') !! b'.

Lemma freelist_valid:
  forall fbl m m' b,
    Mem.free_list m fbl = Some m' ->
    ~ Mem.valid_block m' b ->
    ~ Mem.valid_block m b.

Lemma free_size_mem:
  forall m1 lo hi m1' b1
         (FREE: free m1 b1 lo hi = Some m1')
         (bnds: bounds_of_block m1 b1 = (lo,hi)),
    size_mem m1 =
    size_mem m1' + align (Z.max 0 (hi-lo)) 8.

Definition size_mem_blocks (bl: list (block * Z * Z)) (z:Z) : Z :=
  fold_left
    (fun acc x =>
       align acc 8 + Z.max 0 (snd x - snd (fst x))) bl z.

Definition list_of_blocks (bl: list (block*Z*Z)) : list (block*Z) :=
  map (fun a : ident * Z * Z => (fst (fst a), snd a - snd (fst a))) bl.

Lemma size_mem_blocks_al:
  forall l z,
    size_mem_al 8 (list_of_blocks l) (align z 8) =
    align (size_mem_blocks l z) 8.

Lemma size_mem_al_rev:
  forall l z,
    size_mem_al 8 l z = size_mem_al 8 (rev l) z.

Lemma size_mem_app:
  forall a b c,
    size_mem_al 8 (a++b) c = size_mem_al 8 b (size_mem_al 8 a c).

Lemma freelist_size_mem:
  forall bl m m'
         (bnds: Forall (fun bbnds : ident*Z*Z => Mem.bounds_of_block m (fst (fst bbnds))
                                                 = (snd (fst bbnds), snd (bbnds)) ) bl)
         (FL: Mem.free_list m bl = Some m')
         (NR: list_norepet (map fst (map fst bl))),
    size_mem m = align (size_mem_blocks bl (size_mem m')) 8.

Local Hint Resolve valid_block_free_1 valid_block_free_2
             perm_free_1 perm_free_2 perm_free_3
             valid_access_free_1 valid_access_free_inv_1: mem.

Properties related to drop_perm

Theorem range_perm_drop_1:
  forall m b lo hi p m', drop_perm m b lo hi p = Some m' -> range_perm m b lo hi Cur Freeable.

Theorem range_perm_drop_2:
  forall m b lo hi p,
  range_perm m b lo hi Cur Freeable -> {m' | drop_perm m b lo hi p = Some m' }.

Section DROP.

Variable m: mem.
Variable b: block.
Variable lo hi: Z.
Variable p: permission.
Variable m': mem.
Hypothesis DROP: drop_perm m b lo hi p = Some m'.

Theorem mask_drop_perm:
  forall b',
    mask m b' = mask m' b'.

Theorem size_of_block_drop_perm:
  forall b',
    size_of_block m b' = size_of_block m' b'.

Theorem nextblock_drop:
  nextblock m' = nextblock m.

Theorem drop_perm_valid_block_1:
  forall b', valid_block m b' -> valid_block m' b'.

Theorem drop_perm_valid_block_2:
  forall b', valid_block m' b' -> valid_block m b'.

Theorem perm_drop_1:
  forall ofs k, lo <= ofs < hi -> perm m' b ofs k p.
  
Theorem perm_drop_2:
  forall ofs k p', lo <= ofs < hi -> perm m' b ofs k p' -> perm_order p p'.

Theorem perm_drop_3:
  forall b' ofs k p', b' <> b \/ ofs < lo \/ hi <= ofs -> perm m b' ofs k p' -> perm m' b' ofs k p'.

Theorem perm_drop_4:
  forall b' ofs k p', perm m' b' ofs k p' -> perm m b' ofs k p'.

Lemma valid_access_drop_1:
  forall chunk b' ofs p',
  b' <> b \/ ofs + size_chunk chunk <= lo \/ hi <= ofs \/ perm_order p p' ->
  valid_access m chunk b' ofs p' -> valid_access m' chunk b' ofs p'.

Lemma valid_access_drop_2:
  forall chunk b' ofs p',
  valid_access m' chunk b' ofs p' -> valid_access m chunk b' ofs p'.

Theorem load_drop:
  forall chunk b' ofs,
  b' <> b \/ ofs + size_chunk chunk <= lo \/ hi <= ofs \/ perm_order p Readable ->
  load chunk m' b' ofs = load chunk m b' ofs.

Theorem loadbytes_drop:
  forall b' ofs n,
  b' <> b \/ ofs + n <= lo \/ hi <= ofs \/ perm_order p Readable ->
  loadbytes m' b' ofs n = loadbytes m b' ofs n.

End DROP.

Generic injections

Record mem_inj (map_undef: Val.partial_undef_allocation) (f: meminj) (m1 m2: mem) : Prop :=
  mk_mem_inj {
    mi_perm:
      forall b1 b2 delta ofs k p,
      f b1 = Some(b2, delta) ->
      perm m1 b1 ofs k p ->
      perm m2 b2 (ofs + delta) k p;
    mi_bounds: (* we mimic mi_perm with bounds instead of permissions. *)
      forall b1 b2 delta ofs,
      f b1 = Some(b2, delta) ->
      in_bound ofs (bounds_of_block m1 b1) ->
      in_bound (ofs+delta) (bounds_of_block m2 b2);
    mi_align':
      forall b b0 z, (* we can now talk of the alignment of blocks *)
        f b = Some (b0, z) ->
        (mask m2 b0 >= mask m1 b)%nat /\
        Z.divide (two_p (Z.of_nat (mask m1 b))) z;
    mi_memval:
      forall b1 ofs b2 delta,
      f b1 = Some(b2, delta) ->
      perm m1 b1 ofs Cur Readable ->
      memval_inject map_undef f
                    (ZMap.get ofs m1.(mem_contents)#b1)
                    (ZMap.get (ofs+delta) m2.(mem_contents)#b2);
    mi_size_mem: (* m2 must be ``smaller'' than m1. *)
      (size_mem m2) <= (size_mem m1)
  }.

Lemma perm_bounds:
  forall m b ofs p k,
    Mem.perm m b ofs k p ->
    in_bound ofs (Mem.bounds_of_block m b).


Lemma load_in_bound:
  forall chunk b ofs m v,
    load chunk m b ofs = Some v ->
    in_bound ofs (Mem.bounds_of_block m b).

Lemma mi_align:
  forall f um m1 m2,
    mem_inj um f m1 m2 ->
  forall (b1 b2: block) delta,
    f b1 = Some (b2, delta) ->
    forall chunk ofs p,
      range_perm m1 b1 ofs (ofs + size_chunk chunk) Max p ->
      (align_chunk chunk | delta).


Lemma perm_inj:
  forall m f m1 m2 b1 ofs k p b2 delta,
  mem_inj m f m1 m2 ->
  perm m1 b1 ofs k p ->
  f b1 = Some(b2, delta) ->
  perm m2 b2 (ofs + delta) k p.

Lemma range_perm_inj:
  forall m f m1 m2 b1 lo hi k p b2 delta,
  mem_inj m f m1 m2 ->
  range_perm m1 b1 lo hi k p ->
  f b1 = Some(b2, delta) ->
  range_perm m2 b2 (lo + delta) (hi + delta) k p.

Lemma valid_access_inj:
  forall m f m1 m2 b1 b2 delta chunk ofs p,
  mem_inj m f m1 m2 ->
  f b1 = Some(b2, delta) ->
  valid_access m1 chunk b1 ofs p ->
  valid_access m2 chunk b2 (ofs + delta) p.


