From 2ae43be7b9d4118335c9d2cef6e098f9b9f807fe Mon Sep 17 00:00:00 2001
From: xleroy <xleroy@fca1b0fc-160b-0410-b1d3-a4f43f01ea2e>
Date: Thu, 9 Feb 2006 14:55:48 +0000
Subject: Initial import of compcert

git-svn-id: https://yquem.inria.fr/compcert/svn/compcert/trunk@1 fca1b0fc-160b-0410-b1d3-a4f43f01ea2e
---
 backend/Mem.v | 2253 +++++++++++++++++++++++++++++++++++++++++++++++++++++++++
 1 file changed, 2253 insertions(+)
 create mode 100644 backend/Mem.v

(limited to 'backend/Mem.v')

diff --git a/backend/Mem.v b/backend/Mem.v
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+(** This file develops the memory model that is used in the dynamic
+  semantics of all the languages of the compiler back-end.
+  It defines a type [mem] of memory states, the following 4 basic
+  operations over memory states, and their properties:
+- [alloc]: allocate a fresh memory block;
+- [free]: invalidate a memory block;
+- [load]: read a memory chunk at a given address;
+- [store]: store a memory chunk at a given address.
+*)
+  
+Require Import Coqlib.
+Require Import Maps.
+Require Import AST.
+Require Import Integers.
+Require Import Floats.
+Require Import Values.
+
+(** * Structure of memory states *)
+
+(** A memory state is organized in several disjoint blocks.  Each block
+  has a low and a high bound that defines its size.  Each block map
+  byte offsets to the contents of this byte. *)
+
+Definition update (A: Set) (x: Z) (v: A) (f: Z -> A) : Z -> A :=
+  fun y => if zeq y x then v else f y.
+
+Implicit Arguments update [A].
+
+Lemma update_s:
+  forall (A: Set) (x: Z) (v: A) (f: Z -> A),
+  update x v f x = v.
+Proof.
+  intros; unfold update. apply zeq_true.
+Qed.
+
+Lemma update_o:
+  forall (A: Set) (x: Z) (v: A) (f: Z -> A) (y: Z),
+  x <> y -> update x v f y = f y.
+Proof.
+  intros; unfold update. apply zeq_false; auto.
+Qed.
+
+(** The possible contents of a byte-sized memory cell.  To give intuitions,
+  a 4-byte value [v] stored at offset [d] will be represented by
+  the content [Datum32 v] at offset [d] and [Cont] at offsets [d+1],
+  [d+2] and [d+3].  The [Cont] contents enable detecting future writes
+  that would overlap partially the 4-byte value. *)
+
+Inductive content : Set :=
+  | Undef: content              (**r undefined contents *)
+  | Datum8: val -> content      (**r a value that fits in 1 byte *)
+  | Datum16: val -> content     (**r first byte of a 2-byte value *)
+  | Datum32: val -> content     (**r first byte of a 4-byte value *)
+  | Datum64: val -> content     (**r first byte of a 8-byte value *)
+  | Cont: content.            (**r continuation bytes for a multi-byte value *)
+
+Definition contentmap : Set := Z -> content.
+
+(** A memory block comprises the dimensions of the block (low and high bounds)
+  plus a mapping from byte offsets to contents.  For technical reasons,
+  we also carry around a proof that the mapping is equal to [Undef]
+  outside the range of allowed byte offsets. *)
+
+Record block_contents : Set := mkblock {
+  low: Z;
+  high: Z;
+  contents: contentmap;
+  undef_outside: forall ofs, ofs < low \/ ofs >= high -> contents ofs = Undef
+}.
+
+(** A memory state is a mapping from block addresses (represented by [Z]
+  integers) to blocks.  We also maintain the address of the next 
+  unallocated block, and a proof that this address is positive. *)
+
+Record mem : Set := mkmem {
+  blocks: Z -> block_contents;
+  nextblock: block;
+  nextblock_pos: nextblock > 0
+}.
+
+(** * Operations on memory stores *)
+
+(** Memory reads and writes are performed by quantities called memory chunks,
+  encoding the type, size and signedness of the chunk being addressed.
+  It is useful to extract only the size information as given by the
+  following [memory_size] type. *)
+
+Inductive memory_size : Set :=
+  | Size8: memory_size
+  | Size16: memory_size
+  | Size32: memory_size
+  | Size64: memory_size.
+
+Definition size_mem (sz: memory_size) :=
+  match sz with
+  | Size8 => 1
+  | Size16 => 2
+  | Size32 => 4
+  | Size64 => 8
+  end.
+
+Definition mem_chunk (chunk: memory_chunk) :=
+  match chunk with
+  | Mint8signed => Size8
+  | Mint8unsigned => Size8
+  | Mint16signed => Size16
+  | Mint16unsigned => Size16
+  | Mint32 => Size32
+  | Mfloat32 => Size32
+  | Mfloat64 => Size64
+  end.
+
+Definition size_chunk (chunk: memory_chunk) := size_mem (mem_chunk chunk).
+
+(** The initial store. *)
+
+Remark one_pos: 1 > 0.
+Proof. omega. Qed.
+
+Remark undef_undef_outside:
+  forall lo hi ofs, ofs < lo \/ ofs >= hi -> (fun y => Undef) ofs = Undef.
+Proof.
+  auto.
+Qed.
+
+Definition empty_block (lo hi: Z) : block_contents :=
+  mkblock lo hi (fun y => Undef) (undef_undef_outside lo hi).
+
+Definition empty: mem :=
+  mkmem (fun x => empty_block 0 0) 1 one_pos.
+
+Definition nullptr: block := 0.
+
+(** Allocation of a fresh block with the given bounds.  Return an updated
+  memory state and the address of the fresh block, which initially contains
+  undefined cells.  Note that allocation never fails: we model an
+  infinite memory. *)
+
+Remark succ_nextblock_pos:
+  forall m, Zsucc m.(nextblock) > 0.
+Proof. intro. generalize (nextblock_pos m). omega. Qed.
+
+Definition alloc (m: mem) (lo hi: Z) :=
+  (mkmem (update m.(nextblock) 
+                 (empty_block lo hi)
+                 m.(blocks))
+         (Zsucc m.(nextblock))
+         (succ_nextblock_pos m),
+   m.(nextblock)).
+
+(** Freeing a block.  Return the updated memory state where the given
+  block address has been invalidated: future reads and writes to this
+  address will fail.  Note that we make no attempt to return the block
+  to an allocation pool: the given block address will never be allocated
+  later. *)
+
+Definition free (m: mem) (b: block) :=
+  mkmem (update b 
+                (empty_block 0 0)
+                m.(blocks))
+        m.(nextblock)
+        m.(nextblock_pos).
+
+(** Freeing of a list of blocks. *)
+
+Definition free_list (m:mem) (l:list block) :=
+  List.fold_right (fun b m => free m b) m l.
+
+(** Return the low and high bounds for the given block address.
+   Those bounds are 0 for freed or not yet allocated address. *)
+
+Definition low_bound (m: mem) (b: block) :=
+  low (m.(blocks) b).
+Definition high_bound (m: mem) (b: block) :=
+  high (m.(blocks) b).
+
+(** A block address is valid if it was previously allocated. It remains valid
+  even after being freed. *)
+
+Definition valid_block (m: mem) (b: block) :=
+  b < m.(nextblock).
+
+(** Reading and writing [N] adjacent locations in a [contentmap].
+
+  We define two functions and prove some of their properties:
+  - [getN n ofs m] returns the datum at offset [ofs] in block contents [m]
+    after checking that the contents of offsets [ofs+1] to [ofs+n+1]
+    are [Cont].
+  - [setN n ofs v m] updates the block contents [m], storing the content [v]
+    at offset [ofs] and the content [Cont] at offsets [ofs+1] to [ofs+n+1].
+ *)
+
+Fixpoint check_cont (n: nat) (p: Z) (m: contentmap) {struct n} : bool :=
+  match n with
+  | O => true
+  | S n1 =>
+      match m p with
+      | Cont => check_cont n1 (p + 1) m
+      | _ => false
+      end
+  end.
+
+Definition getN (n: nat) (p: Z) (m: contentmap) : content :=
+  if check_cont n (p + 1) m then m p else Undef.
+
+Fixpoint set_cont (n: nat) (p: Z) (m: contentmap) {struct n} : contentmap :=
+  match n with
+  | O => m
+  | S n1 => update p Cont (set_cont n1 (p + 1) m)
+  end.
+
+Definition setN (n: nat) (p: Z) (v: content) (m: contentmap) : contentmap :=
+  update p v (set_cont n (p + 1) m).
+
+Lemma check_cont_true:
+  forall n p m,
+  (forall q, p <= q < p + Z_of_nat n -> m q = Cont) ->
+  check_cont n p m = true.
+Proof.
+  induction n; intros.
+  reflexivity.
+  simpl. rewrite H. apply IHn. 
+  intros. apply H. rewrite inj_S. omega.
+  rewrite inj_S. omega. 
+Qed.
+
+Hint Resolve check_cont_true.
+
+Lemma check_cont_inv:
+  forall n p m,
+  check_cont n p m = true ->
+  (forall q, p <= q < p + Z_of_nat n -> m q = Cont).
+Proof.
+  induction n; intros until m.
+  unfold Z_of_nat. intros. omegaContradiction.
+  unfold check_cont; fold check_cont. 
+  caseEq (m p); intros; try discriminate.
+  assert (p = q \/ p + 1 <= q < (p + 1) + Z_of_nat n).
+    rewrite inj_S in H1. omega. 
+  elim H2; intro.
+  subst q. auto.
+  apply IHn with (p + 1); auto.
+Qed.
+
+Hint Resolve check_cont_inv.
+
+Lemma check_cont_false:
+  forall n p q m,
+  p <= q < p + Z_of_nat n ->
+  m q <> Cont ->
+  check_cont n p m = false.
+Proof.
+  intros. caseEq (check_cont n p m); intro.
+  generalize (check_cont_inv _ _ _ H1 q H). intro. contradiction.
+  auto.
+Qed.
+
+Hint Resolve check_cont_false.
+
+Lemma set_cont_inside:
+  forall n p m q,
+  p <= q < p + Z_of_nat n ->
+  (set_cont n p m) q = Cont.
+Proof.
+  induction n; intros.
+  unfold Z_of_nat in H. omegaContradiction.
+  simpl. 
+  assert (p = q \/ p + 1 <= q < (p + 1) + Z_of_nat n).
+    rewrite inj_S in H. omega. 
+  elim H0; intro.
+  subst q. apply update_s.
+  rewrite update_o. apply IHn. auto. 
+  red; intro; subst q. omega. 
+Qed.
+
+Hint Resolve set_cont_inside.
+
+Lemma set_cont_outside:
+  forall n p m q,
+  q < p \/ p + Z_of_nat n <= q ->
+  (set_cont n p m) q = m q.
+Proof.
+  induction n; intros.
+  simpl. auto.
+  simpl. rewrite inj_S in H. 
+  rewrite update_o. apply IHn. omega. omega.
+Qed.
+
+Hint Resolve set_cont_outside.
+
+Lemma getN_setN_same:
+  forall n p v m,
+  getN n p (setN n p v m) = v.
+Proof.
+  intros. unfold getN, setN.
+  rewrite check_cont_true. apply update_s.
+  intros. rewrite update_o. apply set_cont_inside. auto.
+  omega. 
+Qed.
+
+Hint Resolve getN_setN_same.
+
+Lemma getN_setN_other:
+  forall n1 n2 p1 p2 v m,
+  p1 + Z_of_nat n1 < p2 \/ p2 + Z_of_nat n2 < p1 ->
+  getN n2 p2 (setN n1 p1 v m) = getN n2 p2 m.
