path: root/common/Events.v

(* *********************************************************************)
(*                                                                     *)
(*              The Compcert verified compiler                         *)
(*                                                                     *)
(*          Xavier Leroy, INRIA Paris-Rocquencourt                     *)
(*                                                                     *)
(*  Copyright Institut National de Recherche en Informatique et en     *)
(*  Automatique.  All rights reserved.  This file is distributed       *)
(*  under the terms of the GNU General Public License as published by  *)
(*  the Free Software Foundation, either version 2 of the License, or  *)
(*  (at your option) any later version.  This file is also distributed *)
(*  under the terms of the INRIA Non-Commercial License Agreement.     *)
(*                                                                     *)
(* *********************************************************************)

(** Observable events, execution traces, and semantics of external calls. *)

Require Import Coqlib.
Require Intv.
Require Import AST.
Require Import Integers.
Require Import Floats.
Require Import Values.
Require Import Memory.
Require Import Globalenvs.

(** * Events and traces *)

(** The observable behaviour of programs is stated in terms of
  input/output events, which represent the actions of the program
  that the external world can observe.  CompCert leaves much flexibility as to
  the exact content of events: the only requirement is that they
  do not expose memory states nor pointer values 
  (other than pointers to global variables), because these
  are not preserved literally during compilation.  For concreteness,
  we use the following type for events.  Each event represents either:

- A system call (e.g. an input/output operation), recording the
  name of the system call, its parameters, and its result.

- A volatile load from a global memory location, recording the chunk
  and address being read and the value just read. 

- A volatile store to a global memory location, recording the chunk
  and address being written and the value stored there. 

- An annotation, recording the text of the annotation and the values
  of the arguments.

  The values attached to these events are of the following form.
  As mentioned above, we do not expose pointer values directly.
  Pointers relative to a global variable are shown with the name
  of the variable instead of the block identifier.
*)

Inductive eventval: Type :=
  | EVint: int -> eventval
  | EVfloat: float -> eventval
  | EVptr_global: ident -> int -> eventval.

Inductive event: Type :=
  | Event_syscall: ident -> list eventval -> eventval -> event
  | Event_vload: memory_chunk -> ident -> int -> eventval -> event
  | Event_vstore: memory_chunk -> ident -> int -> eventval -> event
  | Event_annot: ident -> list eventval -> event.

(** The dynamic semantics for programs collect traces of events.
  Traces are of two kinds: finite (type [trace]) or infinite (type [traceinf]). *)

Definition trace := list event.

Definition E0 : trace := nil.

Definition Eapp (t1 t2: trace) : trace := t1 ++ t2.

CoInductive traceinf : Type :=
  | Econsinf: event -> traceinf -> traceinf.

Fixpoint Eappinf (t: trace) (T: traceinf) {struct t} : traceinf :=
  match t with
  | nil => T
  | ev :: t' => Econsinf ev (Eappinf t' T)
  end.

(** Concatenation of traces is written [**] in the finite case
  or [***] in the infinite case. *)

Infix "**" := Eapp (at level 60, right associativity).
Infix "***" := Eappinf (at level 60, right associativity).

Lemma E0_left: forall t, E0 ** t = t.
Proof. auto. Qed.

Lemma E0_right: forall t, t ** E0 = t.
Proof. intros. unfold E0, Eapp. rewrite <- app_nil_end. auto. Qed.

Lemma Eapp_assoc: forall t1 t2 t3, (t1 ** t2) ** t3 = t1 ** (t2 ** t3).
Proof. intros. unfold Eapp, trace. apply app_ass. Qed.

Lemma Eapp_E0_inv: forall t1 t2, t1 ** t2 = E0 -> t1 = E0 /\ t2 = E0.
Proof (@app_eq_nil event).
  
Lemma E0_left_inf: forall T, E0 *** T = T.
Proof. auto. Qed.

Lemma Eappinf_assoc: forall t1 t2 T, (t1 ** t2) *** T = t1 *** (t2 *** T).
Proof.
  induction t1; intros; simpl. auto. decEq; auto.
Qed.

Hint Rewrite E0_left E0_right Eapp_assoc
             E0_left_inf Eappinf_assoc: trace_rewrite.

Opaque trace E0 Eapp Eappinf.

(** The following [traceEq] tactic proves equalities between traces
  or infinite traces. *)

Ltac substTraceHyp :=
  match goal with
  | [ H: (@eq trace ?x ?y) |- _ ] =>
       subst x || clear H
  end.