Lemma getN_inj:
  forall m f m1 m2 b1 b2 delta,
  mem_inj m f m1 m2 ->
  f b1 = Some(b2, delta) ->
  forall n ofs,
  range_perm m1 b1 ofs (ofs + Z_of_nat n) Cur Readable ->
  list_forall2 (memval_inject m f)
               (getN n ofs (m1.(mem_contents)#b1))
               (getN n (ofs + delta) (m2.(mem_contents)#b2)).

Lemma load_inj:
  forall m f m1 m2 chunk b1 ofs b2 delta v1,
  mem_inj m f m1 m2 ->
  load chunk m1 b1 ofs = Some v1 ->
  f b1 = Some (b2, delta) ->
  exists v2, load chunk m2 b2 (ofs + delta) = Some v2 /\ Val.expr_inject m f v1 v2.

Lemma loadbytes_inj:
  forall m f m1 m2 len b1 ofs b2 delta bytes1,
  mem_inj m f m1 m2 ->
  loadbytes m1 b1 ofs len = Some bytes1 ->
  f b1 = Some (b2, delta) ->
  exists bytes2, loadbytes m2 b2 (ofs + delta) len = Some bytes2
              /\ list_forall2 (memval_inject m f) bytes1 bytes2.


Lemma setN_inj:
  forall (access: Z -> Prop) delta m f vl1 vl2,
  list_forall2 (memval_inject m f) vl1 vl2 ->
  forall p c1 c2,
  (forall q, access q -> memval_inject m f (ZMap.get q c1) (ZMap.get (q + delta) c2)) ->
  (forall q, access q -> memval_inject m f (ZMap.get q (setN vl1 p c1))
                                       (ZMap.get (q + delta) (setN vl2 (p + delta) c2))).

Definition meminj_no_overlap (f: meminj) (m: mem) : Prop :=
  forall b1 b1' delta1 b2 b2' delta2 ofs1 ofs2,
  b1 <> b2 ->
  f b1 = Some (b1', delta1) ->
  f b2 = Some (b2', delta2) ->
  perm m b1 ofs1 Max Nonempty ->
  perm m b2 ofs2 Max Nonempty ->
  b1' <> b2' \/ ofs1 + delta1 <> ofs2 + delta2.

Definition meminj_no_overlap' (f: meminj) (m: mem) : Prop :=
  forall b1 b1' delta1 b2 b2' delta2 ofs1 ofs2,
  b1 <> b2 ->
  f b1 = Some (b1', delta1) ->
  f b2 = Some (b2', delta2) ->
  in_bound ofs1 (bounds_of_block m b1) ->
  in_bound ofs2 (bounds_of_block m b2) ->
  b1' <> b2' \/ ofs1 + delta1 <> ofs2 + delta2.


Lemma encode_val_tot_inject :
  forall (m : Val.partial_undef_allocation) (f : meminj)
         (v1 v2 : expr_sym) (chunk : memory_chunk),
    Val.expr_inject m f v1 v2 ->
    list_forall2 (memval_inject m f) (encode_val_tot chunk v1)
                 (encode_val_tot chunk v2).

Lemma store_size_block:
  forall chunk m b o v m',
    store chunk m b o v = Some m' ->
    size_block m' = size_block m.


Lemma store_mask:
  forall chunk m b o v m',
    store chunk m b o v = Some m' ->
    mask m' = mask m.


Lemma store_mk_block_list:
  forall chunk m b o v m',
    store chunk m b o v = Some m' ->
    mk_block_list m' = mk_block_list m.

Lemma store_size_mem:
  forall chunk m b o v m',
    store chunk m b o v = Some m' ->
    size_mem m' = size_mem m.




Lemma size_block_store:
  forall chunk m1 b1 ofs v1 n1,
    store chunk m1 b1 ofs v1 = Some n1 ->
    forall b',
      size_block n1 b' = size_block m1 b'.

Lemma store_mapped_inj:
  forall m f chunk m1 b1 ofs v1 n1 m2 b2 delta v2,
  mem_inj m f m1 m2 ->
  store chunk m1 b1 ofs v1 = Some n1 ->
  meminj_no_overlap f m1 ->
  f b1 = Some (b2, delta) ->
  Val.expr_inject m f v1 v2 ->
  exists n2,
    store chunk m2 b2 (ofs + delta) v2 = Some n2
    /\ mem_inj m f n1 n2.

Lemma store_unmapped_inj:
  forall m f chunk m1 b1 ofs v1 n1 m2,
  mem_inj m f m1 m2 ->
  store chunk m1 b1 ofs v1 = Some n1 ->
  f b1 = None ->
  mem_inj m f n1 m2.

Lemma store_outside_inj:
  forall m f m1 m2 chunk b ofs v m2',
  mem_inj m f m1 m2 ->
  (forall b' delta ofs',
    f b' = Some(b, delta) ->
    perm m1 b' ofs' Cur Readable ->
    ofs <= ofs' + delta < ofs + size_chunk chunk -> False) ->
  store chunk m2 b ofs v = Some m2' ->
  mem_inj m f m1 m2'.

Lemma storebytes_size_block:
  forall m b o v m',
    storebytes m b o v = Some m' ->
    size_block m' = size_block m.


Lemma storebytes_mask:
  forall m b o v m',
    storebytes m b o v = Some m' ->
    mask m' = mask m.


Lemma storebytes_mk_block_list:
  forall m b o v m',
    storebytes m b o v = Some m' ->
    mk_block_list m' = mk_block_list m.

Lemma storebytes_size_mem:
  forall m b o v m',
    storebytes m b o v = Some m' ->
    size_mem m' = size_mem m.

Lemma size_block_storebytes:
  forall m1 b1 ofs lv1 n1,
    storebytes m1 b1 ofs lv1 = Some n1 ->
    forall b',
      size_block n1 b' = size_block m1 b'.

Lemma storebytes_mapped_inj:
  forall m f m1 b1 ofs bytes1 n1 m2 b2 delta bytes2,
  mem_inj m f m1 m2 ->
  storebytes m1 b1 ofs bytes1 = Some n1 ->
  meminj_no_overlap f m1 ->
  f b1 = Some (b2, delta) ->
  list_forall2 (memval_inject m f) bytes1 bytes2 ->
  Forall wt_memval bytes2 ->
  exists n2,
    storebytes m2 b2 (ofs + delta) bytes2 = Some n2
    /\ mem_inj m f n1 n2.

Lemma storebytes_unmapped_inj:
  forall m f m1 b1 ofs bytes1 n1 m2,
    mem_inj m f m1 m2 ->
    storebytes m1 b1 ofs bytes1 = Some n1 ->
    f b1 = None ->
    mem_inj m f n1 m2.

Lemma storebytes_outside_inj:
  forall m f m1 m2 b ofs bytes2 m2',
    mem_inj m f m1 m2 ->
    (forall b' delta ofs',
       f b' = Some(b, delta) ->
       perm m1 b' ofs' Cur Readable ->
       ofs <= ofs' + delta < ofs + Z_of_nat (length bytes2) -> False) ->
    storebytes m2 b ofs bytes2 = Some m2' ->
    mem_inj m f m1 m2'.

Lemma storebytes_empty_inj:
  forall m f m1 b1 ofs1 m1' m2 b2 ofs2 m2',
    mem_inj m f m1 m2 ->
    storebytes m1 b1 ofs1 nil = Some m1' ->
    storebytes m2 b2 ofs2 nil = Some m2' ->
    mem_inj m f m1' m2'.



Lemma alloc_size_mem:
  forall m1 lo hi m1' b1
         (ALLOC: alloc m1 lo hi = Some (m1', b1)),
    (size_mem m1) <= (size_mem m1').

Lemma alloc_left_unmapped_inj:
  forall m f m1 m2 lo hi m1' b1,
  mem_inj m f m1 m2 ->
  alloc m1 lo hi = Some (m1', b1) ->
  f b1 = None ->
  mem_inj m f m1' m2.