+Proof.
+  intros. unfold getN, setN.
+  caseEq (check_cont n2 (p2 + 1) m); intro.
+  rewrite check_cont_true. rewrite update_o.
+  apply set_cont_outside. omega. omega.
+  intros. rewrite update_o. rewrite set_cont_outside.
+  eapply check_cont_inv. eauto. auto.
+  omega. omega. 
+  caseEq (check_cont n2 (p2 + 1) (update p1 v (set_cont n1 (p1 + 1) m))); intros.
+  assert (check_cont n2 (p2 + 1) m = true).
+  apply check_cont_true. intros. 
+  generalize (check_cont_inv _ _ _ H1 q H2).
+  rewrite update_o. rewrite set_cont_outside. auto. omega. omega.
+  rewrite H0 in H2; discriminate.
+  auto.
+Qed. 
+
+Hint Resolve getN_setN_other.
+
+Lemma getN_setN_overlap:
+  forall n1 n2 p1 p2 v m,
+  p1 <> p2 ->
+  p1 + Z_of_nat n1 >= p2 -> p2 + Z_of_nat n2 >= p1 ->
+  v <> Cont ->
+  getN n2 p2 (setN n1 p1 v m) = Cont \/
+  getN n2 p2 (setN n1 p1 v m) = Undef.
+Proof.
+  intros. unfold getN.
+  caseEq (check_cont n2 (p2 + 1) (setN n1 p1 v m)); intro.
+  case (zlt p2 p1); intro.
+  assert (p2 + 1 <= p1 < p2 + 1 + Z_of_nat n2). omega.
+  generalize (check_cont_inv _ _ _ H3 p1 H4). 
+  unfold setN. rewrite update_s. intro. contradiction.
+  left. unfold setN. rewrite update_o.
+  apply set_cont_inside. omega. auto.
+  right; auto.
+Qed. 
+
+Hint Resolve getN_setN_overlap.
+
+Lemma getN_setN_mismatch:
+  forall n1 n2 p v m,
+  getN n2 p (setN n1 p v m) = v \/ getN n2 p (setN n1 p v m) = Undef.
+Proof.
+  intros. unfold getN. 
+  caseEq (check_cont n2 (p + 1) (setN n1 p v m)); intro.
+  left. unfold setN. apply update_s.
+  right. auto.
+Qed.
+
+Hint Resolve getN_setN_mismatch.
+
+Lemma getN_init:
+  forall n p,
+  getN n p (fun y => Undef) = Undef.
+Proof.
+  intros. unfold getN.
+  case (check_cont n (p + 1) (fun y => Undef)); auto.
+Qed.
+
+Hint Resolve getN_init.
+
+(** The following function checks whether accessing the given memory chunk
+  at the given offset in the given block respects the bounds of the block. *)
+
+Definition in_bounds (chunk: memory_chunk) (ofs: Z) (c: block_contents) : 
+        {c.(low) <= ofs /\ ofs + size_chunk chunk <= c.(high)}
+      + {c.(low) > ofs \/ ofs + size_chunk chunk > c.(high)} :=
+  match zle c.(low) ofs, zle (ofs + size_chunk chunk) c.(high) with
+  | left P1, left P2 => left _ (conj P1 P2)
+  | left P1, right P2 => right _ (or_intror _ P2)
+  | right P1, _ => right _ (or_introl _ P1)
+  end.
+
+Lemma in_bounds_holds:
+  forall (chunk: memory_chunk) (ofs: Z) (c: block_contents)
+         (A: Set) (a b: A),
+  c.(low) <= ofs -> ofs + size_chunk chunk <= c.(high) ->
+  (if in_bounds chunk ofs c then a else b) = a.
+Proof.
+  intros. case (in_bounds chunk ofs c); intro.
+  auto.
+  omegaContradiction.
+Qed.
+
+Lemma in_bounds_exten:
+  forall (chunk: memory_chunk) (ofs: Z) (c: block_contents) (x: contentmap) P,
+  in_bounds chunk ofs (mkblock (low c) (high c) x P) =
+  in_bounds chunk ofs c.
+Proof.
+  intros; reflexivity.
+Qed.
+
+Hint Resolve in_bounds_holds in_bounds_exten.
+
+(** [valid_pointer] holds if the given block address is valid and the
+  given offset falls within the bounds of the corresponding block. *)
+
+Definition valid_pointer (m: mem) (b: block) (ofs: Z) : bool :=
+  if zlt b m.(nextblock) then
+    (let c := m.(blocks) b in
+     if zle c.(low) ofs then if zlt ofs c.(high) then true else false
+                        else false)
+  else false.
+
+(** Read a quantity of size [sz] at offset [ofs] in block contents [c].
+  Return [Vundef] if the requested size does not match that of the
+  current contents, or if the following offsets do not contain [Cont].
+  The first check captures a size mismatch between the read and the
+  latest write at this offset.  The second check captures partial overwriting
+  of the latest write at this offset by a more recent write at a nearby
+  offset. *)
+
+Definition load_contents (sz: memory_size) (c: contentmap) (ofs: Z) : val :=
+  match sz with
+  | Size8 =>
+      match getN 0%nat ofs c with
+      | Datum8 n => n
+      | _ => Vundef
+      end
+  | Size16 =>
+      match getN 1%nat ofs c with
+      | Datum16 n => n
+      | _ => Vundef
+      end
+  | Size32 =>
+      match getN 3%nat ofs c with
+      | Datum32 n => n
+      | _ => Vundef
+      end
+  | Size64 =>
+      match getN 7%nat ofs c with
+      | Datum64 n => n
+      | _ => Vundef
+      end
+  end.
+
+(** [load chunk m b ofs] perform a read in memory state [m], at address
+  [b] and offset [ofs].  [None] is returned if the address is invalid
+  or the memory access is out of bounds. *)
+
+Definition load (chunk: memory_chunk) (m: mem) (b: block) (ofs: Z)
+                : option val :=
+  if zlt b m.(nextblock) then
+    (let c := m.(blocks) b in
+     if in_bounds chunk ofs c
+     then Some (Val.load_result chunk
+                    (load_contents (mem_chunk chunk) c.(contents) ofs))
+     else None)
+  else
+    None.
+
+(** [loadv chunk m addr] is similar, but the address and offset are given
+  as a single value [addr], which must be a pointer value. *)
+
+Definition loadv (chunk: memory_chunk) (m: mem) (addr: val) : option val :=
+  match addr with
+  | Vptr b ofs => load chunk m b (Int.signed ofs)
+  | _ => None
+  end.
+
+Theorem load_in_bounds:
+  forall (chunk: memory_chunk) (m: mem) (b: block) (ofs: Z),
+  valid_block m b ->
+  low_bound m b <= ofs ->
+  ofs + size_chunk chunk <= high_bound m b ->
+  exists v, load chunk m b ofs = Some v.
+Proof.
+  intros. unfold load. rewrite zlt_true; auto.
+  rewrite in_bounds_holds. 
+  exists (Val.load_result chunk
+            (load_contents (mem_chunk chunk)
+                           (contents (m.(blocks) b))
+                           ofs)).
+  auto.
+  exact H0. exact H1.
+Qed.
+
+Lemma load_inv:
+  forall (chunk: memory_chunk) (m: mem) (b: block) (ofs: Z) (v: val),
+  load chunk m b ofs = Some v ->
+  let c := m.(blocks) b in
+  b < m.(nextblock) /\
+  c.(low) <= ofs /\
+  ofs + size_chunk chunk <= c.(high) /\
+  Val.load_result chunk (load_contents (mem_chunk chunk)  c.(contents) ofs) = v.
+Proof.
+  intros until v; unfold load.
+  case (zlt b (nextblock m)); intro.
+  set (c := m.(blocks) b).
+  case (in_bounds chunk ofs c).
+  intuition congruence.
+  intros; discriminate.
+  intros; discriminate.
+Qed.
+Hint Resolve load_in_bounds load_inv.
+
+(* Write the value [v] with size [sz] at offset [ofs] in block contents [c].
+   Return updated block contents.  [Cont] contents are stored at the following
+   offsets. *)
+
+Definition store_contents (sz: memory_size) (c: contentmap)
+                          (ofs: Z) (v: val) : contentmap :=
+  match sz with
+  | Size8 =>
+      setN 0%nat ofs (Datum8 v) c
+  | Size16 =>
+      setN 1%nat ofs (Datum16 v) c
+  | Size32 =>
+      setN 3%nat ofs (Datum32 v) c
+  | Size64 =>
+      setN 7%nat ofs (Datum64 v) c
+  end.
+
+Remark store_contents_undef_outside:
+  forall sz c ofs v lo hi,
+  lo <= ofs /\ ofs + size_mem sz <= hi ->
+  (forall x, x < lo \/ x >= hi -> c x = Undef) ->
+  (forall x, x < lo \/ x >= hi ->
+     store_contents sz c ofs v x = Undef).
+Proof.
+  intros until hi; intros [LO HI] UO.
+  assert (forall n d x, 
+            ofs + (1 + Z_of_nat n) <= hi ->
+            x < lo \/ x >= hi ->
+            setN n ofs d c x = Undef).
+    intros. unfold setN. rewrite update_o.
+    transitivity (c x). apply set_cont_outside. omega. 
+    apply UO. omega. omega.
+  unfold store_contents; destruct sz; unfold size_mem in HI;
+  intros; apply H; auto; simpl Z_of_nat; auto.
+Qed.
+
+Definition unchecked_store
+     (chunk: memory_chunk) (m: mem) (b: block)
+     (ofs: Z) (v: val)
+     (P: (m.(blocks) b).(low) <= ofs /\
+         ofs + size_chunk chunk <= (m.(blocks) b).(high)) : mem :=
+  let c := m.(blocks) b in
+  mkmem
+    (update b
+        (mkblock c.(low) c.(high)
+                 (store_contents (mem_chunk chunk) c.(contents) ofs v)
+                 (store_contents_undef_outside (mem_chunk chunk) c.(contents)
+                      ofs v _ _ P c.(undef_outside)))
+        m.(blocks))
+    m.(nextblock)
+    m.(nextblock_pos).
+
+(** [store chunk m b ofs v] perform a write in memory state [m].
+  Value [v] is stored at address [b] and offset [ofs].
+  Return the updated memory store, or [None] if the address is invalid
+  or the memory access is out of bounds. *)
+
+Definition store (chunk: memory_chunk) (m: mem) (b: block)
+                 (ofs: Z) (v: val) : option mem :=
+  if zlt b m.(nextblock) then
+    match in_bounds chunk ofs (m.(blocks) b) with
+    | left P => Some(unchecked_store chunk m b ofs v P)
+    | right _ => None
+    end
+  else
+    None.
+
+(** [storev chunk m addr v] is similar, but the address and offset are given
+  as a single value [addr], which must be a pointer value. *)
+
+Definition storev (chunk: memory_chunk) (m: mem) (addr v: val) : option mem :=
+  match addr with
+  | Vptr b ofs => store chunk m b (Int.signed ofs) v
+  | _ => None
+  end.
+
+Theorem store_in_bounds:
+  forall (chunk: memory_chunk) (m: mem) (b: block) (ofs: Z) (v: val),
+  valid_block m b ->
+  low_bound m b <= ofs ->
+  ofs + size_chunk chunk <= high_bound m b ->
+  exists m', store chunk m b ofs v = Some m'.
+Proof.
+  intros. unfold store.
+  rewrite zlt_true; auto. 
+  case (in_bounds chunk ofs (blocks m b)); intro P.
+  exists (unchecked_store chunk m b ofs v P). reflexivity.
+  unfold low_bound in H0. unfold high_bound in H1. omegaContradiction.
+Qed.
