Module Memtype


This file defines the interface for the memory model that is used in the dynamic semantics of all the languages used in the compiler. It defines a type mem of memory states, the following 4 basic operations over memory states, and their properties:

Require Import Coqlib.
Require Import AST.
Require Import Integers.
Require Import Floats.
Require Import Values.
Require Import Memdata.

Memory states are accessed by addresses b, ofs: pairs of a block identifier b and a byte offset ofs within that block. Each address is associated to permissions, also known as access rights. The following permissions are expressible: The first four cases are represented by the following type of permissions. Being empty is represented by the absence of any permission.

Inductive permission: Type :=
  | Freeable: permission
  | Writable: permission
  | Readable: permission
  | Nonempty: permission.

In the list, each permission implies the other permissions further down the list. We reflect this fact by the following order over permissions.

Inductive perm_order: permission -> permission -> Prop :=
  | perm_refl: forall p, perm_order p p
  | perm_F_any: forall p, perm_order Freeable p
  | perm_W_R: perm_order Writable Readable
  | perm_any_N: forall p, perm_order p Nonempty.

Hint Constructors perm_order: mem.

Lemma perm_order_trans:
  forall p1 p2 p3, perm_order p1 p2 -> perm_order p2 p3 -> perm_order p1 p3.
Proof.
  intros. inv H; inv H0; constructor.
Qed.

Each address has not one, but two permissions associated with it. The first is the current permission. It governs whether operations (load, store, free, etc) over this address succeed or not. The other is the maximal permission. It is always at least as strong as the current permission. Once a block is allocated, the maximal permission of an address within this block can only decrease, as a result of free or drop_perm operations, or of external calls. In contrast, the current permission of an address can be temporarily lowered by an external call, then raised again by another external call.

Inductive perm_kind: Type :=
  | Max: perm_kind
  | Cur: perm_kind.

Module Type MEM.

The abstract type of memory states.
Parameter mem: Type.

Operations on memory states


empty is the initial memory state.
Parameter empty: mem.

alloc m lo hi allocates a fresh block of size hi - lo bytes. Valid offsets in this block are between lo included and hi excluded. These offsets are writable in the returned memory state. This block is not initialized: its contents are initially undefined. Returns a pair (m', b) of the updated memory state m' and the identifier b of the newly-allocated block. Note that alloc never fails: we are modeling an infinite memory.
Parameter alloc: forall (m: mem) (lo hi: Z), mem * block.

free m b lo hi frees (deallocates) the range of offsets from lo included to hi excluded in block b. Returns the updated memory state, or None if the freed addresses are not writable.
Parameter free: forall (m: mem) (b: block) (lo hi: Z), option mem.

load chunk m b ofs reads a memory quantity chunk from addresses b, ofs to b, ofs + size_chunk chunk - 1 in memory state m. Returns the value read, or None if the accessed addresses are not readable.
Parameter load: forall (chunk: memory_chunk) (m: mem) (b: block) (ofs: Z), option val.

store chunk m b ofs v writes value v as memory quantity chunk from addresses b, ofs to b, ofs + size_chunk chunk - 1 in memory state m. Returns the updated memory state, or None if the accessed addresses are not writable.
Parameter store: forall (chunk: memory_chunk) (m: mem) (b: block) (ofs: Z) (v: val), option mem.

loadv and storev are variants of load and store where the address being accessed is passed as a value (of the Vptr kind).

Definition loadv (chunk: memory_chunk) (m: mem) (addr: val) : option val :=
  match addr with
  | Vptr b ofs => load chunk m b (Ptrofs.unsigned ofs)
  | _ => None
  end.

Definition storev (chunk: memory_chunk) (m: mem) (addr v: val) : option mem :=
  match addr with
  | Vptr b ofs => store chunk m b (Ptrofs.unsigned ofs) v
  | _ => None
  end.

loadbytes m b ofs n reads and returns the byte-level representation of the values contained at offsets ofs to ofs + n - 1 within block b in memory state m. None is returned if the accessed addresses are not readable.
Parameter loadbytes: forall (m: mem) (b: block) (ofs n: Z), option (list memval).

storebytes m b ofs bytes stores the given list of bytes bytes starting at location (b, ofs). Returns updated memory state or None if the accessed locations are not writable.
Parameter storebytes: forall (m: mem) (b: block) (ofs: Z) (bytes: list memval), option mem.

free_list frees all the given (block, lo, hi) triples.
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.

drop_perm m b lo hi p sets the permissions of the byte range (b, lo) ... (b, hi - 1) to p. These bytes must have Freeable permissions in the initial memory state m. Returns updated memory state, or None if insufficient permissions.