Definition inj_offset_aligned (delta: Z) (size: Z) : Prop :=
  forall chunk, size_chunk chunk <= size -> (align_chunk chunk | delta).


Lemma get_setN_inside:
  forall vl ofs lo t,
    lo <= ofs < lo + Z.of_nat (length vl) ->
    Some (ZMap.get ofs (setN vl lo t)) = nth_error vl (Z.to_nat (ofs - lo)).

Lemma nth_error_init_block':
  forall n b cur ofs,
    (ofs < n)%nat ->
    nth_error (init_block' n b cur) ofs = Some (Symbolic (Eval (Eundef b (Int.add cur (Int.repr (Z.of_nat ofs))))) O).

Lemma init_block_length:
  forall n b i,
    length (init_block' n b i) = S n.

Lemma get_init_block:
  forall ofs lo hi b,
    lo <= ofs < hi ->
    ZMap.get ofs (init_block lo hi b) = Symbolic (Eval (Eundef b (Int.repr ofs))) O.

Lemma free_size_mem_wrong_bounds:
  forall m1 lo hi m1' b1
         (FREE: free m1 b1 lo hi = Some m1')
         (bnds: bounds_of_block m1 b1 <> (lo,hi)),
    size_mem m1 = size_mem m1'.

Lemma bounds_dec:
  forall m b lo hi,
    {bounds_of_block m b = (lo,hi)} + {bounds_of_block m b <> (lo,hi)}.


Lemma drop_perm_bounds:
  forall m b lo hi p m',
    drop_perm m b lo hi p = Some m' ->
    forall b',
      bounds_of_block m b' = bounds_of_block m' b'.

Lemma nat_mask_drop_perm:
  forall m b lo hi p m' b',
    drop_perm m b lo hi p = Some m' ->
    nat_mask m b' = nat_mask m' b'.


Lemma drop_perm_size_block:
  forall m b o v p m',
    drop_perm m b o v p = Some m' ->
    size_block m' = size_block m.


Lemma drop_perm_mask:
  forall m b o v m' p,
    drop_perm m b o v p = Some m' ->
    mask m' = mask m.


Lemma drop_perm_nextblock:
  forall m b o v m' p,
    drop_perm m b o v p = Some m' ->
    nextblock m' = nextblock m.



Lemma drop_perm_mk_block_list:
  forall (m m' : mem) (b : block) (lo hi : Z) (p : permission),
    drop_perm m b lo hi p = Some m' ->
    mk_block_list m = mk_block_list m'.


Lemma drop_perm_size_mem:
  forall m m' b lo hi p,
    drop_perm m b lo hi p = Some m' ->
    (size_mem m) = (size_mem m').

Lemma size_block_drop:
  forall m b lo hi p m',
    drop_perm m b lo hi p = Some m' ->
    forall b',
      size_block m' b' = size_block m b'.

Lemma drop_unmapped_inj:
  forall m f m1 m2 b lo hi p m1',
  mem_inj m f m1 m2 ->
  drop_perm m1 b lo hi p = Some m1' ->
  f b = None ->
  mem_inj m f m1' m2.

Lemma drop_mapped_inj:
  forall m f m1 m2 b1 b2 delta lo hi p m1',
    mem_inj m f m1 m2 ->
    drop_perm m1 b1 lo hi p = Some m1' ->
    meminj_no_overlap f m1 ->
    f b1 = Some(b2, delta) ->
    exists m2',
      drop_perm m2 b2 (lo + delta) (hi + delta) p = Some m2'
      /\ mem_inj m f m1' m2'.

Lemma drop_outside_inj:
  forall m f m1 m2 b lo hi p m2',
    mem_inj m f m1 m2 ->
    drop_perm m2 b lo hi p = Some m2' ->
    (forall b' delta ofs' k p,
       f b' = Some(b, delta) ->
       perm m1 b' ofs' k p ->
       lo <= ofs' + delta < hi -> False) ->
    mem_inj m f m1 m2'.

Memory extensions

Record extends' (m: Val.partial_undef_allocation) (m1 m2: mem) : Prop :=
  mk_extends {
    mext_next: nextblock m1 = nextblock m2;
    mext_inj: mem_inj m inject_id m1 m2
  }.

Definition extends := extends'.

Theorem extends_refl:
  forall m, extends (fun _ _ => None) m m.

Lemma expr_inject_lessdef:
  forall v1 v2 m,
    Val.expr_inject m inject_id v1 v2 <->
    Val.lessdef m v1 v2.

Theorem load_extends:
  forall m chunk m1 m2 b ofs v1,
  extends m m1 m2 ->
  load chunk m1 b ofs = Some v1 ->
  exists v2, load chunk m2 b ofs = Some v2 /\ Val.lessdef m v1 v2.

Theorem loadbytes_extends:
  forall m m1 m2 b ofs len bytes1,
  extends m m1 m2 ->
  loadbytes m1 b ofs len = Some bytes1 ->
  exists bytes2, loadbytes m2 b ofs len = Some bytes2
              /\ list_forall2 (memval_lessdef m) bytes1 bytes2.

Theorem store_within_extends:
  forall m chunk m1 m2 b ofs v1 m1' v2,
  extends m m1 m2 ->
  store chunk m1 b ofs v1 = Some m1' ->
  Val.lessdef m v1 v2 ->
  exists m2',
     store chunk m2 b ofs v2 = Some m2'
  /\ extends m m1' m2'.

Theorem store_outside_extends:
  forall m chunk m1 m2 b ofs v m2',
  extends m m1 m2 ->
  store chunk m2 b ofs v = Some m2' ->
  (forall ofs', perm m1 b ofs' Cur Readable -> ofs <= ofs' < ofs + size_chunk chunk -> False) ->
  extends m m1 m2'.

Theorem storebytes_within_extends:
  forall m m1 m2 b ofs bytes1 m1' bytes2,
  extends m m1 m2 ->
  storebytes m1 b ofs bytes1 = Some m1' ->
  list_forall2 (memval_lessdef m) bytes1 bytes2 ->
  Forall wt_memval bytes2 ->
  exists m2',
     storebytes m2 b ofs bytes2 = Some m2'
  /\ extends m m1' m2'.

Theorem storebytes_outside_extends:
  forall m m1 m2 b ofs bytes2 m2',
  extends m m1 m2 ->
  storebytes m2 b ofs bytes2 = Some m2' ->
  (forall ofs', perm m1 b ofs' Cur Readable -> ofs <= ofs' < ofs + Z_of_nat (length bytes2) -> False) ->
  extends m m1 m2'.

Memory injections

Opaque Z.add.
Lemma size_sup_one_block:
  forall m bs b z0
         (BS: bs # b = Some (0, z0))
         (PLE: ((Pos.to_nat b) <= m)%nat)
,
    z0 <=
    size_mem_aux
      (mk_block_list_aux (get_size bs) m).

Lemma in_bound_rep:
  forall m b o,
    in_bound o (bounds_of_block m b) ->
    0 <= o < Int.max_unsigned.

Lemma mi_representable':
  forall m f m1 m2
         (MI: mem_inj m f m1 m2),
  forall b b' delta ofs
         (FB: f b = Some(b', delta))
         (PERM: in_bound (Int.unsigned ofs) (bounds_of_block m1 b) \/
                in_bound (Int.unsigned ofs - 1) (bounds_of_block m1 b))
         (delta_pos: delta >= 0),
    0 <= Int.unsigned ofs + delta <= Int.max_unsigned.