+
+Lemma store_inv:
+  forall (chunk: memory_chunk) (m m': mem) (b: block) (ofs: Z) (v: val),
+  store chunk m b ofs v = Some m' ->
+  let c := m.(blocks) b in
+  b < m.(nextblock) /\
+  c.(low) <= ofs /\
+  ofs + size_chunk chunk <= c.(high) /\
+  m'.(nextblock) = m.(nextblock) /\
+  exists P, m'.(blocks) =
+    update b (mkblock c.(low) c.(high)
+                        (store_contents (mem_chunk chunk) c.(contents) ofs v) P)
+                        m.(blocks).
+Proof.
+  intros until v; unfold store.
+  case (zlt b (nextblock m)); intro.
+  set (c := m.(blocks) b).
+  case (in_bounds chunk ofs c).
+  intros; injection H; intro; subst m'. simpl.
+  intuition. fold c. 
+  exists (store_contents_undef_outside (mem_chunk chunk) 
+            (contents c) ofs v (low c) (high c) a (undef_outside c)).
+  auto.
+  intros; discriminate.
+  intros; discriminate.
+Qed.
+
+Hint Resolve store_in_bounds store_inv.
+
+(** * Properties of the memory operations *)
+
+(** ** Properties related to block validity *)
+
+Lemma valid_not_valid_diff:
+  forall m b b', valid_block m b -> ~(valid_block m b') -> b <> b'.
+Proof.
+  intros; red; intros. subst b'. contradiction.
+Qed.
+
+Theorem fresh_block_alloc:
+  forall (m1 m2: mem) (lo hi: Z) (b: block), 
+  alloc m1 lo hi = (m2, b) -> ~(valid_block m1 b).
+Proof.
+  intros. injection H; intros; subst b. 
+  unfold valid_block. omega.
+Qed.
+
+Theorem valid_new_block:
+  forall (m1 m2: mem) (lo hi: Z) (b: block), 
+  alloc m1 lo hi = (m2, b) -> valid_block m2 b.
+Proof.
+  unfold alloc, valid_block; intros.
+  injection H; intros. subst b; subst m2; simpl. omega.
+Qed.
+
+Theorem valid_block_alloc:
+  forall (m1 m2: mem) (lo hi: Z) (b b': block), 
+  alloc m1 lo hi = (m2, b') ->
+  valid_block m1 b -> valid_block m2 b.
+Proof.
+  unfold alloc, valid_block; intros.
+  injection H; intros. subst m2; simpl. omega.
+Qed.
+
+Theorem valid_block_store:
+  forall (chunk: memory_chunk) (m1 m2: mem) (b b': block) (ofs: Z) (v: val),
+  store chunk m1 b' ofs v = Some m2 ->
+  valid_block m1 b -> valid_block m2 b.
+Proof.
+  intros. generalize (store_inv _ _ _ _ _ _ H). 
+  intros [A [B [C [D [P E]]]]].
+  red. rewrite D. exact H0.
+Qed.
+
+Theorem valid_block_free:
+  forall (m: mem) (b b': block), 
+  valid_block m b -> valid_block (free m b') b.
+Proof.
+  unfold valid_block, free; intros.
+  simpl. auto.
+Qed.
+
+(** ** Properties related to [alloc] *)
+
+Theorem load_alloc_other:
+  forall (chunk: memory_chunk) (m1 m2: mem)
+         (b b': block) (ofs lo hi: Z) (v: val),
+  alloc m1 lo hi = (m2, b') ->
+  load chunk m1 b ofs = Some v ->
+  load chunk m2 b ofs = Some v.
+Proof.
+  unfold alloc; intros.
+  injection H; intros; subst m2; clear H.
+  generalize (load_inv _ _ _ _ _ H0).
+  intros (A, (B, (C, D))).
+  unfold load; simpl.
+  rewrite zlt_true.
+  repeat (rewrite update_o).
+  rewrite in_bounds_holds. congruence. auto. auto.
+  omega. omega.
+Qed.
+
+Lemma load_contents_init:
+  forall (sz: memory_size) (ofs: Z),
+  load_contents sz (fun y => Undef) ofs = Vundef.
+Proof.
+  intros. destruct sz; reflexivity.
+Qed.
+
+Theorem load_alloc_same:
+  forall (chunk: memory_chunk) (m1 m2: mem)
+         (b b': block) (ofs lo hi: Z) (v: val),
+  alloc m1 lo hi = (m2, b') ->
+  load chunk m2 b' ofs = Some v ->
+  v = Vundef.
+Proof.
+  unfold alloc; intros.
+  injection H; intros; subst m2; clear H.
+  generalize (load_inv _ _ _ _ _ H0).
+  simpl. rewrite H1. rewrite update_s. simpl. intros (A, (B, (C, D))).
+  rewrite <- D. rewrite load_contents_init. 
+  destruct chunk; reflexivity.
+Qed.  
+
+Theorem low_bound_alloc:
+  forall (m1 m2: mem) (b b': block) (lo hi: Z),
+  alloc m1 lo hi = (m2, b') ->
+  low_bound m2 b = if zeq b b' then lo else low_bound m1 b.
+Proof.
+  unfold alloc; intros. 
+  injection H; intros; subst m2; clear H.
+  unfold low_bound; simpl. 
+  unfold update.
+  subst b'.
+  case (zeq b (nextblock m1)); reflexivity.
+Qed.
+
+Theorem high_bound_alloc:
+  forall (m1 m2: mem) (b b': block) (lo hi: Z),
+  alloc m1 lo hi = (m2, b') ->
+  high_bound m2 b = if zeq b b' then hi else high_bound m1 b.
+Proof.
+  unfold alloc; intros. 
+  injection H; intros; subst m2; clear H.
+  unfold high_bound; simpl. 
+  unfold update.
+  subst b'.
+  case (zeq b (nextblock m1)); reflexivity.
+Qed.
+
+Theorem store_alloc:
+  forall (chunk: memory_chunk) (m1 m2: mem) (b: block) (ofs lo hi: Z) (v: val),
+  alloc m1 lo hi = (m2, b) ->
+  lo <= ofs -> ofs + size_chunk chunk <= hi ->
+  exists m2', store chunk m2 b ofs v = Some m2'.
+Proof.
+  unfold alloc; intros.
+  injection H; intros.
+  assert (A: b < m2.(nextblock)).
+  subst m2; subst b; simpl; omega.
+  assert (B: low_bound m2 b <= ofs).
+  subst m2; subst b. unfold low_bound; simpl. rewrite update_s.
+  simpl. assumption.
+  assert (C: ofs + size_chunk chunk <= high_bound m2 b).
+  subst m2; subst b. unfold high_bound; simpl. rewrite update_s.
+  simpl. assumption.
+  exact (store_in_bounds chunk m2 b ofs v A B C).
+Qed.
+
+Hint Resolve store_alloc high_bound_alloc low_bound_alloc load_alloc_same
+load_contents_init load_alloc_other.
+
+(** ** Properties related to [free] *)
+
+Theorem load_free:
+  forall (chunk: memory_chunk) (m: mem) (b bf: block) (ofs: Z),
+  b <> bf ->
+  load chunk (free m bf) b ofs = load chunk m b ofs.
+Proof.
+  intros. unfold free, load; simpl.
+  case (zlt b (nextblock m)).
+  repeat (rewrite update_o; auto).
+  reflexivity.
+Qed.
+
+Theorem low_bound_free:
+  forall (m: mem) (b bf: block),
+  b <> bf ->
+  low_bound (free m bf) b = low_bound m b.
+Proof.
+  intros. unfold free, low_bound; simpl.
+  rewrite update_o; auto.
+Qed.
+
+Theorem high_bound_free:
+  forall (m: mem) (b bf: block),
+  b <> bf ->
+  high_bound (free m bf) b = high_bound m b.
+Proof.
+  intros. unfold free, high_bound; simpl.
+  rewrite update_o; auto.
+Qed.
+Hint Resolve load_free low_bound_free high_bound_free.
+
+(** ** Properties related to [store] *)
+
+Lemma store_is_in_bounds:
+  forall chunk m1 b ofs v m2,
+  store chunk m1 b ofs v = Some m2 ->
+  low_bound m1 b <= ofs /\ ofs + size_chunk chunk <= high_bound m1 b.
+Proof.
+  intros. generalize (store_inv _ _ _ _ _ _ H).
+  intros [A [B [C [P D]]]].
+  unfold low_bound, high_bound. tauto.
+Qed.
+
+Lemma load_store_contents_same:
+  forall (sz: memory_size) (c: contentmap) (ofs: Z) (v: val),
+  load_contents sz (store_contents sz c ofs v) ofs = v.
+Proof.
+  intros until v.
+  unfold load_contents, store_contents in |- *; case sz;
+    rewrite getN_setN_same; reflexivity.
+Qed.
+  
+Theorem load_store_same:
+  forall (chunk: memory_chunk) (m1 m2: mem) (b: block) (ofs: Z) (v: val),
+  store chunk m1 b ofs v = Some m2 ->
+  load chunk m2 b ofs = Some (Val.load_result chunk v).
+Proof.
+  intros.
+  generalize (store_inv _ _ _ _ _ _ H).
+  intros (A, (B, (C, (D, (P, E))))).
+  unfold load. rewrite D. 
+  rewrite zlt_true; auto. rewrite E. 
+  repeat (rewrite update_s). simpl.
+  rewrite in_bounds_exten. rewrite in_bounds_holds; auto.
+  rewrite load_store_contents_same; auto.
+Qed.
+           
+Lemma load_store_contents_other:
+  forall (sz1 sz2: memory_size) (c: contentmap) 
+         (ofs1 ofs2: Z) (v: val),
+  ofs2 + size_mem sz2 <= ofs1 \/ ofs1 + size_mem sz1 <= ofs2 ->
+  load_contents sz2 (store_contents sz1 c ofs1 v) ofs2 =
+  load_contents sz2 c ofs2.
+Proof.
+  intros until v.
+  unfold size_mem, store_contents, load_contents;
+  case sz1; case sz2; intros;
+  (rewrite getN_setN_other; 
+   [reflexivity | simpl Z_of_nat; omega]).
+Qed.
+
+Theorem load_store_other:
+  forall (chunk1 chunk2: memory_chunk) (m1 m2: mem)
+         (b1 b2: block) (ofs1 ofs2: Z) (v: val),
+  store chunk1 m1 b1 ofs1 v = Some m2 ->
+  b1 <> b2
+  \/ ofs2 + size_chunk chunk2 <= ofs1
+  \/ ofs1 + size_chunk chunk1 <= ofs2 ->
+  load chunk2 m2 b2 ofs2 = load chunk2 m1 b2 ofs2.
+Proof.
+  intros.
+  generalize (store_inv _ _ _ _ _ _ H).
+  intros (A, (B, (C, (D, (P, E))))).
+  unfold load. rewrite D.
+  case (zlt b2 (nextblock m1)); intro.
+  rewrite E; unfold update; case (zeq b2 b1); intro; simpl.
+  subst b2. rewrite in_bounds_exten. 
+  rewrite load_store_contents_other. auto.
+  tauto.
+  reflexivity. 
+  reflexivity.
+Qed.
+
+Ltac LSCO :=
+  match goal with
+  | |- (match getN ?sz2 ?ofs2 (setN ?sz1 ?ofs1 ?v ?c) with
+        | Undef => _
+        | Datum8 _ => _
+        | Datum16 _ => _
+        | Datum32 _ => _
+        | Datum64 _ => _
+        | Cont => _ 
+        end = _) =>
+    elim (getN_setN_overlap sz1 sz2 ofs1 ofs2 v c);
+    [ let H := fresh in (intro H; rewrite H; reflexivity)
+    | let H := fresh in (intro H; rewrite H; reflexivity)
+    | assumption
+    | simpl Z_of_nat; omega
+    | simpl Z_of_nat; omega
+    | discriminate ]
+  end.