Ltac decomposeTraceEq :=
  match goal with
  | [ |- (_ ** _) = (_ ** _) ] =>
      apply (f_equal2 Eapp); auto; decomposeTraceEq
  | _ =>
      auto
  end.

Ltac traceEq := 
  repeat substTraceHyp; autorewrite with trace_rewrite; decomposeTraceEq.

(** Bisimilarity between infinite traces. *)

CoInductive traceinf_sim: traceinf -> traceinf -> Prop :=
  | traceinf_sim_cons: forall e T1 T2,
      traceinf_sim T1 T2 ->
      traceinf_sim (Econsinf e T1) (Econsinf e T2).

Lemma traceinf_sim_refl:
  forall T, traceinf_sim T T.
Proof.
  cofix COINDHYP; intros.
  destruct T. constructor. apply COINDHYP.
Qed.
 
Lemma traceinf_sim_sym:
  forall T1 T2, traceinf_sim T1 T2 -> traceinf_sim T2 T1.
Proof.
  cofix COINDHYP; intros. inv H; constructor; auto.
Qed.

Lemma traceinf_sim_trans:
  forall T1 T2 T3, 
  traceinf_sim T1 T2 -> traceinf_sim T2 T3 -> traceinf_sim T1 T3.
Proof.
  cofix COINDHYP;intros. inv H; inv H0; constructor; eauto.
Qed.

CoInductive traceinf_sim': traceinf -> traceinf -> Prop :=
  | traceinf_sim'_cons: forall t T1 T2,
      t <> E0 -> traceinf_sim' T1 T2 -> traceinf_sim' (t *** T1) (t *** T2).

Lemma traceinf_sim'_sim:
  forall T1 T2, traceinf_sim' T1 T2 -> traceinf_sim T1 T2.
Proof.
  cofix COINDHYP; intros. inv H. 
  destruct t. elim H0; auto.
Transparent Eappinf.
Transparent E0.
  simpl. 
  destruct t. simpl. constructor. apply COINDHYP; auto.
  constructor. apply COINDHYP.
  constructor. unfold E0; congruence. auto.
Qed.

(** An alternate presentation of infinite traces as
  infinite concatenations of nonempty finite traces. *)

CoInductive traceinf': Type :=
  | Econsinf': forall (t: trace) (T: traceinf'), t <> E0 -> traceinf'.

Program Definition split_traceinf' (t: trace) (T: traceinf') (NE: t <> E0): event * traceinf' :=
  match t with
  | nil => _
  | e :: nil => (e, T)
  | e :: t' => (e, Econsinf' t' T _)
  end.
Next Obligation.
  elimtype False. elim NE. auto. 
Qed.
Next Obligation.
  red; intro. elim (H e). rewrite H0. auto. 
Qed.

CoFixpoint traceinf_of_traceinf' (T': traceinf') : traceinf :=
  match T' with
  | Econsinf' t T'' NOTEMPTY =>
      let (e, tl) := split_traceinf' t T'' NOTEMPTY in
      Econsinf e (traceinf_of_traceinf' tl)
  end.

Remark unroll_traceinf':
  forall T, T = match T with Econsinf' t T' NE => Econsinf' t T' NE end.
Proof.
  intros. destruct T; auto.
Qed.

Remark unroll_traceinf:
  forall T, T = match T with Econsinf t T' => Econsinf t T' end.
Proof.
  intros. destruct T; auto.
Qed.

Lemma traceinf_traceinf'_app:
  forall t T NE,
  traceinf_of_traceinf' (Econsinf' t T NE) = t *** traceinf_of_traceinf' T.
Proof.
  induction t.
  intros. elim NE. auto.
  intros. simpl.   
  rewrite (unroll_traceinf (traceinf_of_traceinf' (Econsinf' (a :: t) T NE))).
  simpl. destruct t. auto.
Transparent Eappinf.
  simpl. f_equal. apply IHt. 
Qed.

(** Prefixes of traces. *)

Definition trace_prefix (t1 t2: trace) :=
  exists t3, t2 = t1 ** t3.

Definition traceinf_prefix (t1: trace) (T2: traceinf) :=
  exists T3, T2 = t1 *** T3.