Lemma mi_representable:
  forall m f m1 m2
         (MI: mem_inj m f m1 m2),
  forall b b' delta ofs
         (FB: f b = Some(b', delta))
         (PERM: perm m1 b (Int.unsigned ofs) Max Nonempty \/
                perm m1 b (Int.unsigned ofs - 1) Max Nonempty)
         (delta_pos: delta >= 0),
    0 <= Int.unsigned ofs + delta <= Int.max_unsigned.

Record inject' (m:Val.partial_undef_allocation) (f: meminj) (m1 m2: mem) : Prop :=
  mk_inject {
    mi_inj:
      mem_inj m f m1 m2;
    mi_freeblocks:
      forall b, ~(valid_block m1 b) -> f b = None;
    mi_mappedblocks:
      forall b b' delta, f b = Some(b', delta) -> valid_block m2 b';
    mi_freeblocks_undef: (* freeblocks may not be specialized *)
      forall b o,
        ~ valid_block m1 b ->
        m b o = None;
    mi_no_overlap':
      meminj_no_overlap' f m1;
    mi_delta_pos: (* the mi_representable is derivable from mi_delta_pos and mi_inj *)
      forall b b' delta,
        f b = Some(b', delta) ->
        delta >= 0
  }.

Definition inject := inject'.

Local Hint Resolve mi_mappedblocks: mem.

Lemma meminj_no_overlap_impl:
  forall f m,
    Mem.meminj_no_overlap' f m ->
    Mem.meminj_no_overlap f m.

Lemma inject_meminj_no_overlap:
  forall p f m1 m2,
    inject p f m1 m2 ->
    meminj_no_overlap f m1.


Theorem valid_block_inject_1:
  forall m f m1 m2 b1 b2 delta,
  f b1 = Some(b2, delta) ->
  inject m f m1 m2 ->
  valid_block m1 b1.

Theorem valid_block_inject_2:
  forall m f m1 m2 b1 b2 delta,
  f b1 = Some(b2, delta) ->
  inject m f m1 m2 ->
  valid_block m2 b2.

Local Hint Resolve valid_block_inject_1 valid_block_inject_2: mem.

Theorem perm_inject:
  forall m f m1 m2 b1 b2 delta ofs k p,
  f b1 = Some(b2, delta) ->
  inject m f m1 m2 ->
  perm m1 b1 ofs k p -> perm m2 b2 (ofs + delta) k p.

Theorem range_perm_inject:
  forall m f m1 m2 b1 b2 delta lo hi k p,
  f b1 = Some(b2, delta) ->
  inject m f m1 m2 ->
  range_perm m1 b1 lo hi k p -> range_perm m2 b2 (lo + delta) (hi + delta) k p.

Theorem valid_access_inject:
  forall m f m1 m2 chunk b1 ofs b2 delta p,
  f b1 = Some(b2, delta) ->
  inject m f m1 m2 ->
  valid_access m1 chunk b1 ofs p ->
  valid_access m2 chunk b2 (ofs + delta) p.

Theorem valid_pointer_inject:
  forall m f m1 m2 b1 ofs b2 delta,
  f b1 = Some(b2, delta) ->
  inject m f m1 m2 ->
  valid_pointer m1 b1 ofs = true ->
  valid_pointer m2 b2 (ofs + delta) = true.

Theorem weak_valid_pointer_inject:
  forall m f m1 m2 b1 ofs b2 delta,
  f b1 = Some(b2, delta) ->
  inject m f m1 m2 ->
  weak_valid_pointer m1 b1 ofs = true ->
  weak_valid_pointer m2 b2 (ofs + delta) = true.


Lemma address_inject:
  forall m f m1 m2 b1 ofs1 b2 delta p,
  inject m f m1 m2 ->
  perm m1 b1 (Int.unsigned ofs1) Cur p ->
  f b1 = Some (b2, delta) ->
  Int.unsigned (Int.add ofs1 (Int.repr delta)) = Int.unsigned ofs1 + delta.

Lemma address_inject':
  forall m f m1 m2 chunk b1 ofs1 b2 delta,
  inject m f m1 m2 ->
  valid_access m1 chunk b1 (Int.unsigned ofs1) Nonempty ->
  f b1 = Some (b2, delta) ->
  Int.unsigned (Int.add ofs1 (Int.repr delta)) = Int.unsigned ofs1 + delta.

Theorem weak_valid_pointer_inject_no_overflow:
  forall m f m1 m2 b ofs b' delta,
  inject m f m1 m2 ->
  weak_valid_pointer m1 b (Int.unsigned ofs) = true ->
  f b = Some(b', delta) ->
  0 <= Int.unsigned ofs + Int.unsigned (Int.repr delta) <= Int.max_unsigned.

Theorem valid_pointer_inject_no_overflow:
  forall m f m1 m2 b ofs b' delta,
  inject m f m1 m2 ->
  valid_pointer m1 b (Int.unsigned ofs) = true ->
  f b = Some(b', delta) ->
  0 <= Int.unsigned ofs + Int.unsigned (Int.repr delta) <= Int.max_unsigned.

Theorem valid_pointer_inject_val:
  forall m f m1 m2 b ofs b' ofs',
  inject m f m1 m2 ->
  valid_pointer m1 b (Int.unsigned ofs) = true ->
  Val.expr_inject m f (Eval (Eptr b ofs)) (Eval (Eptr b' ofs')) ->
  valid_pointer m2 b' (Int.unsigned ofs') = true.

Theorem weak_valid_pointer_inject_val:
  forall m f m1 m2 b ofs b' ofs',
  inject m f m1 m2 ->
  weak_valid_pointer m1 b (Int.unsigned ofs) = true ->
  Val.expr_inject m f (Eval (Eptr b ofs)) (Eval (Eptr b' ofs')) ->
  weak_valid_pointer m2 b' (Int.unsigned ofs') = true.

Theorem inject_no_overlap:
  forall m f m1 m2 b1 b2 b1' b2' delta1 delta2 ofs1 ofs2,
  inject m f m1 m2 ->
  b1 <> b2 ->
  f b1 = Some (b1', delta1) ->
  f b2 = Some (b2', delta2) ->
  perm m1 b1 ofs1 Max Nonempty ->
  perm m1 b2 ofs2 Max Nonempty ->
  b1' <> b2' \/ ofs1 + delta1 <> ofs2 + delta2.

Theorem different_pointers_inject:
  forall p f m m' b1 ofs1 b2 ofs2 b1' delta1 b2' delta2,
  inject p f m m' ->
  b1 <> b2 ->
  valid_pointer m b1 (Int.unsigned ofs1) = true ->
  valid_pointer m b2 (Int.unsigned ofs2) = true ->
  f b1 = Some (b1', delta1) ->
  f b2 = Some (b2', delta2) ->
  b1' <> b2' \/
  Int.unsigned (Int.add ofs1 (Int.repr delta1)) <>
  Int.unsigned (Int.add ofs2 (Int.repr delta2)).

Require Intv.

Theorem disjoint_or_equal_inject:
  forall p f m m' b1 b1' delta1 b2 b2' delta2 ofs1 ofs2 sz,
  inject p f m m' ->
  f b1 = Some(b1', delta1) ->
  f b2 = Some(b2', delta2) ->
  range_perm m b1 ofs1 (ofs1 + sz) Max Nonempty ->
  range_perm m b2 ofs2 (ofs2 + sz) Max Nonempty ->
  sz > 0 ->
  b1 <> b2 \/ ofs1 = ofs2 \/ ofs1 + sz <= ofs2 \/ ofs2 + sz <= ofs1 ->
  b1' <> b2' \/ ofs1 + delta1 = ofs2 + delta2
             \/ ofs1 + delta1 + sz <= ofs2 + delta2
             \/ ofs2 + delta2 + sz <= ofs1 + delta1.