Parameter drop_perm: forall (m: mem) (b: block) (lo hi: Z) (p: permission), option mem.

Permissions, block validity, access validity, and bounds


The next block of a memory state is the block identifier for the next allocation. It increases by one at each allocation. Block identifiers below nextblock are said to be valid, meaning that they have been allocated previously. Block identifiers above nextblock are fresh or invalid, i.e. not yet allocated. Note that a block identifier remains valid after a free operation over this block.

Parameter nextblock: mem -> block.

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

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

perm m b ofs k p holds if the address b, ofs in memory state m has permission p: one of freeable, writable, readable, and nonempty. If the address is empty, perm m b ofs p is false for all values of p. k is the kind of permission we are interested in: either the current permissions or the maximal permissions.
Parameter perm: forall (m: mem) (b: block) (ofs: Z) (k: perm_kind) (p: permission), Prop.

Logical implications between permissions

Axiom 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.

The current permission is always less than or equal to the maximal permission.

Axiom perm_cur_max:
  forall m b ofs p, perm m b ofs Cur p -> perm m b ofs Max p.
Axiom perm_cur:
  forall m b ofs k p, perm m b ofs Cur p -> perm m b ofs k p.
Axiom perm_max:
  forall m b ofs k p, perm m b ofs k p -> perm m b ofs Max p.

Having a (nonempty) permission implies that the block is valid. In other words, invalid blocks, not yet allocated, are all empty.
Axiom perm_valid_block:
  forall m b ofs k p, perm m b ofs k p -> valid_block m b.


range_perm m b lo hi p holds iff the addresses b, lo to b, hi-1 all have permission p of kind k.
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.

Axiom 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.

An access to a memory quantity chunk at address b, ofs with permission p is valid in m if the accessed addresses all have current permission p and moreover the offset is properly aligned.
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).

Axiom 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.

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

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

valid_pointer m b ofs returns true if the address b, ofs is nonempty in m and false if it is empty.

Parameter valid_pointer: forall (m: mem) (b: block) (ofs: Z), bool.

Axiom valid_pointer_nonempty_perm:
  forall m b ofs,
  valid_pointer m b ofs = true <-> perm m b ofs Cur Nonempty.
Axiom valid_pointer_valid_access:
  forall m b ofs,
  valid_pointer m b ofs = true <-> valid_access m Mint8unsigned b ofs Nonempty.

C allows pointers one past the last element of an array. These are not valid according to the previously defined valid_pointer. The property weak_valid_pointer m b ofs holds if address b, ofs is a valid pointer in m, or a pointer one past a valid block in m.

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

Axiom 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.
Axiom valid_pointer_implies:
  forall m b ofs,
  valid_pointer m b ofs = true -> weak_valid_pointer m b ofs = true.

Properties of the memory operations


Properties of the initial memory state.


Axiom nextblock_empty: nextblock empty = 1%positive.
Axiom perm_empty: forall b ofs k p, ~perm empty b ofs k p.
Axiom valid_access_empty:
  forall chunk b ofs p, ~valid_access empty chunk b ofs p.

Properties of load.


A load succeeds if and only if the access is valid for reading
Axiom valid_access_load:
  forall m chunk b ofs,
  valid_access m chunk b ofs Readable ->
  exists v, load chunk m b ofs = Some v.
Axiom load_valid_access:
  forall m chunk b ofs v,
  load chunk m b ofs = Some v ->
  valid_access m chunk b ofs Readable.

The value returned by load belongs to the type of the memory quantity accessed: Vundef, Vint or Vptr for an integer quantity, Vundef or Vfloat for a float quantity.
Axiom load_type:
  forall m chunk b ofs v,
  load chunk m b ofs = Some v ->
  Val.has_type v (type_of_chunk chunk).

For a small integer or float type, the value returned by load is invariant under the corresponding cast.
Axiom 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.

Axiom load_int8_signed_unsigned:
  forall m b ofs,
  load Mint8signed m b ofs = option_map (Val.sign_ext 8) (load Mint8unsigned m b ofs).

Axiom load_int16_signed_unsigned:
  forall m b ofs,
  load Mint16signed m b ofs = option_map (Val.sign_ext 16) (load Mint16unsigned m b ofs).


Properties of loadbytes.


loadbytes succeeds if and only if we have read permissions on the accessed memory area.

Axiom 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.
Axiom 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.

If loadbytes succeeds, the corresponding load succeeds and returns a value that is determined by the bytes read by loadbytes.
Axiom 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).

Conversely, if load returns a value, the corresponding loadbytes succeeds and returns a list of bytes which decodes into the result of load.
Axiom 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.

loadbytes returns a list of length n (the number of bytes read).
Axiom loadbytes_length:
  forall m b ofs n bytes,
  loadbytes m b ofs n = Some bytes ->
  length bytes = nat_of_Z n.