Lemma trace_prefix_app:
  forall t1 t2 t,
  trace_prefix t1 t2 ->
  trace_prefix (t ** t1) (t ** t2).
Proof.
  intros. destruct H as [t3 EQ]. exists t3. traceEq. 
Qed.

Lemma traceinf_prefix_app:
  forall t1 T2 t,
  traceinf_prefix t1 T2 ->
  traceinf_prefix (t ** t1) (t *** T2).
Proof.
  intros. destruct H as [T3 EQ]. exists T3. subst T2. traceEq.
Qed.

(** * Relating values and event values *)

Set Implicit Arguments.

Section EVENTVAL.

(** Global environment used to translate between global variable names and their block identifiers. *)
Variables F V: Type.
Variable ge: Genv.t F V.

(** Translation between values and event values. *)

Inductive eventval_match: eventval -> typ -> val -> Prop :=
  | ev_match_int: forall i,
      eventval_match (EVint i) Tint (Vint i)
  | ev_match_float: forall f,
      eventval_match (EVfloat f) Tfloat (Vfloat f)
  | ev_match_ptr: forall id b ofs,
      Genv.find_symbol ge id = Some b ->
      eventval_match (EVptr_global id ofs) Tint (Vptr b ofs).

Inductive eventval_list_match: list eventval -> list typ -> list val -> Prop :=
  | evl_match_nil:
      eventval_list_match nil nil nil
  | evl_match_cons:
      forall ev1 evl ty1 tyl v1 vl,
      eventval_match ev1 ty1 v1 ->
      eventval_list_match evl tyl vl ->
      eventval_list_match (ev1::evl) (ty1::tyl) (v1::vl).

(** Some properties of these translation predicates. *)

Lemma eventval_match_type:
  forall ev ty v,
  eventval_match ev ty v -> Val.has_type v ty.
Proof.
  intros. inv H; constructor.
Qed.

Lemma eventval_list_match_length:
  forall evl tyl vl, eventval_list_match evl tyl vl -> List.length vl = List.length tyl.
Proof.
  induction 1; simpl; eauto.
Qed.

Lemma eventval_match_lessdef:
  forall ev ty v1 v2,
  eventval_match ev ty v1 -> Val.lessdef v1 v2 -> eventval_match ev ty v2.
Proof.
  intros. inv H; inv H0; constructor; auto.
Qed.

Lemma eventval_list_match_lessdef:
  forall evl tyl vl1, eventval_list_match evl tyl vl1 ->
  forall vl2, Val.lessdef_list vl1 vl2 -> eventval_list_match evl tyl vl2.
Proof.
  induction 1; intros. inv H; constructor.
  inv H1. constructor. eapply eventval_match_lessdef; eauto. eauto.
Qed.

(** Compatibility with memory injections *)

Variable f: block -> option (block * Z).

Definition meminj_preserves_globals : Prop :=
     (forall id b, Genv.find_symbol ge id = Some b -> f b = Some(b, 0))
  /\ (forall b gv, Genv.find_var_info ge b = Some gv -> f b = Some(b, 0))
  /\ (forall b1 b2 delta gv, Genv.find_var_info ge b2 = Some gv -> f b1 = Some(b2, delta) -> b2 = b1).

Hypothesis glob_pres: meminj_preserves_globals.

Lemma eventval_match_inject:
  forall ev ty v1 v2,
  eventval_match ev ty v1 -> val_inject f v1 v2 -> eventval_match ev ty v2.
Proof.
  intros. inv H; inv H0. constructor. constructor.
  destruct glob_pres as [A [B C]].
  exploit A; eauto. intro EQ; rewrite H3 in EQ; inv EQ.
  rewrite Int.add_zero. econstructor; eauto. 
Qed.

Lemma eventval_match_inject_2:
  forall ev ty v,
  eventval_match ev ty v -> val_inject f v v.
Proof.
  induction 1. constructor. constructor.
  destruct glob_pres as [A [B C]].
  exploit A; eauto. intro EQ.
  econstructor; eauto. rewrite Int.add_zero; auto.
Qed.

Lemma eventval_list_match_inject:
  forall evl tyl vl1, eventval_list_match evl tyl vl1 ->
  forall vl2, val_list_inject f vl1 vl2 -> eventval_list_match evl tyl vl2.
Proof.
  induction 1; intros. inv H; constructor.
  inv H1. constructor. eapply eventval_match_inject; eauto. eauto.
Qed.