Theorem aligned_area_inject:
  forall p f m m' b ofs al sz b' delta,
  inject p f m m' ->
  al = 1 \/ al = 2 \/ al = 4 \/ al = 8 -> sz > 0 ->
  (al | sz) ->
  range_perm m b ofs (ofs + sz) Cur Nonempty ->
  (al | ofs) ->
  f b = Some(b', delta) ->
  (al | ofs + delta).


Theorem load_inject:
  forall p f m1 m2 chunk b1 ofs b2 delta v1,
  inject p f m1 m2 ->
  load chunk m1 b1 ofs = Some v1 ->
  f b1 = Some (b2, delta) ->
  exists v2, load chunk m2 b2 (ofs + delta) = Some v2 /\ Val.expr_inject p f v1 v2.
 
Definition id_pua : Val.partial_undef_allocation :=
  fun _ _ => None.

Lemma bounds_of_block_alloc_eq:
  forall m lo hi b m',
    alloc m lo hi = Some (m',b) ->
    forall b',
      bounds_of_block m' b' = if eq_block b' b
                              then (0,hi)
                              else if plt b' (nextblock m')
                                   then bounds_of_block m b'
                                   else (0,0).


Lemma alloc_0_inject:
  forall tm1 tm2 sp f1 m2,
    alloc tm1 0 0 = Some (tm2, sp) ->
    inject id_pua f1 m2 tm1 ->
    inject id_pua f1 m2 tm2.


Lemma alignment_inj:
  forall i i0 i1 i2
         (A : Int.and i (Int.not (Int.repr (two_power_nat i0 - 1))) = i)
         (B : Int.and (Int.not (Int.repr (two_power_nat i0 - 1)))
                      (Int.not (Int.repr (two_power_nat i1 - 1))) =
              Int.not (Int.repr (two_power_nat i0 - 1)))
         (C : Int.and i2 (Int.not (Int.repr (two_power_nat i1 - 1))) = i2),
   Int.and (Int.add i i2) (Int.not (Int.repr (two_power_nat i1 - 1))) =
   Int.add i i2.


Lemma int_add_rew:
  forall a b,
    Int.repr (Int.unsigned a + Int.unsigned b) = Int.add a b.

Lemma two_p_abs:
  forall n n0,
    (n >= n0)%nat ->
    Int.and (Int.not (Int.repr (two_power_nat n - 1)))
            (Int.not (Int.repr (two_power_nat n0 - 1))) =
    Int.not (Int.repr (two_power_nat n - 1)).

Definition all_blocks_injected (f:meminj) (m:mem) : Prop :=
  forall b lo hi,
    bounds_of_block m b = (lo,hi) ->
    hi - lo <> 0 ->
    f b <> None.

Lemma abi_in_bound:
  forall f m b o
         (ABI : all_blocks_injected f m)
         (IB : in_bound (Int.unsigned o) (bounds_of_block m b))
         (Fnone : f b = None),
    False.


Lemma int_add_diff_range_plus:
  forall o z o' z0 i,
    o + z <> o' + z0 ->
    z0 >= 0 -> z >= 0 ->
    0 <= o' + z0 <= Int.modulus - 1 ->
    0 <= o + z <= Int.modulus - 1 ->
    (( i + z) mod Int.modulus + o) mod Int.modulus <>
    (( i + z0) mod Int.modulus + o') mod Int.modulus.

Lemma compat_inject:
  forall p f m m'
         (ABI: all_blocks_injected f m)
         (INJ : inject p f m m')
         al
         (COMP: compat al (bounds_of_block m') (nat_mask m')),
    compat (Val.inj_alloc al f) (bounds_of_block m) (nat_mask m).

Lemma norm_inject:
  forall p f m m' e1 v1 e1'
         (ABI: all_blocks_injected f m)
         (INJ: inject p f m m')
         (EI: Val.expr_inject p f e1 e1')
         (NU: v1 <> Vundef),
  forall r (MN:mem_norm m e1 r = v1),
    mem_norm m' e1' r <> Vundef /\
    Val.expr_inject p f (Eval (sval_of_val v1)) (Eval (sval_of_val (mem_norm m' e1' r))).

Lemma expr_val_inject:
  forall p f v1 v2,
    Val.expr_inject p f (Eval (sval_of_val v1))
                    (Eval (sval_of_val v2)) ->
    val_inject f v1 v2.
  
Lemma norm_inject_val':
  forall p f m m' e1 e1' r
         (ABI: all_blocks_injected f m)
         (INJ: inject p f m m')
         (EI: Val.expr_inject p f e1 e1'),
    val_inject f (mem_norm m e1 r) (mem_norm m' e1' r).

Theorem loadv_inject:
  forall p f m1 m2 chunk a1 a2 v1
    (ABI: all_blocks_injected f m1),
    inject p f m1 m2 ->
    loadv chunk m1 a1 = Some v1 ->
    Val.expr_inject p f a1 a2 ->
    exists v2, loadv chunk m2 a2 = Some v2 /\ Val.expr_inject p f v1 v2.

Theorem loadbytes_inject:
  forall p f m1 m2 b1 ofs len b2 delta bytes1,
  inject p f m1 m2 ->
  loadbytes m1 b1 ofs len = Some bytes1 ->
  f b1 = Some (b2, delta) ->
  exists bytes2, loadbytes m2 b2 (ofs + delta) len = Some bytes2
              /\ list_forall2 (memval_inject p f) bytes1 bytes2.


Theorem store_mapped_inject:
  forall p f chunk m1 b1 ofs v1 n1 m2 b2 delta v2,
  inject p f m1 m2 ->
  store chunk m1 b1 ofs v1 = Some n1 ->
  f b1 = Some (b2, delta) ->
  Val.expr_inject p f v1 v2 ->
  exists n2,
    store chunk m2 b2 (ofs + delta) v2 = Some n2
    /\ inject p f n1 n2.

Theorem storev_inject:
  forall p f m1 m2 chunk a1 a2 b1 b2 v1
    (ABI: all_blocks_injected f m1),
    inject p f m1 m2 ->
    storev chunk m1 a1 b1 = Some v1 ->
    Val.expr_inject p f a1 a2 ->
    Val.expr_inject p f b1 b2 ->
    exists v2, storev chunk m2 a2 b2 = Some v2 /\ inject p f v1 v2.

Theorem store_unmapped_inject:
  forall p f chunk m1 b1 ofs v1 n1 m2,
    inject p f m1 m2 ->
    store chunk m1 b1 ofs v1 = Some n1 ->
    f b1 = None ->
    inject p f n1 m2.

Theorem store_outside_inject:
  forall p f m1 m2 chunk b ofs v m2',
  inject p f m1 m2 ->
  (forall b' delta ofs',
    f b' = Some(b, delta) ->
    perm m1 b' ofs' Cur Readable ->
    ofs <= ofs' + delta < ofs + size_chunk chunk -> False) ->
  store chunk m2 b ofs v = Some m2' ->
  inject p f m1 m2'.

Theorem storev_mapped_inject:
  forall p f chunk m1 a1 v1 n1 m2 a2 v2
         (ABI: all_blocks_injected f m1),
  inject p f m1 m2 ->
  storev chunk m1 a1 v1 = Some n1 ->
  Val.expr_inject p f a1 a2 ->
  Val.expr_inject p f v1 v2 ->
  exists n2,
    storev chunk m2 a2 v2 = Some n2 /\ inject p f n1 n2.