Axiom loadbytes_empty:
  forall m b ofs n,
  n <= 0 -> loadbytes m b ofs n = Some nil.

Composing or decomposing loadbytes operations at adjacent addresses.
Axiom 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).
Axiom 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.

Properties of store.


store preserves block validity, permissions, access validity, and bounds. Moreover, a store succeeds if and only if the corresponding access is valid for writing.

Axiom nextblock_store:
  forall chunk m1 b ofs v m2, store chunk m1 b ofs v = Some m2 ->
  nextblock m2 = nextblock m1.
Axiom store_valid_block_1:
  forall chunk m1 b ofs v m2, store chunk m1 b ofs v = Some m2 ->
  forall b', valid_block m1 b' -> valid_block m2 b'.
Axiom store_valid_block_2:
  forall chunk m1 b ofs v m2, store chunk m1 b ofs v = Some m2 ->
  forall b', valid_block m2 b' -> valid_block m1 b'.

Axiom perm_store_1:
  forall chunk m1 b ofs v m2, store chunk m1 b ofs v = Some m2 ->
  forall b' ofs' k p, perm m1 b' ofs' k p -> perm m2 b' ofs' k p.
Axiom perm_store_2:
  forall chunk m1 b ofs v m2, store chunk m1 b ofs v = Some m2 ->
  forall b' ofs' k p, perm m2 b' ofs' k p -> perm m1 b' ofs' k p.

Axiom valid_access_store:
  forall m1 chunk b ofs v,
  valid_access m1 chunk b ofs Writable ->
  { m2: mem | store chunk m1 b ofs v = Some m2 }.
Axiom store_valid_access_1:
  forall chunk m1 b ofs v m2, store chunk m1 b ofs v = Some m2 ->
  forall chunk' b' ofs' p,
  valid_access m1 chunk' b' ofs' p -> valid_access m2 chunk' b' ofs' p.
Axiom store_valid_access_2:
  forall chunk m1 b ofs v m2, store chunk m1 b ofs v = Some m2 ->
  forall chunk' b' ofs' p,
  valid_access m2 chunk' b' ofs' p -> valid_access m1 chunk' b' ofs' p.
Axiom store_valid_access_3:
  forall chunk m1 b ofs v m2, store chunk m1 b ofs v = Some m2 ->
  valid_access m1 chunk b ofs Writable.

Load-store properties.

Axiom load_store_similar:
  forall chunk m1 b ofs v m2, store chunk m1 b ofs v = Some m2 ->
  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_val v chunk chunk' v'.

Axiom load_store_same:
  forall chunk m1 b ofs v m2, store chunk m1 b ofs v = Some m2 ->
  load chunk m2 b ofs = Some (Val.load_result chunk v).

Axiom load_store_other:
  forall chunk m1 b ofs v m2, store chunk m1 b ofs v = Some m2 ->
  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'.

Integrity of pointer values.

Definition compat_pointer_chunks (chunk1 chunk2: memory_chunk) : Prop :=
  match chunk1, chunk2 with
  | (Mint32 | Many32), (Mint32 | Many32) => True
  | (Mint64 | Many64), (Mint64 | Many64) => True
  | _, _ => False
  end.

Axiom load_store_pointer_overlap:
  forall chunk m1 b ofs v_b v_o m2 chunk' ofs' v,
  store chunk m1 b ofs (Vptr v_b v_o) = Some m2 ->
  load chunk' m2 b ofs' = Some v ->
  ofs' <> ofs ->
  ofs' + size_chunk chunk' > ofs ->
  ofs + size_chunk chunk > ofs' ->
  v = Vundef.
Axiom load_store_pointer_mismatch:
  forall chunk m1 b ofs v_b v_o m2 chunk' v,
  store chunk m1 b ofs (Vptr v_b v_o) = Some m2 ->
  load chunk' m2 b ofs = Some v ->
  ~compat_pointer_chunks chunk chunk' ->
  v = Vundef.
Axiom load_pointer_store:
  forall chunk m1 b ofs v m2 chunk' b' ofs' v_b v_o,
  store chunk m1 b ofs v = Some m2 ->
  load chunk' m2 b' ofs' = Some(Vptr v_b v_o) ->
  (v = Vptr v_b v_o /\ compat_pointer_chunks chunk chunk' /\ b' = b /\ ofs' = ofs)
  \/ (b' <> b \/ ofs' + size_chunk chunk' <= ofs \/ ofs + size_chunk chunk <= ofs').

Load-store properties for loadbytes.