(** Determinism *)

Lemma eventval_match_determ_1:
  forall ev ty v1 v2, eventval_match ev ty v1 -> eventval_match ev ty v2 -> v1 = v2.
Proof.
  intros. inv H; inv H0; auto. congruence.
Qed.

Lemma eventval_match_determ_2:
  forall ev1 ev2 ty v, eventval_match ev1 ty v -> eventval_match ev2 ty v -> ev1 = ev2.
Proof.
  intros. inv H; inv H0; auto.
  decEq. eapply Genv.genv_vars_inj; eauto. 
Qed.

Lemma eventval_list_match_determ_2:
  forall evl1 tyl vl, eventval_list_match evl1 tyl vl ->
  forall evl2, eventval_list_match evl2 tyl vl -> evl1 = evl2.
Proof.
  induction 1; intros. inv H. auto. inv H1. f_equal; eauto.
  eapply eventval_match_determ_2; eauto.
Qed.

(** Validity *)

Definition eventval_valid (ev: eventval) : Prop :=
  match ev with
  | EVint _ => True
  | EVfloat _ => True
  | EVptr_global id ofs => exists b, Genv.find_symbol ge id = Some b
  end.

Definition eventval_type (ev: eventval) : typ :=
  match ev with
  | EVint _ => Tint
  | EVfloat _ => Tfloat
  | EVptr_global id ofs => Tint
  end.

Lemma eventval_valid_match:
  forall ev ty,
  eventval_valid ev -> eventval_type ev = ty -> exists v, eventval_match ev ty v.
Proof.
  intros. subst ty. destruct ev; simpl in *.
  exists (Vint i); constructor.
  exists (Vfloat f0); constructor.
  destruct H as [b A]. exists (Vptr b i0); constructor; auto.
Qed.

Lemma eventval_match_valid:
  forall ev ty v,
  eventval_match ev ty v -> eventval_valid ev /\ eventval_type ev = ty.
Proof.
  induction 1; simpl; auto. split; auto. exists b; auto.
Qed.

End EVENTVAL.

(** Invariance under changes to the global environment *)

Section EVENTVAL_INV.

Variables F1 V1 F2 V2: Type.
Variable ge1: Genv.t F1 V1.
Variable ge2: Genv.t F2 V2.

Hypothesis symbols_preserved:
  forall id, Genv.find_symbol ge2 id = Genv.find_symbol ge1 id.

Lemma eventval_match_preserved:
  forall ev ty v,
  eventval_match ge1 ev ty v -> eventval_match ge2 ev ty v.
Proof.
  induction 1; constructor. rewrite symbols_preserved; auto. 
Qed.

Lemma eventval_list_match_preserved:
  forall evl tyl vl,
  eventval_list_match ge1 evl tyl vl -> eventval_list_match ge2 evl tyl vl.
Proof.
  induction 1; constructor; auto. eapply eventval_match_preserved; eauto.
Qed.

Lemma eventval_valid_preserved:
  forall ev, eventval_valid ge1 ev -> eventval_valid ge2 ev.
Proof.
  unfold eventval_valid; destruct ev; auto. 
  intros [b A]. exists b; rewrite symbols_preserved; auto.
Qed.

End EVENTVAL_INV.

(** * Matching traces. *)

Section MATCH_TRACES.

Variables F V: Type.
Variable ge: Genv.t F V.

(** Matching between traces corresponding to single transitions.
  Arguments (provided by the program) must be equal.
  Results (provided by the outside world) can vary as long as they
  can be converted safely to values. *)

Inductive match_traces: trace -> trace -> Prop :=
  | match_traces_E0:
      match_traces nil nil
  | match_traces_syscall: forall id args res1 res2,
      eventval_valid ge res1 -> eventval_valid ge res2 -> eventval_type res1 = eventval_type res2 ->
      match_traces (Event_syscall id args res1 :: nil) (Event_syscall id args res2 :: nil)
  | match_traces_vload: forall chunk id ofs res1 res2,
      eventval_valid ge res1 -> eventval_valid ge res2 -> eventval_type res1 = eventval_type res2 ->
      match_traces (Event_vload chunk id ofs res1 :: nil) (Event_vload chunk id ofs res2 :: nil)
  | match_traces_vstore: forall chunk id ofs arg,
      match_traces (Event_vstore chunk id ofs arg :: nil) (Event_vstore chunk id ofs arg :: nil)
  | match_traces_annot: forall id args,
      match_traces (Event_annot id args :: nil) (Event_annot id args :: nil).