+
+Lemma load_store_contents_overlap:
+  forall (sz1 sz2: memory_size) (c: contentmap) 
+         (ofs1 ofs2: Z) (v: val),
+  ofs1 <> ofs2 ->
+  ofs2 + size_mem sz2 > ofs1 -> ofs1 + size_mem sz1 > ofs2 ->
+  load_contents sz2 (store_contents sz1 c ofs1 v) ofs2 = Vundef.
+Proof.
+  intros.
+  destruct sz1; destruct sz2; simpl in H0; simpl in H1; simpl; LSCO.
+Qed.
+
+Ltac LSCM :=
+  match goal with
+  | H:(?x <> ?x) |- _ =>
+    elim H; reflexivity
+  | |- (match getN ?sz2 ?ofs (setN ?sz1 ?ofs ?v ?c) with
+        | Undef => _
+        | Datum8 _ => _
+        | Datum16 _ => _
+        | Datum32 _ => _
+        | Datum64 _ => _
+        | Cont => _ 
+        end = _) =>
+    elim (getN_setN_mismatch sz1 sz2 ofs v c);
+    [ let H := fresh in (intro H; rewrite H; reflexivity)
+    | let H := fresh in (intro H; rewrite H; reflexivity) ]
+  end.
+
+Lemma load_store_contents_mismatch:
+  forall (sz1 sz2: memory_size) (c: contentmap) 
+         (ofs: Z) (v: val),
+  sz1 <> sz2 ->
+  load_contents sz2 (store_contents sz1 c ofs v) ofs = Vundef.
+Proof.
+  intros.
+  destruct sz1; destruct sz2; simpl; LSCM.
+Qed.  
+
+Theorem low_bound_store:
+  forall (chunk: memory_chunk) (m1 m2: mem) (b b': block) (ofs: Z) (v: val),
+  store chunk m1 b ofs v = Some m2 ->
+  low_bound m2 b' = low_bound m1 b'.
+Proof.
+  intros.
+  generalize (store_inv _ _ _ _ _ _ H).
+  intros (A, (B, (C, (D, (P, E))))).
+  unfold low_bound. rewrite E; unfold update.
+  case (zeq b' b); intro.
+  subst b'. reflexivity.
+  reflexivity.
+Qed.
+
+Theorem high_bound_store:
+  forall (chunk: memory_chunk) (m1 m2: mem) (b b': block) (ofs: Z) (v: val),
+  store chunk m1 b ofs v = Some m2 ->
+  high_bound m2 b' = high_bound m1 b'.
+Proof.
+  intros.
+  generalize (store_inv _ _ _ _ _ _ H).
+  intros (A, (B, (C, (D, (P, E))))).
+  unfold high_bound. rewrite E; unfold update.
+  case (zeq b' b); intro.
+  subst b'. reflexivity.
+  reflexivity. 
+Qed.
+
+Hint Resolve high_bound_store low_bound_store load_store_contents_mismatch
+  load_store_contents_overlap load_store_other store_is_in_bounds  
+  load_store_contents_same  load_store_same load_store_contents_other.
+
+(** * Agreement between memory blocks. *)
+
+(** Two memory blocks [c1] and [c2] agree on a range [lo] to [hi]
+  if they associate the same contents to byte offsets in the range
+  [lo] (included) to [hi] (excluded). *)
+
+Definition contentmap_agree (lo hi: Z) (c1 c2: contentmap) :=
+  forall (ofs: Z),
+    lo <= ofs < hi -> c1 ofs = c2 ofs.
+
+Definition block_contents_agree (lo hi: Z) (c1 c2: block_contents) :=
+  contentmap_agree lo hi c1.(contents) c2.(contents).
+
+Definition block_agree (b: block) (lo hi: Z) (m1 m2: mem) :=
+  block_contents_agree lo hi
+     (m1.(blocks) b) (m2.(blocks) b).
+
+Theorem block_agree_refl:
+  forall (m: mem) (b: block) (lo hi: Z),
+  block_agree b lo hi m m.
+Proof.
+  intros. hnf. auto.
+Qed.
+
+Theorem block_agree_sym:
+  forall (m1 m2: mem) (b: block) (lo hi: Z),
+  block_agree b lo hi m1 m2 ->
+  block_agree b lo hi m2 m1.
+Proof.
+  intros. hnf. intros. symmetry. apply H. auto.
+Qed.
+
+Theorem block_agree_trans:
+  forall (m1 m2 m3: mem) (b: block) (lo hi: Z),
+  block_agree b lo hi m1 m2 ->
+  block_agree b lo hi m2 m3 ->
+  block_agree b lo hi m1 m3.
+Proof.
+  intros. hnf. intros. 
+  transitivity (contents (blocks m2 b) ofs).
+  apply H; auto. apply H0; auto.
+Qed.
+
+Lemma check_cont_agree:
+  forall (c1 c2: contentmap) (lo hi: Z),
+  contentmap_agree lo hi c1 c2 ->
+  forall (n: nat) (ofs: Z),
+  lo <= ofs -> ofs + Z_of_nat n <= hi ->
+  check_cont n ofs c2 = check_cont n ofs c1.
+Proof.
+  induction n; intros; simpl.
+  auto.
+  rewrite inj_S in H1.
+  rewrite H. case (c2 ofs); intros; auto. 
+  apply IHn; omega.
+  omega.
+Qed.
+
+Lemma getN_agree:
+  forall (c1 c2: contentmap) (lo hi: Z),
+  contentmap_agree lo hi c1 c2 ->
+  forall (n: nat) (ofs: Z),
+  lo <= ofs -> ofs + Z_of_nat n < hi ->
+  getN n ofs c2 = getN n ofs c1.
+Proof.
+  intros. unfold getN. 
+  rewrite (check_cont_agree c1 c2 lo hi H n (ofs + 1)).
+  case (check_cont n (ofs + 1) c1).
+  symmetry. apply H. omega. auto.
+  omega. omega.
+Qed.
+
+Lemma load_contentmap_agree:
+  forall (sz: memory_size) (c1 c2: contentmap) (lo hi ofs: Z),
+  contentmap_agree lo hi c1 c2 ->
+  lo <= ofs -> ofs + size_mem sz <= hi ->
+  load_contents sz c2 ofs = load_contents sz c1 ofs.
+Proof.
+  intros sz c1 c2 lo hi ofs EX LO.
+  unfold load_contents, size_mem; case sz; intro HI;
+  rewrite (getN_agree c1 c2 lo hi EX); auto; simpl Z_of_nat; omega.
+Qed.
+
+Lemma set_cont_agree:
+  forall (lo hi: Z) (n: nat) (c1 c2: contentmap) (ofs: Z),
+  contentmap_agree lo hi c1 c2 ->
+  contentmap_agree lo hi (set_cont n ofs c1) (set_cont n ofs c2).
+Proof.
+  induction n; simpl; intros.
+  auto.
+  red. intros p B. 
+  case (zeq p ofs); intro.
+  subst p. repeat (rewrite update_s). reflexivity.
+  repeat (rewrite update_o). apply IHn. assumption.
+  assumption. auto. auto.
+Qed.
+
+Lemma setN_agree:
+  forall (lo hi: Z) (n: nat) (c1 c2: contentmap) (ofs: Z) (v: content),
+  contentmap_agree lo hi c1 c2 ->
+  contentmap_agree lo hi (setN n ofs v c1) (setN n ofs v c2).
+Proof.
+  intros. unfold setN. red; intros p B.
+  case (zeq p ofs); intro.
+  subst p. repeat (rewrite update_s). reflexivity.
+  repeat (rewrite update_o; auto). 
+  exact (set_cont_agree lo hi n c1 c2 (ofs + 1) H p B).  
+Qed.
+
+Lemma store_contentmap_agree:
+  forall (sz: memory_size) (c1 c2: contentmap) (lo hi ofs: Z) (v: val),
+  contentmap_agree lo hi c1 c2 ->
+  contentmap_agree lo hi
+       (store_contents sz c1 ofs v) (store_contents sz c2 ofs v).
+Proof.
+  intros. unfold store_contents; case sz; apply setN_agree; auto.
+Qed.
+
+Lemma set_cont_outside_agree:
+  forall (lo hi: Z) (n: nat) (c1 c2: contentmap) (ofs: Z),
+  contentmap_agree lo hi c1 c2 ->
+  ofs + Z_of_nat n <= lo \/ hi <= ofs ->
+  contentmap_agree lo hi c1 (set_cont n ofs c2).
+Proof.
+  induction n; intros; simpl.
+  auto.
+  red; intros p R. rewrite inj_S in H0.
+  unfold update. case (zeq p ofs); intro.
+  subst p. omegaContradiction.
+  apply IHn. auto. omega. auto. 
+Qed.
+
+Lemma setN_outside_agree:
+  forall (lo hi: Z) (n: nat) (c1 c2: contentmap) (ofs: Z) (v: content),
+  contentmap_agree lo hi c1 c2 ->
+  ofs + Z_of_nat n < lo \/ hi <= ofs ->
+  contentmap_agree lo hi c1 (setN n ofs v c2).
+Proof.
+  intros. unfold setN. red; intros p R.
+  unfold update. case (zeq p ofs); intro.
+  omegaContradiction.
+  apply (set_cont_outside_agree lo hi n c1 c2 (ofs + 1) H).
+  omega. auto.
+Qed.
+
+Lemma store_contentmap_outside_agree:
+  forall (sz: memory_size) (c1 c2: contentmap) (lo hi ofs: Z) (v: val),
+  contentmap_agree lo hi c1 c2 ->
+  ofs + size_mem sz <= lo \/ hi <= ofs ->
+  contentmap_agree lo hi c1 (store_contents sz c2 ofs v).
+Proof.
+  intros until v.
+  unfold store_contents; case sz; simpl; intros;
+  apply setN_outside_agree; auto; simpl Z_of_nat; omega.
+Qed.
+
+Theorem load_agree:
+  forall (chunk: memory_chunk) (m1 m2: mem) 
+         (b: block) (lo hi: Z) (ofs: Z) (v1 v2: val),
+  block_agree b lo hi m1 m2 ->
+  lo <= ofs -> ofs + size_chunk chunk <= hi ->
+  load chunk m1 b ofs = Some v1 ->
+  load chunk m2 b ofs = Some v2 ->
+  v1 = v2.
+Proof.
+  intros. 
+  generalize (load_inv _ _ _ _ _ H2). intros [K [L [M N]]].
+  generalize (load_inv _ _ _ _ _ H3). intros [P [Q [R S]]].
+  subst v1; subst v2. symmetry.
+  decEq. eapply load_contentmap_agree. 
+  apply H. auto. auto. 
+Qed.
+
+Theorem store_agree:
+  forall (chunk: memory_chunk) (m1 m2 m1' m2': mem)
+         (b: block) (lo hi: Z)
+         (b': block) (ofs: Z) (v: val),
+  block_agree b lo hi m1 m2 ->
+  store chunk m1 b' ofs v = Some m1' ->
+  store chunk m2 b' ofs v = Some m2' ->
+  block_agree b lo hi m1' m2'.
+Proof.
+  intros.
+  generalize (store_inv _ _ _ _ _ _ H0).
+  intros [I [J [K [L [x M]]]]].
+  generalize (store_inv _ _ _ _ _ _ H1).
+  intros [P [Q [R [S [y T]]]]].
+  red. red. 
+  rewrite M; rewrite T; unfold update.
+  case (zeq b b'); intro; simpl.
+  subst b'. apply store_contentmap_agree. assumption.
+  apply H. 