Axiom loadbytes_store_same:
  forall chunk m1 b ofs v m2, store chunk m1 b ofs v = Some m2 ->
  loadbytes m2 b ofs (size_chunk chunk) = Some(encode_val chunk v).
Axiom loadbytes_store_other:
  forall chunk m1 b ofs v m2, store chunk m1 b ofs v = Some m2 ->
  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.

store is insensitive to the signedness or the high bits of small integer quantities.

Axiom store_signed_unsigned_8:
  forall m b ofs v,
  store Mint8signed m b ofs v = store Mint8unsigned m b ofs v.
Axiom store_signed_unsigned_16:
  forall m b ofs v,
  store Mint16signed m b ofs v = store Mint16unsigned m b ofs v.
Axiom store_int8_zero_ext:
  forall m b ofs n,
  store Mint8unsigned m b ofs (Vint (Int.zero_ext 8 n)) =
  store Mint8unsigned m b ofs (Vint n).
Axiom store_int8_sign_ext:
  forall m b ofs n,
  store Mint8signed m b ofs (Vint (Int.sign_ext 8 n)) =
  store Mint8signed m b ofs (Vint n).
Axiom store_int16_zero_ext:
  forall m b ofs n,
  store Mint16unsigned m b ofs (Vint (Int.zero_ext 16 n)) =
  store Mint16unsigned m b ofs (Vint n).
Axiom store_int16_sign_ext:
  forall m b ofs n,
  store Mint16signed m b ofs (Vint (Int.sign_ext 16 n)) =
  store Mint16signed m b ofs (Vint n).

Properties of storebytes.


storebytes preserves block validity, permissions, access validity, and bounds. Moreover, a storebytes succeeds if and only if we have write permissions on the addressed memory area.

Axiom range_perm_storebytes:
  forall m1 b ofs bytes,
  range_perm m1 b ofs (ofs + Z_of_nat (length bytes)) Cur Writable ->
  { m2 : mem | storebytes m1 b ofs bytes = Some m2 }.
Axiom storebytes_range_perm:
  forall m1 b ofs bytes m2, storebytes m1 b ofs bytes = Some m2 ->
  range_perm m1 b ofs (ofs + Z_of_nat (length bytes)) Cur Writable.
Axiom perm_storebytes_1:
  forall m1 b ofs bytes m2, storebytes m1 b ofs bytes = Some m2 ->
  forall b' ofs' k p, perm m1 b' ofs' k p -> perm m2 b' ofs' k p.
Axiom perm_storebytes_2:
  forall m1 b ofs bytes m2, storebytes m1 b ofs bytes = Some m2 ->
  forall b' ofs' k p, perm m2 b' ofs' k p -> perm m1 b' ofs' k p.
Axiom storebytes_valid_access_1:
  forall m1 b ofs bytes m2, storebytes m1 b ofs bytes = Some m2 ->
  forall chunk' b' ofs' p,
  valid_access m1 chunk' b' ofs' p -> valid_access m2 chunk' b' ofs' p.
Axiom storebytes_valid_access_2:
  forall m1 b ofs bytes m2, storebytes m1 b ofs bytes = Some m2 ->
  forall chunk' b' ofs' p,
  valid_access m2 chunk' b' ofs' p -> valid_access m1 chunk' b' ofs' p.
Axiom nextblock_storebytes:
  forall m1 b ofs bytes m2, storebytes m1 b ofs bytes = Some m2 ->
  nextblock m2 = nextblock m1.
Axiom storebytes_valid_block_1:
  forall m1 b ofs bytes m2, storebytes m1 b ofs bytes = Some m2 ->
  forall b', valid_block m1 b' -> valid_block m2 b'.
Axiom storebytes_valid_block_2:
  forall m1 b ofs bytes m2, storebytes m1 b ofs bytes = Some m2 ->
  forall b', valid_block m2 b' -> valid_block m1 b'.

Connections between store and storebytes.

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

Axiom store_storebytes:
  forall m1 b ofs chunk v m2,
  store chunk m1 b ofs v = Some m2 ->
  storebytes m1 b ofs (encode_val chunk v) = Some m2.

Load-store properties.

Axiom loadbytes_storebytes_same:
  forall m1 b ofs bytes m2, storebytes m1 b ofs bytes = Some m2 ->
  loadbytes m2 b ofs (Z_of_nat (length bytes)) = Some bytes.
Axiom loadbytes_storebytes_other:
  forall m1 b ofs bytes m2, storebytes m1 b ofs bytes = Some m2 ->
  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.
Axiom load_storebytes_other:
  forall m1 b ofs bytes m2, storebytes m1 b ofs bytes = Some m2 ->
  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'.