End MATCH_TRACES.

(** Invariance by change of global environment *)

Section MATCH_TRACES_INV.

Variables F1 V1 F2 V2: Type.
Variable ge1: Genv.t F1 V1.
Variable ge2: Genv.t F2 V2.

Hypothesis symbols_preserved:
  forall id, Genv.find_symbol ge2 id = Genv.find_symbol ge1 id.

Lemma match_traces_preserved:
  forall t1 t2, match_traces ge1 t1 t2 -> match_traces ge2 t1 t2.
Proof.
  induction 1; constructor; auto; eapply eventval_valid_preserved; eauto.
Qed.

End MATCH_TRACES_INV.

(** An output trace is a trace composed only of output events,
  that is, events that do not take any result from the outside world. *)

Definition output_event (ev: event) : Prop :=
  match ev with
  | Event_syscall _ _ _ => False
  | Event_vload _ _ _ _ => False
  | Event_vstore _ _ _ _ => True
  | Event_annot _ _ => True
  end.
  
Fixpoint output_trace (t: trace) : Prop :=
  match t with
  | nil => True
  | ev :: t' => output_event ev /\ output_trace t'
  end.

(** * Semantics of volatile memory accesses *)

Definition block_is_volatile (F V: Type) (ge: Genv.t F V) (b: block) : bool :=
  match Genv.find_var_info ge b with
  | None => false
  | Some gv => gv.(gvar_volatile)
  end.

Inductive volatile_load (F V: Type) (ge: Genv.t F V):
                   memory_chunk -> mem -> block -> int -> trace -> val -> Prop :=
  | volatile_load_vol: forall chunk m b ofs id ev v,
      block_is_volatile ge b = true ->
      Genv.find_symbol ge id = Some b ->
      eventval_match ge ev (type_of_chunk chunk) v ->
      volatile_load ge chunk m b ofs
                      (Event_vload chunk id ofs ev :: nil)
                      (Val.load_result chunk v)
  | volatile_load_nonvol: forall chunk m b ofs v,
      block_is_volatile ge b = false ->
      Mem.load chunk m b (Int.unsigned ofs) = Some v ->
      volatile_load ge chunk m b ofs E0 v.

Inductive volatile_store (F V: Type) (ge: Genv.t F V):
                  memory_chunk -> mem -> block -> int -> val -> trace -> mem -> Prop :=
  | volatile_store_vol: forall chunk m b ofs id ev v,
      block_is_volatile ge b = true ->
      Genv.find_symbol ge id = Some b ->
      eventval_match ge ev (type_of_chunk chunk) v ->
      volatile_store ge chunk m b ofs v
                      (Event_vstore chunk id ofs ev :: nil)
                      m
  | volatile_store_nonvol: forall chunk m b ofs v m',
      block_is_volatile ge b = false ->
      Mem.store chunk m b (Int.unsigned ofs) v = Some m' ->
      volatile_store ge chunk m b ofs v E0 m'.

(** * Semantics of external functions *)

(** For each external function, its behavior is defined by a predicate relating:
- the global environment
- the values of the arguments passed to this function
- the memory state before the call
- the result value of the call
- the memory state after the call
- the trace generated by the call (can be empty).
*)

Definition extcall_sem : Type :=
  forall (F V: Type), Genv.t F V -> list val -> mem -> trace -> val -> mem -> Prop.

(** We now specify the expected properties of this predicate. *)

Definition mem_unchanged_on (P: block -> Z -> Prop) (m_before m_after: mem): Prop :=
  (forall b ofs p,
   P b ofs -> Mem.perm m_before b ofs p -> Mem.perm m_after b ofs p)
/\(forall chunk b ofs v,
   (forall i, ofs <= i < ofs + size_chunk chunk -> P b i) ->
   Mem.load chunk m_before b ofs = Some v ->
   Mem.load chunk m_after b ofs = Some v).

Definition loc_out_of_bounds (m: mem) (b: block) (ofs: Z) : Prop :=
  ofs < Mem.low_bound m b \/ ofs > Mem.high_bound m b.

Definition loc_unmapped (f: meminj) (b: block) (ofs: Z): Prop :=
  f b = None.