Theorem storebytes_mapped_inject:
  forall p f m1 b1 ofs bytes1 n1 m2 b2 delta bytes2,
  inject p f m1 m2 ->
  storebytes m1 b1 ofs bytes1 = Some n1 ->
  f b1 = Some (b2, delta) ->
  list_forall2 (memval_inject p f) bytes1 bytes2 ->
  Forall wt_memval bytes2 ->
  exists n2,
    storebytes m2 b2 (ofs + delta) bytes2 = Some n2
    /\ inject p f n1 n2.

Theorem storebytes_unmapped_inject:
  forall p f m1 b1 ofs bytes1 n1 m2,
  inject p f m1 m2 ->
  storebytes m1 b1 ofs bytes1 = Some n1 ->
  f b1 = None ->
  inject p f n1 m2.

Theorem storebytes_outside_inject:
  forall p f m1 m2 b ofs bytes2 m2',
  inject p f m1 m2 ->
  (forall b' delta ofs',
    f b' = Some(b, delta) ->
    perm m1 b' ofs' Cur Readable ->
    ofs <= ofs' + delta < ofs + Z_of_nat (length bytes2) -> False) ->
  storebytes m2 b ofs bytes2 = Some m2' ->
  inject p f m1 m2'.

Theorem storebytes_empty_inject:
  forall p f m1 b1 ofs1 m1' m2 b2 ofs2 m2',
  inject p f m1 m2 ->
  storebytes m1 b1 ofs1 nil = Some m1' ->
  storebytes m2 b2 ofs2 nil = Some m2' ->
  inject p f m1' m2'.


Theorem alloc_left_unmapped_inject:
  forall p f m1 m2 lo hi m1' b1,
  inject p f m1 m2 ->
  alloc m1 lo hi = Some (m1', b1) ->
  exists f',
     inject p f' m1' m2
  /\ inject_incr f f'
  /\ f' b1 = None
  /\ (forall b, b <> b1 -> f' b = f b).

Lemma alignment_of_size_inj:
  forall n n0,
    Z.max 0 n >= Z.max 0 n0 ->
    (alignment_of_size n >= alignment_of_size n0)%nat.

Lemma ioa_div:
  forall sz delta,
    inj_offset_aligned delta sz ->
    (two_p (Z.of_nat (alignment_of_size sz)) | delta).

Lemma alloc_left_mapped_inj:
  forall m f m1 m2 lo hi m1' b1 b2 delta
         (Hmi: forall ofs b2 delta,
                 lo <= ofs < hi ->
                 f (nextblock m1) = Some (b2, delta) ->
                 memval_inject
                   m f
                   (Symbolic (Eval (Eundef (nextblock m1) (Int.repr ofs))) 0)
                   (ZMap.get (ofs + delta) (mem_contents m2) # b2))
         (msk_b2: (mem_mask m2) # b2 <> None),
    mem_inj m f m1 m2 ->
    lo = 0 ->
    alloc m1 lo hi = Some (m1', b1) ->
    valid_block m2 b2 ->
    inj_offset_aligned delta 8 ->
    (forall ofs k p, lo <= ofs < hi -> perm m2 b2 (ofs + delta) k p) ->
    (forall ofs, lo <= ofs < hi -> in_bound (ofs+delta) (bounds_of_block m2 b2)) ->
    f b1 = Some(b2, delta) ->
    mem_inj m f m1' m2.

Theorem memval_inject_alloc_left_mapped:
  forall p f m1 m2 lo hi m1' b1 b2 delta
         (ALLOC: alloc m1 lo hi = Some (m1', b1))
         (VB: valid_block m2 b2)
         (RANGE: 0 <= delta <= Int.max_unsigned)
         (INJ: mem_inj p f m1 m2)
         (VB_none: forall b : block, ~ valid_block m1 b -> f b = None)
         (vb_none: forall b o, ~ valid_block m1 b -> p b o = None)
         (INJ': mem_inj p (fun b : positive => if eq_block b b1 then Some (b2, delta) else f b) m1 m2)
         (ofs : Z) (b0 : block) (delta0 : Z)
         (RANGE': lo <= ofs < hi)
         (f'eq: (if eq_block (nextblock m1) b1 then Some (b2, delta) else f (nextblock m1)) =
                Some (b0, delta0))
         (contents: forall ofs,
                      lo <= ofs < hi ->
                      ZMap.get (ofs + delta) (mem_contents m2) # b0 =
                      Symbolic (Eval (Eundef b0 (Int.repr (ofs+delta)))) O),
    memval_inject p
                  (fun b : block => if eq_block b b1 then Some (b2, delta) else f b)
                  (Symbolic (Eval (Eundef (nextblock m1) (Int.repr ofs))) 0)
                  (ZMap.get (ofs + delta0) (mem_contents m2) # b0).

Lemma mask_alloc_old:
  forall m1 lo hi m1' b1,
    alloc m1 lo hi = Some (m1', b1) ->
    mask m1 b1 = O.

Lemma nat_mask_alloc_old:
  forall m1 lo hi m1' b1,
    alloc m1 lo hi = Some (m1', b1) ->
    nat_mask m1 b1 = Int.mone.

Theorem alloc_left_mapped_inject_aux_mem_inj:
  forall p f m1 m2 lo hi m1' b1 b2 delta
         (INJ: inject p f m1 m2)
         (lo0: lo = 0)
         (ALLOC: alloc m1 lo hi = Some (m1', b1))
         (VB: valid_block m2 b2)
         (MSKb2: (mem_mask m2) # b2 <> None)
         (DMU: 0 <= delta <= Int.max_unsigned)
         (PERM_MU:
            forall ofs k p, perm m2 b2 ofs k p -> delta = 0 \/ 0 <= ofs < Int.max_unsigned)
         (BND_PERM: forall ofs k p, lo <= ofs < hi -> perm m2 b2 (ofs + delta) k p)
         (IOA: inj_offset_aligned delta 8)
         (space_free:
            forall b delta' ofs k p,
              f b = Some (b2, delta') ->
              perm m1 b ofs k p ->
              lo + delta <= ofs + delta' < hi + delta ->
              False)
         (Hmi:
            forall ofs, lo <= ofs < hi ->
                        ZMap.get (ofs + delta) (mem_contents m2) # b2 =
                        Symbolic (Eval (Eundef b2 (Int.repr (ofs+delta)))) O),
    mem_inj p (fun b => if eq_block b b1 then Some(b2, delta) else f b) m1' m2.

Theorem alloc_left_mapped_inject_aux:
  forall p f m1 m2 lo hi m1' b1 b2 delta
         (INJ: inject p f m1 m2)
         (lo0: lo = 0)
         (ALLOC: alloc m1 lo hi = Some (m1', b1))
         (VB: valid_block m2 b2)
         (MSKb2: (mem_mask m2) #b2 <> None)
         (DMU: 0 <= delta <= Int.max_unsigned)
         (PERM_MU:
            forall ofs k p, perm m2 b2 ofs k p -> delta = 0 \/ 0 <= ofs < Int.max_unsigned)
         (BND_PERM:
            forall ofs k p, lo <= ofs < hi -> perm m2 b2 (ofs + delta) k p)
           (IOA: inj_offset_aligned delta 8)
           (space_free_perm:
              forall b delta' ofs k p,
                f b = Some (b2, delta') ->
                perm m1 b ofs k p ->
                lo + delta <= ofs + delta' < hi + delta -> False)
           (space_free_bnd:
              forall b delta' ofs,
                f b = Some (b2, delta') ->
                in_bound ofs (bounds_of_block m1 b) ->
                lo + delta <= ofs + delta' < hi + delta -> False)
           (Hmi:
              forall ofs : Z,
                lo <= ofs < hi ->
                ZMap.get (ofs + delta) (mem_contents m2) # b2 =
                Symbolic (Eval (Eundef b2 (Int.repr (ofs + delta)))) 0)
           (BND_MU:
              forall ofs, in_bound ofs (bounds_of_block m2 b2) ->
                          delta = 0 \/ 0 <= ofs < Int.max_unsigned)
           (BND_bound:
              forall ofs, lo <= ofs < hi -> in_bound (ofs+delta) (bounds_of_block m2 b2)),
  inject p (fun b => if eq_block b b1 then Some(b2, delta) else f b) m1' m2.