Composing or decomposing storebytes operations at adjacent addresses.

Axiom 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.
Axiom 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.

Properties of alloc.


The identifier of the freshly allocated block is the next block of the initial memory state.

Axiom alloc_result:
  forall m1 lo hi m2 b, alloc m1 lo hi = (m2, b) ->
  b = nextblock m1.

Effect of alloc on block validity.

Axiom nextblock_alloc:
  forall m1 lo hi m2 b, alloc m1 lo hi = (m2, b) ->
  nextblock m2 = Psucc (nextblock m1).

Axiom valid_block_alloc:
  forall m1 lo hi m2 b, alloc m1 lo hi = (m2, b) ->
  forall b', valid_block m1 b' -> valid_block m2 b'.
Axiom fresh_block_alloc:
  forall m1 lo hi m2 b, alloc m1 lo hi = (m2, b) ->
  ~(valid_block m1 b).
Axiom valid_new_block:
  forall m1 lo hi m2 b, alloc m1 lo hi = (m2, b) ->
  valid_block m2 b.
Axiom valid_block_alloc_inv:
  forall m1 lo hi m2 b, alloc m1 lo hi = (m2, b) ->
  forall b', valid_block m2 b' -> b' = b \/ valid_block m1 b'.

Effect of alloc on permissions.

Axiom perm_alloc_1:
  forall m1 lo hi m2 b, alloc m1 lo hi = (m2, b) ->
  forall b' ofs k p, perm m1 b' ofs k p -> perm m2 b' ofs k p.
Axiom perm_alloc_2:
  forall m1 lo hi m2 b, alloc m1 lo hi = (m2, b) ->
  forall ofs k, lo <= ofs < hi -> perm m2 b ofs k Freeable.
Axiom perm_alloc_3:
  forall m1 lo hi m2 b, alloc m1 lo hi = (m2, b) ->
  forall ofs k p, perm m2 b ofs k p -> lo <= ofs < hi.
Axiom perm_alloc_4:
  forall m1 lo hi m2 b, alloc m1 lo hi = (m2, b) ->
  forall b' ofs k p, perm m2 b' ofs k p -> b' <> b -> perm m1 b' ofs k p.
Axiom perm_alloc_inv:
  forall m1 lo hi m2 b, alloc m1 lo hi = (m2, b) ->
  forall b' ofs k p,
  perm m2 b' ofs k p ->
  if eq_block b' b then lo <= ofs < hi else perm m1 b' ofs k p.

Effect of alloc on access validity.

Axiom valid_access_alloc_other:
  forall m1 lo hi m2 b, alloc m1 lo hi = (m2, b) ->
  forall chunk b' ofs p,
  valid_access m1 chunk b' ofs p ->
  valid_access m2 chunk b' ofs p.
Axiom valid_access_alloc_same:
  forall m1 lo hi m2 b, alloc m1 lo hi = (m2, b) ->
  forall chunk ofs,
  lo <= ofs -> ofs + size_chunk chunk <= hi -> (align_chunk chunk | ofs) ->
  valid_access m2 chunk b ofs Freeable.
Axiom valid_access_alloc_inv:
  forall m1 lo hi m2 b, alloc m1 lo hi = (m2, b) ->
  forall chunk b' ofs p,
  valid_access m2 chunk b' ofs p ->
  if eq_block b' b
  then lo <= ofs /\ ofs + size_chunk chunk <= hi /\ (align_chunk chunk | ofs)
  else valid_access m1 chunk b' ofs p.

Load-alloc properties.

Axiom load_alloc_unchanged:
  forall m1 lo hi m2 b, alloc m1 lo hi = (m2, b) ->
  forall chunk b' ofs,
  valid_block m1 b' ->
  load chunk m2 b' ofs = load chunk m1 b' ofs.
Axiom load_alloc_other:
  forall m1 lo hi m2 b, alloc m1 lo hi = (m2, b) ->
  forall chunk b' ofs v,
  load chunk m1 b' ofs = Some v ->
  load chunk m2 b' ofs = Some v.
Axiom load_alloc_same:
  forall m1 lo hi m2 b, alloc m1 lo hi = (m2, b) ->
  forall chunk ofs v,
  load chunk m2 b ofs = Some v ->
  v = Vundef.
Axiom load_alloc_same':
  forall m1 lo hi m2 b, alloc m1 lo hi = (m2, b) ->
  forall chunk ofs,
  lo <= ofs -> ofs + size_chunk chunk <= hi -> (align_chunk chunk | ofs) ->
  load chunk m2 b ofs = Some Vundef.

Properties of free.


free succeeds if and only if the correspond range of addresses has Freeable current permission.