+Qed.
+
+Theorem store_outside_agree:
+  forall (chunk: memory_chunk) (m1 m2 m2': mem)
+         (b: block) (lo hi: Z)
+         (b': block) (ofs: Z) (v: val),
+  block_agree b lo hi m1 m2 ->
+  b <> b' \/ ofs + size_chunk chunk <= lo \/ hi <= ofs ->
+  store chunk m2 b' ofs v = Some m2' ->
+  block_agree b lo hi m1 m2'.
+Proof.
+  intros.
+  generalize (store_inv _ _ _ _ _ _ H1).
+  intros [I [J [K [L [x M]]]]].
+  red. red. rewrite M; unfold update;
+  case (zeq b b'); intro; simpl. 
+  subst b'. apply store_contentmap_outside_agree.
+  assumption. 
+  elim H0; intro. tauto. exact H2.
+  apply H.
+Qed.
+
+(** * Store extensions *)
+
+(** A store [m2] extends a store [m1] if [m2] can be obtained from [m1]
+  by increasing the sizes of the memory blocks of [m1] (decreasing
+  the low bounds, increasing the high bounds), while still keeping the
+  same contents for block offsets that are valid in [m1]. 
+  This notion is used in the proof of semantic equivalence in 
+  module [Machenv]. *)
+
+Definition block_contents_extends (c1 c2: block_contents) :=
+  c2.(low) <= c1.(low) /\ c1.(high) <= c2.(high) /\
+  contentmap_agree c1.(low) c1.(high) c1.(contents) c2.(contents).
+
+Definition extends (m1 m2: mem) :=
+  m1.(nextblock) = m2.(nextblock) /\
+  forall (b: block),
+  b < m1.(nextblock) ->
+  block_contents_extends (m1.(blocks) b) (m2.(blocks) b).
+
+Theorem extends_refl:
+  forall (m: mem), extends m m.
+Proof.
+  intro. red. split.
+  reflexivity.
+  intros. red. 
+  split. omega. split. omega. 
+  red. intros. reflexivity.
+Qed.
+
+Theorem alloc_extends:
+  forall (m1 m2 m1' m2': mem) (lo1 hi1 lo2 hi2: Z) (b1 b2: block),
+  extends m1 m2 ->
+  lo2 <= lo1 -> hi1 <= hi2 ->
+  alloc m1 lo1 hi1 = (m1', b1) ->
+  alloc m2 lo2 hi2 = (m2', b2) ->
+  b1 = b2 /\ extends m1' m2'.
+Proof.
+  unfold extends, alloc; intros.
+  injection H2; intros; subst m1'; subst b1.
+  injection H3; intros; subst m2'; subst b2.
+  simpl. intuition.
+  rewrite <- H4. case (zeq b (nextblock m1)); intro.
+  subst b. repeat rewrite update_s.
+  red; simpl. intuition. 
+  red; intros; reflexivity.
+  repeat rewrite update_o.  apply H5.  omega.
+  auto. auto.
+Qed.
+
+Theorem free_extends:
+  forall (m1 m2: mem) (b: block),
+  extends m1 m2 ->
+  extends (free m1 b) (free m2 b).
+Proof.
+  unfold extends, free; intros.
+  simpl. intuition. 
+  red; intros; simpl. 
+  case (zeq b0 b); intro.
+  subst b0; repeat (rewrite update_s). 
+  simpl. split. omega. split. omega. 
+  red. intros. omegaContradiction. 
+  repeat (rewrite update_o). 
+  exact (H1 b0 H).
+  auto. auto.  
+Qed.
+
+Theorem load_extends:
+  forall (chunk: memory_chunk) (m1 m2: mem) (b: block) (ofs: Z) (v: val),
+  extends m1 m2 ->
+  load chunk m1 b ofs = Some v ->
+  load chunk m2 b ofs = Some v.
+Proof.
+  intros sz m1 m2 b ofs v (X, Y) L.
+  generalize (load_inv _ _ _ _ _ L).
+  intros (A, (B, (C, D))).
+  unfold load. rewrite <- X. rewrite zlt_true; auto.
+  generalize (Y b A); intros [M [P Q]].
+  rewrite in_bounds_holds.
+  rewrite <- D. 
+  decEq. decEq.
+  apply load_contentmap_agree with
+    (lo := low((blocks m1) b))
+    (hi := high((blocks m1) b)); auto.
+  omega. omega. 
+Qed.
+
+Theorem store_within_extends:
+  forall (chunk: memory_chunk) (m1 m2 m1': mem) (b: block) (ofs: Z) (v: val),
+  extends m1 m2 ->
+  store chunk m1 b ofs v = Some m1' ->
+  exists m2', store chunk m2 b ofs v = Some m2' /\ extends m1' m2'.
+Proof.
+  intros sz m1 m2 m1' b ofs v (X, Y) STORE.
+  generalize (store_inv _ _ _ _ _ _ STORE).
+  intros (A, (B, (C, (D, (x, E))))).
+  generalize (Y b A); intros [M [P Q]].
+  unfold store. rewrite <- X. rewrite zlt_true; auto.
+  case (in_bounds sz ofs (blocks m2 b)); intro.
+  exists (unchecked_store sz m2 b ofs v a).
+  split. auto.
+  split. 
+  unfold unchecked_store; simpl. congruence.
+  intros b' LT. 
+  unfold block_contents_extends, unchecked_store, block_contents_agree.
+  rewrite E; unfold update; simpl.
+  case (zeq b' b); intro; simpl.
+  subst b'. split. omega. split. omega. 
+  apply store_contentmap_agree. auto.
+  apply Y. rewrite <- D. auto.
+  omegaContradiction.
+Qed.
+
+Theorem store_outside_extends:
+  forall (chunk: memory_chunk) (m1 m2 m2': mem) (b: block) (ofs: Z) (v: val),
+  extends m1 m2 ->
+  ofs + size_chunk chunk <= low_bound m1 b \/ high_bound m1 b <= ofs ->
+  store chunk m2 b ofs v = Some m2' ->
+  extends m1 m2'.
+Proof.
+  intros sz m1 m2 m2' b ofs v (X, Y) BOUNDS STORE.
+  generalize (store_inv _ _ _ _ _ _ STORE).
+  intros (A, (B, (C, (D, (x, E))))).
+  split.
+  congruence.
+  intros b' LT.
+  rewrite E; unfold update; case (zeq b' b); intro.
+  subst b'. generalize (Y b LT). intros [M [P Q]].
+  red; simpl. split. omega. split. omega. 
+  apply store_contentmap_outside_agree.
+  assumption. exact BOUNDS. 
+  auto. 
+Qed.
+
+(** * Memory extensionality properties *)
+
+Lemma block_contents_exten:
+  forall (c1 c2: block_contents),
+  c1.(low) = c2.(low) ->
+  c1.(high) = c2.(high) ->
+  block_contents_agree c1.(low) c1.(high) c1 c2 ->
+  c1 = c2.
+Proof.
+  intros. caseEq c1; intros lo1 hi1 m1 UO1 EQ1. subst c1.
+  caseEq c2; intros lo2 hi2 m2 UO2 EQ2. subst c2.
+  simpl in *. subst lo2 hi2. 
+  assert (m1 = m2). 
+    unfold contentmap. apply extensionality. intro.
+    case (zlt x lo1); intro.
+    rewrite UO1. rewrite UO2. auto. tauto. tauto.
+    case (zlt x hi1); intro.
+    apply H1. omega.
+    rewrite UO1. rewrite UO2. auto. tauto. tauto.
+  subst m2. 
+  assert (UO1 = UO2).
+    apply proof_irrelevance.
+  subst UO2. reflexivity.
+Qed.
+
+Theorem mem_exten:
+  forall m1 m2,
+  m1.(nextblock) = m2.(nextblock) ->
+  (forall b, m1.(blocks) b = m2.(blocks) b) ->
+  m1 = m2.
+Proof.
+  intros. destruct m1 as [map1 nb1 nextpos1]. destruct m2 as [map2 nb2 nextpos2].
+  simpl in *. subst nb2. 
+  assert (map1 = map2). apply extensionality. assumption.
+  assert (nextpos1 = nextpos2). apply proof_irrelevance. 
+  congruence.
+Qed.
+
+(** * Store 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]:
+- if [f b = None], the block [b] of [m1] has no equivalent in [m2];
+- if [f b = Some(b', ofs)], the block [b] of [m2] corresponds to
+  a sub-block at offset [ofs] of the block [b'] in [m2].
+*)
+
+Definition meminj := (block -> option (block * Z))%type.
+
+Section MEM_INJECT.
+
+Variable f: meminj.
+
+(** A memory injection defines a relation between values that is the
+  identity relation, except for pointer values which are shifted
+  as prescribed by the memory injection. *)
+
+Inductive val_inject: val -> val -> Prop :=
+  | val_inject_int:
+      forall i, val_inject (Vint i) (Vint i)
+  | val_inject_float:
+      forall f, val_inject (Vfloat f) (Vfloat f)
+  | val_inject_ptr:
+      forall b1 ofs1 b2 ofs2 x,
+      f b1 = Some (b2, x) ->
+      ofs2 = Int.add ofs1 (Int.repr x) ->
+      val_inject (Vptr b1 ofs1) (Vptr b2 ofs2)
+  | val_inject_undef: forall v,
+      val_inject Vundef v.
+
+Hint Resolve val_inject_int val_inject_float val_inject_ptr 
+val_inject_undef.
+
+Inductive val_list_inject: list val -> list val-> Prop:= 
+  | val_nil_inject :
+      val_list_inject nil nil
+  | val_cons_inject : forall v v' vl vl' , 
+      val_inject v v' -> val_list_inject vl vl'->
+      val_list_inject (v::vl) (v':: vl').  
+
+Inductive val_content_inject: memory_size -> val -> val -> Prop :=
+  | val_content_inject_base:
+      forall sz v1 v2,
+      val_inject v1 v2 ->
+      val_content_inject sz v1 v2
+  | val_content_inject_8:
+      forall n1 n2,
+      Int.cast8unsigned n1 = Int.cast8unsigned n2 ->
+      val_content_inject Size8 (Vint n1) (Vint n2)
+  | val_content_inject_16:
+      forall n1 n2,
+      Int.cast16unsigned n1 = Int.cast16unsigned n2 ->
+      val_content_inject Size16 (Vint n1) (Vint n2)
+  | val_content_inject_32:
+      forall f1 f2,
+      Float.singleoffloat f1 = Float.singleoffloat f2 ->
+      val_content_inject Size32 (Vfloat f1) (Vfloat f2).
+
+Hint Resolve val_content_inject_base.
+
+Inductive content_inject: content -> content -> Prop :=
+  | content_inject_undef: 
+      forall c,
+      content_inject Undef c
+  | content_inject_datum8:
+      forall v1 v2,
+      val_content_inject Size8 v1 v2 ->
+      content_inject (Datum8 v1) (Datum8 v2)
+  | content_inject_datum16:
+      forall v1 v2,
+      val_content_inject Size16 v1 v2 ->
+      content_inject (Datum16 v1) (Datum16 v2)
+  | content_inject_datum32:
+      forall v1 v2,
+      val_content_inject Size32 v1 v2 ->
+      content_inject (Datum32 v1) (Datum32 v2)
+  | content_inject_datum64:
+      forall v1 v2,
+      val_content_inject Size64 v1 v2 ->
+      content_inject (Datum64 v1) (Datum64 v2)
+  | content_inject_cont:
+      content_inject Cont Cont.
+
+Hint Resolve content_inject_undef content_inject_datum8 
+content_inject_datum16 content_inject_datum32 content_inject_datum64
+content_inject_cont.