Definition block_undefs lo hi delta m2 b2 :=
  forall ofs : Z,
    lo <= ofs < hi ->
    ZMap.get (ofs + delta) (mem_contents m2) # b2 =
    Symbolic (Eval (Eundef b2 (Int.repr (ofs + delta)))) 0.

Theorem alloc_left_mapped_inject:
  forall p f m1 m2 lo hi m1' b1 b2 delta,
  inject p f m1 m2 ->
  lo = 0 ->
  alloc m1 lo hi = Some (m1', b1) ->
  valid_block m2 b2 ->
  (mem_mask m2) # b2 <> None ->
  0 <= delta <= Int.max_unsigned ->
  (forall ofs, in_bound ofs (bounds_of_block m2 b2) -> delta = 0 \/ 0 <= ofs < Int.max_unsigned) ->
  (forall ofs k p, lo <= ofs < hi -> perm m2 b2 (ofs + delta) k p) ->
  inj_offset_aligned delta 8 ->
  (forall b delta' ofs,
   f b = Some (b2, delta') ->
   in_bound ofs (bounds_of_block m1 b) ->
   lo + delta <= ofs + delta' < hi + delta -> False) ->
  block_undefs lo (Z.max 0 hi) delta m2 b2 ->
  exists f',
     inject p f' m1' m2
  /\ inject_incr f f'
  /\ f' b1 = Some(b2, delta)
  /\ (forall b, b <> b1 -> f' b = f b).

Lemma mem_alloc_block_undefs:
  forall m lo hi m' b,
    alloc m lo hi = Some (m', b) ->
    block_undefs 0 hi 0 m' b.

Lemma alloc_contents_old:
  forall m1 lo hi m2 b
         (ALLOC: alloc m1 lo hi = Some (m2, b))
         b'
         (DIFF: b' <> b),
    (mem_contents m2) # b' = (mem_contents m1) # b'.
 
Lemma align_distr:
  forall z z1,
    align z 8 + align z1 8 = align (align z 8 + z1) 8.

Opaque align.

Lemma mk_block_list_aux_eq:
  forall lo hi b' bs ,
    mk_block_list_aux
      (fun b0 : block =>
         get_size
           (PMap.set b' (Some (lo, hi)) bs)
           b0)
      (pred (Pos.to_nat (Pos.succ b'))) =
    (b',hi-lo):: mk_block_list_aux
               (fun b0 => get_size bs b0)
               (pred (Pos.to_nat b')).

Lemma size_mem_inf_add_larger:
  forall (m1 m2 : mem) (p1 p2: block) (lo hi lo' hi': Z),
    hi >= 0 -> hi' >= 0 -> hi >= lo ->
    hi' - lo' >= hi - lo ->
    size_mem_aux
      (mk_block_list_aux (size_block m2)
                         (pred (Pos.to_nat p2))) <=
   size_mem_aux
     (mk_block_list_aux (size_block m1)
                        (pred (Pos.to_nat p1))) ->
   size_mem_aux
     (mk_block_list_aux
        (get_size
           (PMap.set (p2) (Some (lo, hi)) (mem_blocksize m2)))
        (Pos.to_nat p2)) <=
   size_mem_aux
     (mk_block_list_aux
        (get_size
           (PMap.set (p1) (Some (lo', hi')) (mem_blocksize m1)))
        (Pos.to_nat p1)).

Lemma alloc_less_ok:
  forall m z z0,
    Mem.alloc m 0 z = None ->
    Mem.alloc m 0 z0 <> None ->
    0 <= z <= z0 ->
    False.

Lemma size_mem_inf_add':
  forall (m1 m2 : mem) (p1 p2: block) (lo hi : Z),
    hi >= 0 -> lo <= hi ->
    size_mem_aux
      (mk_block_list_aux (size_block m2)
                         (pred (Pos.to_nat p2))) <=
   size_mem_aux
     (mk_block_list_aux (size_block m1)
                        (pred (Pos.to_nat p1))) ->
   size_mem_aux
     (mk_block_list_aux
        (get_size
           (PMap.set (p2) (Some (lo, hi)) (mem_blocksize m2)))
         (Pos.to_nat (p2))) <=
   size_mem_aux
     (mk_block_list_aux
        (get_size
           (PMap.set (p1) (Some (lo, hi)) (mem_blocksize m1)))
        (Pos.to_nat (p1))).

Lemma alloc_same_inj_inf:
  forall m1 m2 lo hi m1' b1
         (ALLOC : alloc m1 lo hi = Some (m1', b1))
         m b
         (ALLOC2 : alloc m2 lo hi = Some (m, b))
         (inf : size_mem m2 <= size_mem m1)
         (Hpos: hi >= 0),
    size_mem m <= size_mem m1'.

Lemma alloc_parallel_inj:
  forall p f m1 m2 lo hi m1' b1 m b
         (INJ: inject p f m1 m2)
         (ALLOC: alloc m1 lo hi = Some (m1', b1))
         (HIpos : hi >= 0)
         (ALLOC2 : alloc m2 lo hi = Some (m, b)),
    mem_inj p (fun b0 : positive => if peq b0 b1 then Some (b, 0) else f b0) m1' m.

Lemma less_alloc_less:
  forall m1 m2 hi
         (ALLOC : alloc m1 0 hi <> None)
         (HIpos : hi >= 0)
         (NALLOC : alloc m2 0 hi = None)
         (sizes: size_mem m2 <= size_mem m1),
    False.

Theorem alloc_parallel_inject:
  forall p f m1 m2 lo1 hi1 m1' b1 (ORD: lo1 <= hi1)
         (INJ: inject p f m1 m2)
         (ALLOC: alloc m1 lo1 hi1 = Some (m1', b1))
         (HIpos: hi1 >= 0),
  exists f', exists m2', exists b2,
                           alloc m2 lo1 hi1 = Some (m2', b2)
                           /\ inject p f' m1' m2'
                           /\ inject_incr f f'
                           /\ f' b1 = Some(b2, 0)
                           /\ (forall b, b <> b1 -> f' b = f b).


Lemma free_bounds:
  forall m b lo hi m' b',
    free m b lo hi = Some m' ->
    bounds_of_block m' b' = bounds_of_block m b' \/ bounds_of_block m' b' = (0,0).

Lemma free_mask:
  forall m b lo hi m' b',
    free m b lo hi = Some m' ->
    mask m' b' = mask m b' \/ (mask m' b' = O /\
                               bounds_of_block m b' = (lo,hi)).

Lemma perm_free_list:
  forall l m m' b ofs k p,
  free_list m l = Some m' ->
  perm m' b ofs k p ->
  perm m b ofs k p /\
  (forall lo hi, In (b, lo, hi) l -> lo <= ofs < hi -> False).

Lemma drop_outside_inject:
  forall m f m1 m2 b lo hi p m2',
    inject m f m1 m2 ->
    drop_perm m2 b lo hi p = Some m2' ->
    (forall b' delta ofs k p,
       f b' = Some(b, delta) ->
       perm m1 b' ofs k p -> lo <= ofs + delta < hi -> False) ->
    inject m f m1 m2'.