Axiom 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 }.
Axiom free_range_perm:
  forall m1 bf lo hi m2, free m1 bf lo hi = Some m2 ->
  range_perm m1 bf lo hi Cur Freeable.

Block validity is preserved by free.

Axiom nextblock_free:
  forall m1 bf lo hi m2, free m1 bf lo hi = Some m2 ->
  nextblock m2 = nextblock m1.
Axiom valid_block_free_1:
  forall m1 bf lo hi m2, free m1 bf lo hi = Some m2 ->
  forall b, valid_block m1 b -> valid_block m2 b.
Axiom valid_block_free_2:
  forall m1 bf lo hi m2, free m1 bf lo hi = Some m2 ->
  forall b, valid_block m2 b -> valid_block m1 b.

Effect of free on permissions.

Axiom perm_free_1:
  forall m1 bf lo hi m2, free m1 bf lo hi = Some m2 ->
  forall b ofs k p,
  b <> bf \/ ofs < lo \/ hi <= ofs ->
  perm m1 b ofs k p ->
  perm m2 b ofs k p.
Axiom perm_free_2:
  forall m1 bf lo hi m2, free m1 bf lo hi = Some m2 ->
  forall ofs k p, lo <= ofs < hi -> ~ perm m2 bf ofs k p.
Axiom perm_free_3:
  forall m1 bf lo hi m2, free m1 bf lo hi = Some m2 ->
  forall b ofs k p,
  perm m2 b ofs k p -> perm m1 b ofs k p.

Effect of free on access validity.

Axiom valid_access_free_1:
  forall m1 bf lo hi m2, free m1 bf lo hi = Some m2 ->
  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.
Axiom valid_access_free_2:
  forall m1 bf lo hi m2, free m1 bf lo hi = Some m2 ->
  forall chunk ofs p,
  lo < hi -> ofs + size_chunk chunk > lo -> ofs < hi ->
  ~(valid_access m2 chunk bf ofs p).
Axiom valid_access_free_inv_1:
  forall m1 bf lo hi m2, free m1 bf lo hi = Some m2 ->
  forall chunk b ofs p,
  valid_access m2 chunk b ofs p ->
  valid_access m1 chunk b ofs p.
Axiom valid_access_free_inv_2:
  forall m1 bf lo hi m2, free m1 bf lo hi = Some m2 ->
  forall chunk ofs p,
  valid_access m2 chunk bf ofs p ->
  lo >= hi \/ ofs + size_chunk chunk <= lo \/ hi <= ofs.

Load-free properties

Axiom load_free:
  forall m1 bf lo hi m2, free m1 bf lo hi = Some m2 ->
  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.

Properties of drop_perm.


Axiom nextblock_drop:
  forall m b lo hi p m', drop_perm m b lo hi p = Some m' ->
  nextblock m' = nextblock m.
Axiom drop_perm_valid_block_1:
  forall m b lo hi p m', drop_perm m b lo hi p = Some m' ->
  forall b', valid_block m b' -> valid_block m' b'.
Axiom drop_perm_valid_block_2:
  forall m b lo hi p m', drop_perm m b lo hi p = Some m' ->
  forall b', valid_block m' b' -> valid_block m b'.

Axiom 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.
Axiom 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' }.

Axiom perm_drop_1:
  forall m b lo hi p m', drop_perm m b lo hi p = Some m' ->
  forall ofs k, lo <= ofs < hi -> perm m' b ofs k p.
Axiom perm_drop_2:
  forall m b lo hi p m', drop_perm m b lo hi p = Some m' ->
  forall ofs k p', lo <= ofs < hi -> perm m' b ofs k p' -> perm_order p p'.
Axiom perm_drop_3:
  forall m b lo hi p m', drop_perm m b lo hi p = Some m' ->
  forall b' ofs k p', b' <> b \/ ofs < lo \/ hi <= ofs -> perm m b' ofs k p' -> perm m' b' ofs k p'.
Axiom perm_drop_4:
  forall m b lo hi p m', drop_perm m b lo hi p = Some m' ->
  forall b' ofs k p', perm m' b' ofs k p' -> perm m b' ofs k p'.

Axiom load_drop:
  forall m b lo hi p m', drop_perm m b lo hi p = Some m' ->
  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.

Relating two memory states.


Memory extensions


A store m2 extends a store m1 if m2 can be obtained from m1 by relaxing the permissions of m1 (for instance, allocating larger blocks) and replacing some of the Vundef values stored in m1 by more defined values stored in m2 at the same addresses.

Parameter extends: mem -> mem -> Prop.

Axiom extends_refl:
  forall m, extends m m.