Definition loc_out_of_reach (f: meminj) (m: mem) (b: block) (ofs: Z): Prop :=
  forall b0 delta,
  f b0 = Some(b, delta) ->
  ofs < Mem.low_bound m b0 + delta \/ ofs >= Mem.high_bound m b0 + delta.

Definition inject_separated (f f': meminj) (m1 m2: mem): Prop :=
  forall b1 b2 delta,
  f b1 = None -> f' b1 = Some(b2, delta) ->
  ~Mem.valid_block m1 b1 /\ ~Mem.valid_block m2 b2.

Record extcall_properties (sem: extcall_sem)
                          (sg: signature) : Prop := mk_extcall_properties {

(** The return value of an external call must agree with its signature. *)
  ec_well_typed:
    forall F V (ge: Genv.t F V) vargs m1 t vres m2,
    sem F V ge vargs m1 t vres m2 ->
    Val.has_type vres (proj_sig_res sg);

(** The number of arguments of an external call must agree with its signature. *)
  ec_arity:
    forall F V (ge: Genv.t F V) vargs m1 t vres m2,
    sem F V ge vargs m1 t vres m2 ->
    List.length vargs = List.length sg.(sig_args);

(** The semantics is invariant under change of global environment that preserves symbols. *)
  ec_symbols_preserved:
    forall F1 V1 (ge1: Genv.t F1 V1) F2 V2 (ge2: Genv.t F2 V2) vargs m1 t vres m2,
    (forall id, Genv.find_symbol ge2 id = Genv.find_symbol ge1 id) ->
    (forall b, block_is_volatile ge2 b = block_is_volatile ge1 b) ->
    sem F1 V1 ge1 vargs m1 t vres m2 ->
    sem F2 V2 ge2 vargs m1 t vres m2;

(** External calls cannot invalidate memory blocks.  (Remember that
  freeing a block does not invalidate its block identifier.) *)
  ec_valid_block:
    forall F V (ge: Genv.t F V) vargs m1 t vres m2 b,
    sem F V ge vargs m1 t vres m2 ->
    Mem.valid_block m1 b -> Mem.valid_block m2 b;

(** External calls preserve the bounds of valid blocks. *)
  ec_bounds:
    forall F V (ge: Genv.t F V) vargs m1 t vres m2 b,
    sem F V ge vargs m1 t vres m2 ->
    Mem.valid_block m1 b -> Mem.bounds m2 b = Mem.bounds m1 b;

(** External calls must commute with memory extensions, in the
  following sense. *)
  ec_mem_extends:
    forall F V (ge: Genv.t F V) vargs m1 t vres m2 m1' vargs',
    sem F V ge vargs m1 t vres m2 ->
    Mem.extends m1 m1' ->
    Val.lessdef_list vargs vargs' ->
    exists vres', exists m2',
       sem F V ge vargs' m1' t vres' m2'
    /\ Val.lessdef vres vres'
    /\ Mem.extends m2 m2'
    /\ mem_unchanged_on (loc_out_of_bounds m1) m1' m2';

(** External calls must commute with memory injections,
  in the following sense. *)
  ec_mem_inject:
    forall F V (ge: Genv.t F V) vargs m1 t vres m2 f m1' vargs',
    meminj_preserves_globals ge f ->
    sem F V ge vargs m1 t vres m2 ->
    Mem.inject f m1 m1' ->
    val_list_inject f vargs vargs' ->
    exists f', exists vres', exists m2',
       sem F V ge vargs' m1' t vres' m2'
    /\ val_inject f' vres vres'
    /\ Mem.inject f' m2 m2'
    /\ mem_unchanged_on (loc_unmapped f) m1 m2
    /\ mem_unchanged_on (loc_out_of_reach f m1) m1' m2'
    /\ inject_incr f f'
    /\ inject_separated f f' m1 m1';

(** External calls produce at most one event. *)
  ec_trace_length:
    forall F V ge vargs m t vres m',
    sem F V ge vargs m t vres m' -> (length t <= 1)%nat;

(** External calls must be receptive to changes of traces by another, matching trace. *)
  ec_receptive:
    forall F V ge vargs m t1 vres1 m1 t2,
    sem F V ge vargs m t1 vres1 m1 -> match_traces ge t1 t2 ->
    exists vres2, exists m2, sem F V ge vargs m t2 vres2 m2;

(** External calls must be deterministic up to matching between traces. *)
  ec_determ:
    forall F V ge vargs m t1 vres1 m1 t2 vres2 m2,
    sem F V ge vargs m t1 vres1 m1 -> sem F V ge vargs m t2 vres2 m2 ->
    match_traces ge t1 t2 /\ (t1 = t2 -> vres1 = vres2 /\ m1 = m2)
}.