Lemma mem_inj_compose:
  forall p p' f f' m1 m2 m3
         (Ffree: forall b, ~ valid_block m1 b -> match f b with
                                                     | None => True
                                                     | Some (b',delta) => ~ valid_block m2 b'
                                                 end),
  mem_inj p f m1 m2 -> mem_inj p' f' m2 m3 -> mem_inj (Val.inj_m f p p') (compose_meminj f f') m1 m3.

Theorem inject_compose:
  forall p p' f f' m1 m2 m3,
  inject p f m1 m2 -> inject p' f' m2 m3 ->
  inject (Val.inj_m f p p') (compose_meminj f f') m1 m3.

Lemma val_lessdef_inject_compose:
  forall p p' f v1 v2 v3,
  Val.lessdef p v1 v2 -> Val.expr_inject p' f v2 v3 -> Val.expr_inject (Val.inj_m inject_id p p') f v1 v3.

Lemma val_inject_lessdef_compose:
  forall p p' f v1 v2 v3,
    Val.expr_inject p f v1 v2 ->
    Val.lessdef p' v2 v3 ->
    Val.expr_inject (Val.inj_m f p p') f v1 v3.


Definition flat_inj (thr: block) : meminj :=
  fun (b: block) => if plt b thr then Some(b, 0) else None.

Definition inject_neutral (thr: block) (m: mem) :=
  mem_inj id_pua (flat_inj thr) m m.

Remark flat_inj_no_overlap:
  forall thr m, meminj_no_overlap (flat_inj thr) m.

Theorem neutral_inject:
  forall m, inject_neutral (nextblock m) m ->
            inject id_pua (flat_inj (nextblock m)) m m.

Theorem empty_inject_neutral:
  forall thr, inject_neutral thr empty.

Theorem alloc_inject_neutral:
  forall thr m lo hi b m'
         (ALLOC: alloc m lo hi = Some (m', b))
         (LO: lo <= 0)
         (INJ: inject_neutral thr m)
         (LT: Plt (nextblock m) thr),
    inject_neutral thr m'.

Theorem store_inject_neutral:
  forall chunk m b ofs v m' thr,
  store chunk m b ofs v = Some m' ->
  inject_neutral thr m ->
  Plt b thr ->
  Val.expr_inject id_pua (flat_inj thr) v v ->
  inject_neutral thr m'.

Theorem drop_inject_neutral:
  forall m b lo hi p m' thr,
  drop_perm m b lo hi p = Some m' ->
  inject_neutral thr m ->
  Plt b thr ->
  inject_neutral thr m'.

Invariance properties between two memory states

Section UNCHANGED_ON.

Variable P: block -> Z -> Prop.

Record unchanged_on (m_before m_after: mem) : Prop := mk_unchanged_on {
  unchanged_on_perm:
    forall b ofs k p,
    P b ofs -> valid_block m_before b ->
    (perm m_before b ofs k p <-> perm m_after b ofs k p);
  unchanged_on_contents:
    forall b ofs,
    P b ofs -> perm m_before b ofs Cur Readable ->
    ZMap.get ofs (PMap.get b m_after.(mem_contents)) =
    ZMap.get ofs (PMap.get b m_before.(mem_contents))
}.

Lemma unchanged_on_refl:
  forall m, unchanged_on m m.

Lemma perm_unchanged_on:
  forall m m' b ofs k p,
  unchanged_on m m' -> P b ofs -> valid_block m b ->
  perm m b ofs k p -> perm m' b ofs k p.

Lemma perm_unchanged_on_2:
  forall m m' b ofs k p,
  unchanged_on m m' -> P b ofs -> valid_block m b ->
  perm m' b ofs k p -> perm m b ofs k p.

Lemma loadbytes_unchanged_on_1:
  forall m m' b ofs n,
  unchanged_on m m' ->
  valid_block m b ->
  (forall i, ofs <= i < ofs + n -> P b i) ->
  loadbytes m' b ofs n = loadbytes m b ofs n.

Lemma loadbytes_unchanged_on:
  forall m m' b ofs n bytes,
  unchanged_on m m' ->
  (forall i, ofs <= i < ofs + n -> P b i) ->
  loadbytes m b ofs n = Some bytes ->
  loadbytes m' b ofs n = Some bytes.

Lemma load_unchanged_on_1:
  forall m m' chunk b ofs,
  unchanged_on m m' ->
  valid_block m b ->
  (forall i, ofs <= i < ofs + size_chunk chunk -> P b i) ->
  load chunk m' b ofs = load chunk m b ofs.

Lemma load_unchanged_on:
  forall m m' chunk b ofs v,
  unchanged_on m m' ->
  (forall i, ofs <= i < ofs + size_chunk chunk -> P b i) ->
  load chunk m b ofs = Some v ->
  load chunk m' b ofs = Some v.

Lemma store_unchanged_on:
  forall chunk m b ofs v m',
  store chunk m b ofs v = Some m' ->
  (forall i, ofs <= i < ofs + size_chunk chunk -> ~ P b i) ->
  unchanged_on m m'.

Lemma storebytes_unchanged_on:
  forall m b ofs bytes m',
  storebytes m b ofs bytes = Some m' ->
  (forall i, ofs <= i < ofs + Z_of_nat (length bytes) -> ~ P b i) ->
  unchanged_on m m'.

Lemma alloc_unchanged_on:
  forall m lo hi m' b,
  alloc m lo hi = Some (m', b) ->
  unchanged_on m m'.

Lemma free_unchanged_on:
  forall m b lo hi m',
  free m b lo hi = Some m' ->
  (forall i, lo <= i < hi -> ~ P b i) ->
  unchanged_on m m'.

End UNCHANGED_ON.



End Mem.

Notation mem := Mem.mem.



Global Opaque Mem.alloc Mem.free Mem.store Mem.load Mem.storebytes Mem.loadbytes.

Hint Resolve
  Mem.valid_not_valid_diff
  Mem.perm_implies
  Mem.perm_cur
  Mem.perm_max
  Mem.perm_valid_block
  Mem.range_perm_implies
  Mem.range_perm_cur
  Mem.range_perm_max
  Mem.valid_access_implies
  Mem.valid_access_valid_block
  Mem.valid_access_perm
  Mem.valid_access_load
  Mem.load_valid_access
  Mem.loadbytes_range_perm
  Mem.valid_access_store
  Mem.perm_store_1
  Mem.perm_store_2
  Mem.nextblock_store
  Mem.store_valid_block_1
  Mem.store_valid_block_2
  Mem.store_valid_access_1
  Mem.store_valid_access_2
  Mem.store_valid_access_3
  Mem.storebytes_range_perm
  Mem.perm_storebytes_1
  Mem.perm_storebytes_2
  Mem.storebytes_valid_access_1
  Mem.storebytes_valid_access_2
  Mem.nextblock_storebytes
  Mem.storebytes_valid_block_1
  Mem.storebytes_valid_block_2
  Mem.nextblock_alloc
  Mem.alloc_result
  Mem.valid_block_alloc
  Mem.fresh_block_alloc
  Mem.valid_new_block
  Mem.perm_alloc_1
  Mem.perm_alloc_2
  Mem.perm_alloc_3
  Mem.perm_alloc_4
  Mem.perm_alloc_inv
  Mem.valid_access_alloc_other
  Mem.valid_access_alloc_same
  Mem.valid_access_alloc_inv
  Mem.range_perm_free
  Mem.free_range_perm
  Mem.nextblock_free
  Mem.valid_block_free_1
  Mem.valid_block_free_2
  Mem.perm_free_1
  Mem.perm_free_2
  Mem.perm_free_3
  Mem.valid_access_free_1
  Mem.valid_access_free_2
  Mem.valid_access_free_inv_1
  Mem.valid_access_free_inv_2
  Mem.unchanged_on_refl
: mem.