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

Axiom loadv_extends:
  forall chunk m1 m2 addr1 addr2 v1,
  extends m1 m2 ->
  loadv chunk m1 addr1 = Some v1 ->
  Val.lessdef addr1 addr2 ->
  exists v2, loadv chunk m2 addr2 = Some v2 /\ Val.lessdef v1 v2.

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

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

Axiom store_outside_extends:
  forall chunk m1 m2 b ofs v m2',
  extends 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 m1 m2'.

Axiom storev_extends:
  forall chunk m1 m2 addr1 v1 m1' addr2 v2,
  extends m1 m2 ->
  storev chunk m1 addr1 v1 = Some m1' ->
  Val.lessdef addr1 addr2 ->
  Val.lessdef v1 v2 ->
  exists m2',
     storev chunk m2 addr2 v2 = Some m2'
  /\ extends m1' m2'.

Axiom storebytes_within_extends:
  forall m1 m2 b ofs bytes1 m1' bytes2,
  extends m1 m2 ->
  storebytes m1 b ofs bytes1 = Some m1' ->
  list_forall2 memval_lessdef bytes1 bytes2 ->
  exists m2',
     storebytes m2 b ofs bytes2 = Some m2'
  /\ extends m1' m2'.

Axiom storebytes_outside_extends:
  forall m1 m2 b ofs bytes2 m2',
  extends 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 m1 m2'.

Axiom alloc_extends:
  forall m1 m2 lo1 hi1 b m1' lo2 hi2,
  extends m1 m2 ->
  alloc m1 lo1 hi1 = (m1', b) ->
  lo2 <= lo1 -> hi1 <= hi2 ->
  exists m2',
     alloc m2 lo2 hi2 = (m2', b)
  /\ extends m1' m2'.

Axiom free_left_extends:
  forall m1 m2 b lo hi m1',
  extends m1 m2 ->
  free m1 b lo hi = Some m1' ->
  extends m1' m2.

Axiom free_right_extends:
  forall m1 m2 b lo hi m2',
  extends m1 m2 ->
  free m2 b lo hi = Some m2' ->
  (forall ofs k p, perm m1 b ofs k p -> lo <= ofs < hi -> False) ->
  extends m1 m2'.

Axiom free_parallel_extends:
  forall m1 m2 b lo hi m1',
  extends m1 m2 ->
  free m1 b lo hi = Some m1' ->
  exists m2',
     free m2 b lo hi = Some m2'
  /\ extends m1' m2'.

Axiom valid_block_extends:
  forall m1 m2 b,
  extends m1 m2 ->
  (valid_block m1 b <-> valid_block m2 b).
Axiom perm_extends:
  forall m1 m2 b ofs k p,
  extends m1 m2 -> perm m1 b ofs k p -> perm m2 b ofs k p.
Axiom valid_access_extends:
  forall m1 m2 chunk b ofs p,
  extends m1 m2 -> valid_access m1 chunk b ofs p -> valid_access m2 chunk b ofs p.
Axiom valid_pointer_extends:
  forall m1 m2 b ofs,
  extends m1 m2 -> valid_pointer m1 b ofs = true -> valid_pointer m2 b ofs = true.
Axiom weak_valid_pointer_extends:
  forall m1 m2 b ofs,
  extends m1 m2 ->
  weak_valid_pointer m1 b ofs = true -> weak_valid_pointer m2 b ofs = true.

Memory injections


A memory injection f is a function from addresses to either None or Some of an address and an offset. It defines a correspondence between the blocks of two memory states m1 and m2: A memory injection f defines a relation Val.inject between values that is the identity for integer and float values, and relocates pointer values as prescribed by f. (See module Values.) Likewise, a memory injection f defines a relation between memory states that we now axiomatize.

Parameter inject: meminj -> mem -> mem -> Prop.

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

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

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

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

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

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

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

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

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

Axiom valid_pointer_inject_val:
  forall f m1 m2 b ofs b' ofs',
  inject f m1 m2 ->
  valid_pointer m1 b (Ptrofs.unsigned ofs) = true ->
  Val.inject f (Vptr b ofs) (Vptr b' ofs') ->
  valid_pointer m2 b' (Ptrofs.unsigned ofs') = true.

Axiom weak_valid_pointer_inject_val:
  forall f m1 m2 b ofs b' ofs',
  inject f m1 m2 ->
  weak_valid_pointer m1 b (Ptrofs.unsigned ofs) = true ->
  Val.inject f (Vptr b ofs) (Vptr b' ofs') ->
  weak_valid_pointer m2 b' (Ptrofs.unsigned ofs') = true.

Axiom inject_no_overlap:
  forall f m1 m2 b1 b2 b1' b2' delta1 delta2 ofs1 ofs2,
  inject 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.