+
+Definition contentmap_inject (c1 c2: contentmap) (lo hi delta: Z) : Prop :=
+  forall x, lo <= x < hi ->
+    content_inject (c1 x) (c2 (x + delta)).
+
+(** A block contents [c1] injects into another block content [c2] at
+  offset [delta] if the contents of [c1] at all valid offsets
+  correspond to the contents of [c2] at the same offsets shifted by [delta].
+  Some additional conditions are imposed to guard against arithmetic
+  overflow in address computations. *)
+
+Record block_contents_inject (c1 c2: block_contents) (delta: Z) : Prop :=
+  mk_block_contents_inject {
+    bci_range1: Int.min_signed <= delta <= Int.max_signed;
+    bci_range2: delta = 0 \/
+                (Int.min_signed <= c2.(low) /\ c2.(high) <= Int.max_signed);
+    bci_lowhigh:forall x, c1.(low) <= x < c1.(high) ->
+                          c2.(low) <= x+delta < c2.(high) ;
+    bci_contents: 
+      contentmap_inject c1.(contents) c2.(contents) c1.(low) c1.(high) delta
+  }.
+
+(** A memory state [m1] injects into another memory state [m2] via the
+  memory injection [f] if the following conditions hold:
+- unallocated blocks in [m1] must be mapped to [None] by [f];
+- if [f b = Some(b', delta)], [b'] must be valid in [m2] and
+  the contents of [b] in [m1] must inject into the contents of [b'] in [m2]
+  with offset [delta];
+- distinct blocks in [m1] cannot be mapped to overlapping sub-blocks in [m2].
+*)
+
+Record mem_inject (m1 m2: mem) : Prop :=
+  mk_mem_inject {
+    mi_freeblocks:
+      forall b, b >= m1.(nextblock) -> f b = None;
+    mi_mappedblocks:
+      forall b b' delta,
+      f b = Some(b', delta) ->
+      b' < m2.(nextblock) /\
+      block_contents_inject (m1.(blocks) b) 
+                            (m2.(blocks) b') 
+                            delta;
+    mi_no_overlap:
+      forall b1 b2 b1' b2' delta1 delta2,
+      b1 <> b2 ->
+      f b1 = Some (b1', delta1) ->
+      f b2 = Some (b2', delta2) ->
+      b1' <> b2' 
+      \/ (forall x1 x2, low_bound m1 b1 <= x1 < high_bound m1 b1 ->
+                              low_bound m1 b2 <= x2 < high_bound m1 b2 ->
+                              x1+delta1 <> x2+delta2)
+ }.
+
+(** The following lemmas establish the absence of machine integer overflow
+  during address computations. *)
+
+Lemma size_mem_pos: forall sz, size_mem sz > 0.
+Proof.  destruct sz; simpl; omega. Qed.
+
+Lemma size_chunk_pos: forall chunk, size_chunk chunk > 0.
+Proof. intros. unfold size_chunk. apply size_mem_pos. Qed.
+
+Lemma address_inject:
+  forall m1 m2 chunk b1 ofs1 b2 ofs2 x,
+  mem_inject m1 m2 ->
+  (m1.(blocks) b1).(low) <= Int.signed ofs1 ->
+  Int.signed ofs1 + size_chunk chunk <= (m1.(blocks) b1).(high) ->
+  f b1 = Some (b2, x) ->
+  ofs2 = Int.add ofs1 (Int.repr x) ->
+  Int.signed ofs2 = Int.signed ofs1 + x.
+Proof.
+  intros. 
+  generalize (size_chunk_pos chunk). intro.
+  generalize (mi_mappedblocks m1 m2 H _ _ _ H2). intros [C D].
+  inversion D. 
+  elim  bci_range4 ; intro. 
+  (**x=0**)
+  subst x .  rewrite Zplus_0_r.  rewrite Int.add_zero in H3. 
+  subst ofs2 ; auto . 
+  (**x<>0**)
+  rewrite H3. rewrite Int.add_signed. repeat rewrite Int.signed_repr.
+  auto.  auto.
+  assert (low (blocks m1 b1) <= Int.signed ofs1 < high (blocks m1 b1)).
+  omega.
+  generalize (bci_lowhigh0 (Int.signed ofs1) H6). omega.
+  auto.
+Qed.
+
+Lemma valid_pointer_inject_no_overflow:
+  forall m1 m2 b ofs b' x,
+  mem_inject m1 m2 ->
+  valid_pointer m1 b (Int.signed ofs) = true ->
+  f b = Some(b', x) ->
+  Int.min_signed <= Int.signed ofs + Int.signed (Int.repr x) <= Int.max_signed.
+Proof.
+  intros. unfold valid_pointer in H0.
+  destruct (zlt b (nextblock m1)); try discriminate.
+  destruct (zle (low (blocks m1 b)) (Int.signed ofs)); try discriminate.
+  destruct (zlt (Int.signed ofs) (high (blocks m1 b))); try discriminate.
+  inversion H. generalize (mi_mappedblocks0 _ _ _ H1).
+  intros [A B]. inversion B. 
+  elim  bci_range4 ; intro. 
+  (**x=0**)
+  rewrite Int.signed_repr ; auto.  
+  subst x .  rewrite Zplus_0_r.  apply Int.signed_range . 
+  (**x<>0**)
+  generalize (bci_lowhigh0 _ (conj z0 z1)). intro.
+  rewrite Int.signed_repr. omega. auto.
+Qed.
+
+(** Relation between injections and loads. *)
+
+Lemma check_cont_inject:
+  forall c1 c2 lo hi delta,
+  contentmap_inject c1 c2 lo hi delta ->
+  forall n p,
+  lo <= p -> p + Z_of_nat n <= hi ->
+  check_cont n p c1 = true ->
+  check_cont n (p + delta) c2 = true.
+Proof.
+  induction n.
+  intros. simpl. reflexivity.
+  simpl check_cont. rewrite inj_S. intros p H0 H1.
+  assert (lo <= p < hi). omega.  
+  generalize (H p H2). intro. inversion H3; intros; try discriminate.
+  replace (p + delta + 1) with ((p + 1) + delta). 
+  apply IHn. omega. omega. auto. 
+  omega.
+Qed.
+
+Hint Resolve check_cont_inject.
+
+Lemma getN_inject:
+  forall c1 c2 lo hi delta,
+  contentmap_inject c1 c2 lo hi delta ->
+  forall n p,
+  lo <= p -> p + Z_of_nat n < hi ->
+  content_inject (getN n p c1) (getN n (p + delta) c2).
+Proof.
+  intros. simpl in H1.
+  assert (lo <= p < hi). omega.
+  unfold getN.
+  caseEq (check_cont n (p + 1) c1); intro.
+  replace (check_cont n (p + delta + 1) c2) with true.
+  apply H. assumption. 
+  symmetry. replace (p + delta + 1) with ((p + 1) + delta).
+  eapply check_cont_inject; eauto.
+  omega. omega. omega.
+  constructor. 
+Qed.
+
+Hint Resolve getN_inject.
+
+Definition ztonat (z:Z): nat:=
+match z with
+| Z0 => O
+| Zpos p =>  (nat_of_P p)
+| Zneg p =>O 
+end.
+
+Lemma load_contents_inject:
+  forall sz c1 c2 lo hi delta p,
+  contentmap_inject c1 c2 lo hi delta ->
+  lo <= p -> p + size_mem sz <= hi ->
+  val_content_inject sz (load_contents sz c1 p) (load_contents sz c2 (p + delta)).
+Proof.
+intros.
+assert (content_inject (getN (ztonat (size_mem sz)-1) p c1) 
+(getN (ztonat(size_mem sz)-1) (p + delta) c2)).
+induction sz; assert (lo<= p< hi); simpl in H1; try omega;
+apply getN_inject with lo hi; try assumption; simpl ; try omega. 
+induction sz ; simpl; inversion H2; auto.
+Qed.
+
+Hint Resolve load_contents_inject.
+
+Lemma load_result_inject:
+  forall chunk v1 v2,
+  val_content_inject (mem_chunk chunk) v1 v2 ->
+  val_inject (Val.load_result chunk v1) (Val.load_result chunk v2).
+Proof.
+intros.
+destruct chunk; simpl in H; inversion H; simpl; auto;
+try (inversion H0; simpl; auto; fail).
+replace (Int.cast8signed n2) with (Int.cast8signed n1). constructor. 
+apply Int.cast8_signed_equal_if_unsigned_equal; auto.
+rewrite H0; constructor.
+replace (Int.cast16signed n2) with (Int.cast16signed n1). constructor. 
+apply Int.cast16_signed_equal_if_unsigned_equal; auto.
+rewrite H0; constructor.
+inversion H0; simpl; auto. 
+apply val_inject_ptr with x; auto.
+Qed.
+
+Lemma in_bounds_inject:
+  forall chunk c1 c2 delta p,
+  block_contents_inject c1 c2 delta ->
+  c1.(low) <= p /\ p + size_chunk chunk <= c1.(high) ->
+  c2.(low) <= p + delta /\ (p + delta) + size_chunk chunk <= c2.(high).
+Proof.
+  intros.
+  inversion H. 
+  generalize (size_chunk_pos chunk); intro.
+  assert (low c1 <= p + size_chunk chunk - 1 < high c1).
+  omega.
+  generalize (bci_lowhigh0 _ H2). intro.
+  assert (low c1 <= p < high c1).
+  omega.
+  generalize (bci_lowhigh0 _ H4). intro.
+  omega. 
+Qed.
+
+Lemma block_cont_val:
+forall c1 c2 delta p sz,
+block_contents_inject c1 c2 delta ->
+c1.(low) <= p -> p + size_mem sz <= c1.(high) ->
+  val_content_inject sz (load_contents sz c1.(contents) p) 
+    (load_contents sz c2.(contents) (p + delta)).
+Proof.
+intros.
+inversion H .
+apply load_contents_inject with c1.(low) c1.(high) ;assumption. 
+Qed.
+
+Lemma load_inject:
+  forall m1 m2 chunk b1 ofs b2 delta v1,
+  mem_inject 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 v1 v2.
+Proof.
+  intros.
+  generalize (load_inv _ _ _ _ _ H0).  intros [A [B [C D]]].
+  inversion H.
+  generalize (mi_mappedblocks0 _ _ _ H1). intros [U V].
+  inversion V.
+  exists (Val.load_result chunk (load_contents (mem_chunk chunk)
+           (m2.(blocks) b2).(contents) (ofs+delta))). 
+  split.
+  unfold load. 
+  generalize (size_chunk_pos chunk); intro. 
+  rewrite zlt_true. rewrite in_bounds_holds. auto.
+  assert (low (blocks m1 b1) <= ofs < high (blocks m1 b1)).
+    omega.
+  generalize (bci_lowhigh0 _ H3). tauto.
+  assert (low (blocks m1 b1) <= ofs + size_chunk chunk - 1 < high(blocks m1 b1)).
+    omega.
+  generalize (bci_lowhigh0 _ H3). omega.
+  assumption.
+  rewrite <- D. apply load_result_inject. 
+  eapply load_contents_inject; eauto.
+Qed.
+
+Lemma loadv_inject:
+  forall m1 m2 chunk a1 a2 v1,
+  mem_inject m1 m2 ->
+  loadv chunk m1 a1 = Some v1 ->
+  val_inject a1 a2 ->
+  exists v2, loadv chunk m2 a2 = Some v2 /\ val_inject v1 v2.
+Proof.
+  intros.
+  induction H1 ; simpl in H0 ; try discriminate H0.
+  simpl. 
+  replace (Int.signed ofs2) with (Int.signed ofs1 + x).
+  apply load_inject with m1 b1 ; assumption.
+  symmetry. generalize (load_inv _ _ _ _ _ H0). intros [A [B [C D]]].