(** ** Semantics of volatile loads *)

Inductive volatile_load_sem (chunk: memory_chunk) (F V: Type) (ge: Genv.t F V):
              list val -> mem -> trace -> val -> mem -> Prop :=
  | volatile_load_sem_intro: forall b ofs m t v,
      volatile_load ge chunk m b ofs t v ->
      volatile_load_sem chunk ge (Vptr b ofs :: nil) m t v m.

Lemma volatile_load_preserved:
  forall F1 V1 (ge1: Genv.t F1 V1) F2 V2 (ge2: Genv.t F2 V2) chunk m b ofs t v,
  (forall id, Genv.find_symbol ge2 id = Genv.find_symbol ge1 id) ->
  (forall b, block_is_volatile ge2 b = block_is_volatile ge1 b) ->
  volatile_load ge1 chunk m b ofs t v ->
  volatile_load ge2 chunk m b ofs t v.
Proof.
  intros. inv H1; constructor; auto.
  rewrite H0; auto.
  rewrite H; auto.
  eapply eventval_match_preserved; eauto.
  rewrite H0; auto.
Qed.

Lemma volatile_load_extends:
  forall F V (ge: Genv.t F V) chunk m b ofs t v m',
  volatile_load ge chunk m b ofs t v ->
  Mem.extends m m' ->
  exists v', volatile_load ge chunk m' b ofs t v' /\ Val.lessdef v v'.
Proof.
  intros. inv H.
  econstructor; split; eauto. econstructor; eauto.
  exploit Mem.load_extends; eauto. intros [v' [A B]]. exists v'; split; auto. constructor; auto.
Qed.

Remark meminj_preserves_block_is_volatile:
  forall F V (ge: Genv.t F V) f b1 b2 delta,
  meminj_preserves_globals ge f ->
  f b1 = Some (b2, delta) ->
  block_is_volatile ge b2 = block_is_volatile ge b1.
Proof.
  intros. destruct H as [A [B C]]. unfold block_is_volatile. 
  case_eq (Genv.find_var_info ge b1); intros.
  exploit B; eauto. intro EQ; rewrite H0 in EQ; inv EQ. rewrite H; auto.
  case_eq (Genv.find_var_info ge b2); intros.
  exploit C; eauto. intro EQ. congruence.
  auto.
Qed.

Lemma volatile_load_inject:
  forall F V (ge: Genv.t F V) f chunk m b ofs t v b' ofs' m',
  meminj_preserves_globals ge f ->
  volatile_load ge chunk m b ofs t v ->
  val_inject f (Vptr b ofs) (Vptr b' ofs') ->
  Mem.inject f m m' ->
  exists v', volatile_load ge chunk m' b' ofs' t v' /\ val_inject f v v'.
Proof.
  intros. inv H0.
  inv H1. exploit (proj1 H); eauto. intros EQ; rewrite H8 in EQ; inv EQ.
  rewrite Int.add_zero. exists (Val.load_result chunk v0); split.
  constructor; auto. 
  apply val_load_result_inject. eapply eventval_match_inject_2; eauto.
  exploit Mem.loadv_inject; eauto. simpl; eauto. simpl; intros [v' [A B]]. exists v'; split; auto. 
  constructor; auto. rewrite <- H3. inv H1. eapply meminj_preserves_block_is_volatile; eauto. 
Qed.

Lemma volatile_load_receptive:
  forall F V (ge: Genv.t F V) chunk m b ofs t1 t2 v1,
  volatile_load ge chunk m b ofs t1 v1 -> match_traces ge t1 t2 ->
  exists v2, volatile_load ge chunk m b ofs t2 v2.
Proof.
  intros. inv H; inv H0.
  exploit eventval_match_valid; eauto. intros [A B]. 
  exploit eventval_valid_match. eexact H9. rewrite <- H10; eauto. 
  intros [v' EVM]. exists (Val.load_result chunk v'). constructor; auto.
  exists v1; constructor; auto.
Qed.