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

Axiom load_inject:
  forall f m1 m2 chunk b1 ofs b2 delta v1,
  inject 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.inject f v1 v2.

Axiom loadv_inject:
  forall f m1 m2 chunk a1 a2 v1,
  inject f m1 m2 ->
  loadv chunk m1 a1 = Some v1 ->
  Val.inject f a1 a2 ->
  exists v2, loadv chunk m2 a2 = Some v2 /\ Val.inject f v1 v2.

Axiom loadbytes_inject:
  forall f m1 m2 b1 ofs len b2 delta bytes1,
  inject 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 f) bytes1 bytes2.

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

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

Axiom store_outside_inject:
  forall f m1 m2 chunk b ofs v m2',
  inject 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 f m1 m2'.

Axiom storev_mapped_inject:
  forall f chunk m1 a1 v1 n1 m2 a2 v2,
  inject f m1 m2 ->
  storev chunk m1 a1 v1 = Some n1 ->
  Val.inject f a1 a2 ->
  Val.inject f v1 v2 ->
  exists n2,
    storev chunk m2 a2 v2 = Some n2 /\ inject f n1 n2.

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

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

Axiom storebytes_outside_inject:
  forall f m1 m2 b ofs bytes2 m2',
  inject 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 f m1 m2'.

Axiom alloc_right_inject:
  forall f m1 m2 lo hi b2 m2',
  inject f m1 m2 ->
  alloc m2 lo hi = (m2', b2) ->
  inject f m1 m2'.

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

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

Axiom alloc_left_mapped_inject:
  forall f m1 m2 lo hi m1' b1 b2 delta,
  inject f m1 m2 ->
  alloc m1 lo hi = (m1', b1) ->
  valid_block m2 b2 ->
  0 <= delta <= Ptrofs.max_unsigned ->
  (forall ofs k p, perm m2 b2 ofs k p -> delta = 0 \/ 0 <= ofs < Ptrofs.max_unsigned) ->
  (forall ofs k p, lo <= ofs < hi -> perm m2 b2 (ofs + delta) k p) ->
  inj_offset_aligned delta (hi-lo) ->
  (forall b delta' ofs k p,
   f b = Some (b2, delta') ->
   perm m1 b ofs k p ->
   lo + delta <= ofs + delta' < hi + delta -> False) ->
  exists f',
     inject f' m1' m2
  /\ inject_incr f f'
  /\ f' b1 = Some(b2, delta)
  /\ (forall b, b <> b1 -> f' b = f b).

Axiom alloc_parallel_inject:
  forall f m1 m2 lo1 hi1 m1' b1 lo2 hi2,
  inject f m1 m2 ->
  alloc m1 lo1 hi1 = (m1', b1) ->
  lo2 <= lo1 -> hi1 <= hi2 ->
  exists f', exists m2', exists b2,
  alloc m2 lo2 hi2 = (m2', b2)
  /\ inject f' m1' m2'
  /\ inject_incr f f'
  /\ f' b1 = Some(b2, 0)
  /\ (forall b, b <> b1 -> f' b = f b).

Axiom free_inject:
  forall f m1 l m1' m2 b lo hi m2',
  inject f m1 m2 ->
  free_list m1 l = Some m1' ->
  free m2 b lo hi = Some m2' ->
  (forall b1 delta ofs k p,
    f b1 = Some(b, delta) -> perm m1 b1 ofs k p -> lo <= ofs + delta < hi ->
    exists lo1, exists hi1, In (b1, lo1, hi1) l /\ lo1 <= ofs < hi1) ->
  inject f m1' m2'.

Axiom free_parallel_inject:
  forall f m1 m2 b lo hi m1' b' delta,
  inject f m1 m2 ->
  free m1 b lo hi = Some m1' ->
  f b = Some(b', delta) ->
  exists m2',
     free m2 b' (lo + delta) (hi + delta) = Some m2'
  /\ inject f m1' m2'.

Axiom drop_outside_inject:
  forall f m1 m2 b lo hi p m2',
  inject 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 f m1 m2'.

Memory states that inject into themselves.

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

Parameter inject_neutral: forall (thr: block) (m: mem), Prop.

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

Axiom empty_inject_neutral:
  forall thr, inject_neutral thr empty.

Axiom alloc_inject_neutral:
  forall thr m lo hi b m',
  alloc m lo hi = (m', b) ->
  inject_neutral thr m ->
  Plt (nextblock m) thr ->
  inject_neutral thr m'.

Axiom 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.inject (flat_inj thr) v v ->
  inject_neutral thr m'.

Axiom 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'.

End MEM.