+  apply address_inject with m1 m2 chunk b1 b2; auto.
+Qed.
+
+(** Relation between injections and stores. *)
+
+Lemma set_cont_inject:
+  forall c1 c2 lo hi delta,
+  contentmap_inject c1 c2 lo hi delta ->
+  forall n p,
+  lo <= p -> p + Z_of_nat n <= hi ->
+  contentmap_inject (set_cont n p c1) (set_cont n (p + delta) c2) lo hi delta.
+Proof.
+induction n. intros. simpl. assumption.
+intros. simpl. unfold contentmap_inject.
+intros p2 Hp2. unfold update.
+case (zeq p2 p); intro.
+subst p2. rewrite zeq_true. constructor.
+rewrite zeq_false. replace (p + delta + 1) with ((p + 1) + delta).
+apply IHn. omega. rewrite inj_S in H1. omega. assumption.
+omega. omega.
+Qed.
+
+Lemma setN_inject:
+  forall c1 c2 lo hi delta n p v1 v2,
+  contentmap_inject c1 c2 lo hi delta ->
+  content_inject v1 v2 ->
+  lo <= p -> p + Z_of_nat n < hi ->
+  contentmap_inject (setN n p v1 c1) (setN n (p + delta) v2 c2)
+                    lo hi delta.
+Proof.
+  intros. unfold setN. red; intros.
+  unfold update. 
+  case (zeq x p); intro. 
+  subst p. rewrite zeq_true. assumption.
+  rewrite zeq_false. 
+  replace (p + delta + 1) with ((p + 1) + delta).
+  apply set_cont_inject with lo hi. 
+  assumption. omega. omega. assumption. omega.
+  omega.
+Qed.
+
+Lemma store_contents_inject:
+  forall c1 c2 lo hi delta sz p v1 v2,
+  contentmap_inject c1 c2 lo hi delta ->
+  val_content_inject sz v1 v2 ->
+  lo <= p -> p + size_mem sz <= hi ->
+  contentmap_inject (store_contents sz c1 p v1) 
+                    (store_contents sz c2 (p + delta) v2)
+                    lo hi delta.
+Proof.
+  intros.
+  destruct sz; simpl in *; apply setN_inject; auto; simpl; omega.
+Qed.
+
+Lemma set_cont_outside1 :
+  forall n  p m q ,
+  (forall x , p <= x < p + Z_of_nat n -> x <> q) ->
+  (set_cont n p m) q = m q.
+Proof.
+  induction n; intros; simpl.
+  auto.
+  rewrite inj_S in H. rewrite update_o. apply IHn.
+  intros. apply H.  omega. 
+  apply H. omega.
+Qed.
+
+Lemma set_cont_outside_inject:
+  forall c1 c2 lo hi delta,
+  contentmap_inject c1 c2 lo hi delta ->
+  forall n p,
+  (forall x1 x2, p <= x1 < p + Z_of_nat n ->
+                 lo <= x2 < hi -> 
+                 x1 <> x2 + delta) ->
+  contentmap_inject c1 (set_cont n p c2) lo hi delta.
+Proof.
+  unfold contentmap_inject. intros.  
+  rewrite set_cont_outside1. apply H. assumption. 
+  intros. apply H0. auto. auto.
+Qed.
+
+Lemma setN_outside_inject:
+  forall n c1 c2 lo hi delta  p v,
+  contentmap_inject c1 c2 lo hi delta ->
+  (forall x1 x2, p <= x1 < p + Z_of_nat (S n) ->
+                 lo <= x2 < hi ->
+                 x1 <> x2 + delta) ->
+  contentmap_inject c1 (setN n p v c2) lo hi delta.
+Proof.
+  intros. unfold setN. red; intros. rewrite update_o.
+  apply set_cont_outside_inject with lo hi. auto.
+  intros. apply H0. rewrite inj_S. omega. auto. auto. 
+  apply H0. rewrite inj_S. omega. auto.
+Qed.
+
+Lemma store_contents_outside_inject:
+  forall c1 c2 lo hi delta sz p v,
+  contentmap_inject c1 c2 lo hi delta ->
+  (forall x1 x2, p <= x1 < p + size_mem sz ->
+                 lo <= x2 < hi ->
+                 x1 <> x2 + delta)->
+  contentmap_inject c1 (store_contents sz c2 p v) lo hi delta.
+Proof.
+  intros c1 c2 lo hi delta sz. generalize (size_mem_pos sz) . intro . 
+  induction sz ;intros ;simpl ; apply setN_outside_inject ; assumption . 
+Qed.
+
+Lemma store_mapped_inject_1:
+  forall chunk m1 b1 ofs v1 n1 m2 b2 delta v2,
+  mem_inject m1 m2 ->
+  store chunk m1 b1 ofs v1 = Some n1 ->
+  f b1 = Some (b2, delta) ->
+  val_content_inject (mem_chunk chunk) v1 v2 ->
+  exists n2,
+    store chunk m2 b2 (ofs + delta) v2 = Some n2
+    /\ mem_inject n1 n2.
+Proof.
+intros. 
+generalize (size_chunk_pos chunk); intro SIZEPOS.
+generalize (store_inv _ _ _ _ _ _ H0).
+intros [A [B [C [D [P E]]]]].
+inversion H.
+generalize (mi_mappedblocks0 _ _ _ H1). intros [U V].
+inversion V.
+generalize (in_bounds_inject _ _ _ _ _ V (conj B C)). intro BND.
+exists  (unchecked_store chunk m2 b2 (ofs+delta) v2 BND).
+split. unfold store.
+rewrite zlt_true; auto.
+case (in_bounds chunk (ofs + delta) (blocks m2 b2)); intro.
+assert (a = BND). apply proof_irrelevance. congruence.
+omegaContradiction.
+constructor.
+intros. apply mi_freeblocks0. rewrite <- D. assumption.
+intros. generalize (mi_mappedblocks0 b b' delta0 H3).
+intros [W Y]. split. simpl. auto.
+rewrite E; simpl. unfold update.
+(* Cas 1: memes blocs b = b1  b'= b2 *)
+case (zeq b b1); intro.
+subst b. assert (b'= b2). congruence. subst b'. 
+assert (delta0 = delta). congruence. subst delta0.
+rewrite zeq_true. inversion Y. constructor; simpl; auto.
+apply store_contents_inject; auto.
+(* Cas 2: blocs differents dans m1 mais mappes sur le meme bloc de m2 *)
+case (zeq b' b2); intro.
+subst b'.  
+inversion Y. constructor; simpl; auto.
+generalize (store_contents_outside_inject _ _ _ _ _ (mem_chunk chunk) 
+  (ofs+delta) v2 bci_contents1).
+intros.
+apply H4. 
+elim (mi_no_overlap0 b b1 b2 b2 delta0 delta n H3 H1).
+tauto.
+unfold high_bound, low_bound. intros. 
+apply sym_not_equal. replace x1 with ((x1 - delta) + delta).
+apply H5. assumption. unfold size_chunk in C. omega. omega. 
+(* Cas 3: blocs differents dans m1 et dans m2 *)
+auto.
+
+unfold high_bound, low_bound. 
+rewrite E; simpl; intros.
+unfold high_bound, low_bound in mi_no_overlap0. 
+unfold update.
+case (zeq b0 b1).
+intro. subst b1. simpl.
+rewrite zeq_false; auto.
+intro. case (zeq b3 b1); intro.
+subst b3. simpl. auto.
+auto.
+Qed.
+
+Lemma store_mapped_inject:
+  forall chunk m1 b1 ofs v1 n1 m2 b2 delta v2,
+  mem_inject m1 m2 ->
+  store chunk m1 b1 ofs v1 = Some n1 ->
+  f b1 = Some (b2, delta) ->
+  val_inject v1 v2 ->
+  exists n2,
+    store chunk m2 b2 (ofs + delta) v2 = Some n2
+    /\ mem_inject n1 n2.
+Proof.
+  intros. eapply store_mapped_inject_1; eauto.
+Qed.
+
+Lemma store_unmapped_inject:
+  forall chunk m1 b1 ofs v1 n1 m2,
+  mem_inject m1 m2 ->
+  store chunk m1 b1 ofs v1 = Some n1 ->
+  f b1 = None ->
+  mem_inject n1 m2.
+Proof.
+intros.
+inversion H.
+generalize (store_inv _ _ _ _ _ _ H0).
+intros [A [B [C [D [P E]]]]].
+constructor.
+rewrite D. assumption.
+intros. elim (mi_mappedblocks0 _ _ _ H2); intros.
+split. auto. 
+rewrite E; unfold update.
+rewrite zeq_false. assumption.
+congruence.
+intros. 
+assert (forall b, low_bound n1 b = low_bound m1 b).
+  intros. unfold low_bound; rewrite E; unfold update.
+  case (zeq b b1); intros. subst b1; reflexivity. reflexivity.
+assert (forall b, high_bound n1 b = high_bound m1 b).
+  intros. unfold high_bound; rewrite E; unfold update.
+  case (zeq b b1); intros. subst b1; reflexivity. reflexivity.
+repeat rewrite H5. repeat rewrite H6. auto.
+Qed.
+
+Lemma storev_mapped_inject_1:
+  forall chunk m1 a1 v1 n1 m2 a2 v2,
+  mem_inject m1 m2 ->
+  storev chunk m1 a1 v1 = Some n1 ->
+  val_inject a1 a2 ->
+  val_content_inject (mem_chunk chunk) v1 v2 ->
+  exists n2,
+    storev chunk m2 a2 v2 = Some n2 /\ mem_inject n1 n2.
+Proof.
+  intros. destruct a1; simpl in H0; try discriminate.
+  inversion H1. subst b.
+  simpl. replace (Int.signed ofs2) with (Int.signed i + x).
+  eapply store_mapped_inject_1; eauto.
+  symmetry. generalize (store_inv _ _ _ _ _ _ H0). intros [A [B [C [D [P E]]]]].
+  apply address_inject with m1 m2 chunk b1 b2; auto.
+Qed.
+
+Lemma storev_mapped_inject:
+  forall chunk m1 a1 v1 n1 m2 a2 v2,
+  mem_inject m1 m2 ->
+  storev chunk m1 a1 v1 = Some n1 ->
+  val_inject a1 a2 ->
+  val_inject v1 v2 ->
+  exists n2,
+    storev chunk m2 a2 v2 = Some n2 /\ mem_inject n1 n2.
+Proof.
+  intros. eapply storev_mapped_inject_1; eauto.
+Qed.
+
+(** Relation between injections and [free] *)
+
+Lemma free_first_inject :
+  forall m1 m2 b,
+  mem_inject m1 m2 ->
+  mem_inject (free m1 b) m2.
+Proof.
+  intros.  inversion H.  constructor . auto.
+  simpl. intros. 
+  generalize (mi_mappedblocks0 b0 b' delta H0).
+  intros [A B] . split. assumption . unfold update.
+  case (zeq b0 b); intro.  inversion B. constructor; simpl; auto.
+  intros.  omega.
+  unfold contentmap_inject.
+  intros. omegaContradiction.
+  auto.  intros. 
+  unfold free . unfold low_bound , high_bound.  simpl. unfold update.
+  case (zeq b1 b);intro.  simpl.
+  right. intros. omegaContradiction.
+  case (zeq b2 b);intro.  simpl.
+  right . intros. omegaContradiction. 
+  unfold low_bound, high_bound in mi_no_overlap0. auto.
+Qed.  
+
+Lemma free_first_list_inject :
+  forall l m1 m2,
+  mem_inject m1 m2 ->
+  mem_inject (free_list m1 l) m2.
+Proof.
+  induction l; simpl; intros.
+  auto.
+  apply free_first_inject. auto.
+Qed.
+
+Lemma free_snd_inject:
+  forall m1 m2 b,
+  (forall b1 delta, f b1 = Some(b, delta) -> 
+                    low_bound m1 b1 >= high_bound m1 b1) ->
+  mem_inject m1 m2 ->
+  mem_inject m1 (free m2 b).
+Proof.
+  intros. inversion H0. constructor. auto.
+  simpl. intros. 
+  generalize (mi_mappedblocks0 b0 b' delta H1).
+  intros [A B] . split. assumption . 
+  inversion B. unfold update.
+  case (zeq b' b); intro.
+  subst b'. generalize (H _ _ H1). unfold low_bound, high_bound; intro.
+  constructor. auto.  elim bci_range4 ; intro.
+  (**delta=0**)
+  left ; auto .  
+  (** delta<>0**)
+  right . 
+  simpl. compute. split; intro; congruence. 
+ intros. omegaContradiction.
+  red; intros. omegaContradiction.
+  auto.
+  auto.
+Qed.
+
+Lemma bounds_free_block: 
+  forall m1 b m1' , free m1 b = m1' ->
+  low_bound m1' b >= high_bound m1' b.
+Proof.
+  intros.  unfold free in H.
+  rewrite<- H . unfold low_bound , high_bound .
+  simpl . rewrite update_s. simpl.  omega.  
+Qed.
+
+Lemma free_empty_bounds:
+  forall l m1 ,
+  let m1' := free_list m1 l in
+  (forall b, In b l -> low_bound m1' b >= high_bound m1' b).
+Proof.
+  induction l . intros .  inversion H.  
+  intros.
+  generalize (in_inv H).
+  intro . elim H0.   intro .  subst b.  simpl in m1' .  
+  generalize ( bounds_free_block  
+  (fold_right (fun (b : block) (m : mem) => free m b) m1 l) a m1') . 
+  intros.  apply H1. auto .  intros.  simpl in m1'. unfold m1' . 
+  unfold low_bound , high_bound .  simpl . 
+  unfold update; case (zeq b a); intro; simpl.
+  omega . 
+  unfold low_bound , high_bound in IHl . auto.
+Qed.       
+
+Lemma free_inject:
+  forall m1 m2 l b,
+  (forall b1 delta, f b1 = Some(b, delta) -> In b1 l) ->
+  mem_inject m1 m2 ->
+  mem_inject (free_list m1 l) (free m2 b).
+Proof.
+   intros. apply free_snd_inject. 
+   intros. apply free_empty_bounds. apply H with delta. auto. 
+   apply free_first_list_inject. auto.
+Qed. 
+
+End MEM_INJECT.
+
+Hint Resolve val_inject_int val_inject_float val_inject_ptr val_inject_undef.
+Hint Resolve val_nil_inject val_cons_inject.
+
+(** Monotonicity properties of memory injections. *)
+
+Definition inject_incr (f1 f2: meminj) : Prop :=
+  forall b, f1 b = f2 b \/ f1 b = None.
+
+Lemma inject_incr_refl :
+   forall f , inject_incr f f .
+Proof. unfold inject_incr . intros. left . auto . Qed.
+
+Lemma inject_incr_trans :
+  forall f1 f2 f3  , 
+  inject_incr f1 f2 -> inject_incr f2 f3 -> inject_incr f1 f3 .
+Proof .
+  unfold inject_incr . intros . 
+  generalize (H b) . intro . generalize (H0 b) . intro . 
+  elim H1 ; elim H2 ; intros.  
+  rewrite H3 in H4 . left . auto .
+  rewrite H3 in H4 . right . auto .
+  right ; auto . 
+  right ; auto .
+Qed.
+
+Lemma val_inject_incr:
+  forall f1 f2 v v',
+  inject_incr f1 f2 ->
+  val_inject f1 v v' ->
+  val_inject f2 v v'.
+Proof.
+  intros. inversion H0.
+  constructor.
+  constructor.
+  elim (H b1); intro. 
+    apply val_inject_ptr with x. congruence. auto. 
+    congruence.
+  constructor.
+Qed.
+
+Lemma val_list_inject_incr:
+  forall f1 f2 vl vl' ,
+  inject_incr f1 f2 -> val_list_inject f1 vl vl' ->
+  val_list_inject f2 vl vl'.
+Proof.
+  induction vl ;  intros;  inversion H0. auto . 
+  subst a . generalize (val_inject_incr f1 f2 v v' H H3) .  
+  intro .  auto .
+Qed.       
+
+Hint Resolve inject_incr_refl val_inject_incr val_list_inject_incr.
+
+Section MEM_INJECT_INCR.
+
+Variable f1 f2: meminj.
+Hypothesis INCR: inject_incr f1 f2.
+
+Lemma val_content_inject_incr:
+  forall chunk v v',
+  val_content_inject f1 chunk v v' ->
+  val_content_inject f2 chunk v v'.
+Proof.
+  intros. inversion H.
+  apply val_content_inject_base. eapply val_inject_incr; eauto.
+  apply val_content_inject_8; auto.
+  apply val_content_inject_16; auto.
+  apply val_content_inject_32; auto.
+Qed.
+
+Lemma content_inject_incr:
+  forall c c', content_inject f1 c c' -> content_inject f2 c c'.
+Proof.
+  induction 1; constructor; eapply val_content_inject_incr; eauto.
+Qed.
+
+Lemma contentmap_inject_incr:
+  forall c c' lo hi delta,
+  contentmap_inject f1 c c' lo hi delta -> 
+  contentmap_inject f2 c c' lo hi delta.
+Proof.
+  unfold contentmap_inject; intros.
+  apply content_inject_incr. auto.
+Qed.
+
+Lemma block_contents_inject_incr:
+  forall c c' delta,
+  block_contents_inject f1 c c' delta ->
+  block_contents_inject f2 c c' delta.
+Proof.
+  intros. inversion H. constructor; auto.
+  apply contentmap_inject_incr; auto.
+Qed.
+
+End MEM_INJECT_INCR.
+
+(** Properties of injections and allocations. *)
+
+Definition extend_inject
+       (b: block) (x: option (block * Z)) (f: meminj) : meminj :=
+  fun b' => if eq_block b' b then x else f b'.
+
+Lemma extend_inject_incr:
+  forall f b x,
+  f b = None ->
+  inject_incr f (extend_inject b x f).
+Proof.
+  intros; red; intros. unfold extend_inject. 
+  case (eq_block b0 b); intro. subst b0. right; auto. left; auto.
+Qed.
+
+Lemma alloc_right_inject:
+  forall f m1 m2 lo hi m2' b,
+  mem_inject f m1 m2 ->
+  alloc m2 lo hi = (m2', b) ->
+  mem_inject f m1 m2'.
+Proof.
+  intros. unfold alloc in H0. injection H0; intros A B; clear H0.
+  inversion H.
+  constructor.
+  assumption.
+  intros. generalize (mi_mappedblocks0 _ _ _ H0). intros [C D].
+  rewrite <- B; simpl. split. omega. 
+  rewrite update_o. auto. omega.
+  assumption.
+Qed.
+
+Lemma alloc_unmapped_inject:
+  forall f m1 m2 lo hi m1' b,
+  mem_inject f m1 m2 ->
+  alloc m1 lo hi = (m1', b) ->
+  mem_inject (extend_inject b None f) m1' m2 /\
+  inject_incr f (extend_inject b None f).
+Proof.
+  intros. unfold alloc in H0. injection H0; intros; clear H0.
+  inversion H.
+  assert (inject_incr f (extend_inject b None f)).
+  apply extend_inject_incr. apply mi_freeblocks0. rewrite H1. omega. 
+  split; auto.
+  constructor. 
+  rewrite <- H2; simpl; intros. unfold extend_inject.
+  case (eq_block b0 b); intro. auto. apply mi_freeblocks0. omega.
+  intros until delta. unfold extend_inject at 1. case (eq_block b0 b); intro.
+  intros; discriminate.
+  intros. rewrite <- H2; simpl. rewrite H1. 
+  rewrite update_o; auto. 
+  elim (mi_mappedblocks0 _ _ _ H3). intros A B.
+  split. auto. apply block_contents_inject_incr with f. auto. auto.
+  intros until delta2. unfold extend_inject.
+  case (eq_block b1 b); intro. congruence.
+  case (eq_block b2 b); intro. congruence.
+  rewrite <- H2. unfold low_bound, high_bound; simpl. rewrite H1.
+  repeat rewrite update_o; auto.
+  exact (mi_no_overlap0 b1 b2 b1' b2' delta1 delta2).
+Qed.
+
+Lemma alloc_mapped_inject:
+  forall f m1 m2 lo hi m1' b b' ofs,
+  mem_inject f m1 m2 ->
+  alloc m1 lo hi = (m1', b) ->
+  valid_block m2 b' ->
+  Int.min_signed <= ofs <= Int.max_signed ->
+  Int.min_signed <= low_bound m2 b' ->
+  high_bound m2 b' <= Int.max_signed ->
+  low_bound m2 b' <= lo + ofs ->
+  hi + ofs <= high_bound m2 b' ->
+  (forall b0 ofs0, 
+   f b0 = Some (b', ofs0) -> 
+   high_bound m1 b0 + ofs0 <= lo + ofs \/
+   hi + ofs <= low_bound m1 b0 + ofs0) ->
+  mem_inject (extend_inject b (Some (b', ofs)) f) m1' m2 /\
+  inject_incr f (extend_inject b (Some (b', ofs)) f).
+Proof.
+  intros. 
+  generalize (fun b' => low_bound_alloc _ _ b' _ _ _ H0).
+  intro LOW.
+  generalize (fun b' => high_bound_alloc _ _ b' _ _ _ H0).
+  intro HIGH.
+  unfold alloc in H0. injection H0; intros A B; clear H0.
+  inversion H.
+  (* inject_incr *)
+  assert (inject_incr f (extend_inject b (Some (b', ofs)) f)).
+  apply extend_inject_incr. apply mi_freeblocks0. rewrite A. omega. 
+  split; auto.
+  constructor. 
+  (* mi_freeblocks *)
+  rewrite <- B; simpl; intros. unfold extend_inject.
+  case (eq_block b0 b); intro. unfold block in *. omegaContradiction.
+  apply mi_freeblocks0. omega.
+  (* mi_mappedblocks *)
+  intros until delta. unfold extend_inject at 1. 
+  case (eq_block b0 b); intro.
+  intros. subst b0. inversion H8. subst b'0; subst delta. 
+  split. assumption. 
+  rewrite <- B; simpl. rewrite A. rewrite update_s.
+  constructor; auto.
+  unfold empty_block. simpl. intros. unfold low_bound in H5. unfold high_bound in H6. omega.
+  simpl. red; intros. constructor.
+  intros. 
+  generalize (mi_mappedblocks0 _ _ _ H8). intros [C D].
+  split. auto. 
+  rewrite <- B; simpl; rewrite A; rewrite update_o; auto.
+  apply block_contents_inject_incr with f; auto.
+  (* no overlap *)
+  intros until delta2. unfold extend_inject.
+  repeat rewrite LOW. repeat rewrite HIGH. unfold eq_block.
+  case (zeq b1 b); case (zeq b2 b); intros.
+  congruence.
+  inversion H9. subst b1'; subst delta1.
+  case (eq_block b' b2'); intro.
+  subst b2'. generalize (H7 _ _ H10). intro. 
+  right. intros. omega. left. auto.
+  inversion H10. subst b2'; subst delta2.
+  case (eq_block b' b1'); intro.
+  subst b1'. generalize (H7 _ _ H9). intro.
+  right. intros. omega. left. auto.
+  apply mi_no_overlap0; auto.
+Qed.
-- 
cgit