Lemma volatile_load_ok:
  forall chunk,
  extcall_properties (volatile_load_sem chunk) 
                     (mksignature (Tint :: nil) (Some (type_of_chunk chunk))).
Proof.
  intros; constructor; intros.
(* well typed *)
  unfold proj_sig_res; simpl. inv H. inv H0.
  destruct chunk; destruct v; simpl; constructor.
  eapply Mem.load_type; eauto. 
(* arity *)
  inv H; inv H0; auto.
(* symbols *)
  inv H1. constructor. eapply volatile_load_preserved; eauto. 
(* valid blocks *)
  inv H; auto.
(* bounds *)
  inv H; auto.
(* mem extends *)
  inv H. inv H1. inv H6. inv H4. 
  exploit volatile_load_extends; eauto. intros [v' [A B]].
  exists v'; exists m1'; intuition. constructor; auto. red; auto.
(* mem injects *)
  inv H0. inv H2. inv H7. inversion H5; subst. 
  exploit volatile_load_inject; eauto. intros [v' [A B]]. 
  exists f; exists v'; exists m1'; intuition. constructor; auto.
  red; auto. red; auto. red; intros. congruence.
(* trace length *)
  inv H; inv H0; simpl; omega.
(* receptive *)
  inv H. exploit volatile_load_receptive; eauto. intros [v2 A].
  exists v2; exists m1; constructor; auto.
(* determ *)
  inv H; inv H0. inv H1; inv H7; try congruence.
  assert (id = id0) by (eapply Genv.genv_vars_inj; eauto). subst id0.
  exploit eventval_match_valid. eexact H2. intros [V1 T1].
  exploit eventval_match_valid. eexact H4. intros [V2 T2].
  split. constructor; auto. congruence.
  intros EQ; inv EQ. 
  assert (v = v0) by (eapply eventval_match_determ_1; eauto). subst v0.
  auto.
  split. constructor. intuition congruence.
Qed.

Inductive volatile_load_global_sem (chunk: memory_chunk) (id: ident) (ofs: int)
                                   (F V: Type) (ge: Genv.t F V):
              list val -> mem -> trace -> val -> mem -> Prop :=
  | volatile_load_global_sem_intro: forall b t v m,
      Genv.find_symbol ge id = Some b ->
      volatile_load ge chunk m b ofs t v ->
      volatile_load_global_sem chunk id ofs ge nil m t v m.

Remark volatile_load_global_charact:
  forall chunk id ofs (F V: Type) (ge: Genv.t F V) vargs m t vres m',
  volatile_load_global_sem chunk id ofs ge vargs m t vres m' <->
  exists b, Genv.find_symbol ge id = Some b /\ volatile_load_sem chunk ge (Vptr b ofs :: vargs) m t vres m'.
Proof.
  intros; split.
  intros. inv H. exists b; split; auto. constructor; auto. 
  intros [b [P Q]]. inv Q. econstructor; eauto.
Qed.

Lemma volatile_load_global_ok:
  forall chunk id ofs,
  extcall_properties (volatile_load_global_sem chunk id ofs) 
                     (mksignature nil (Some (type_of_chunk chunk))).
Proof.
  intros; constructor; intros.
(* well typed *)
  unfold proj_sig_res; simpl. inv H. inv H1.
  destruct chunk; destruct v; simpl; constructor.
  eapply Mem.load_type; eauto. 
(* arity *)
  inv H; inv H1; auto.
(* symbols *)
  inv H1. econstructor. rewrite H; eauto. eapply volatile_load_preserved; eauto. 
(* valid blocks *)
  inv H; auto.
(* bounds *)
  inv H; auto.
(* extends *)
  inv H. inv H1. exploit volatile_load_extends; eauto. intros [v' [A B]].
  exists v'; exists m1'; intuition. econstructor; eauto. red; auto.
(* inject *)
  inv H0. inv H2.
  assert (val_inject f (Vptr b ofs) (Vptr b ofs)). 
    exploit (proj1 H); eauto. intros EQ. econstructor. eauto. rewrite Int.add_zero; auto.
  exploit volatile_load_inject; eauto. intros [v' [A B]].
  exists f; exists v'; exists m1'; intuition. econstructor; eauto. 
  red; auto. red; auto. red; intros; congruence. 
(* trace length *)
  inv H; inv H1; simpl; omega.
(* receptive *)
  inv H. exploit volatile_load_receptive; eauto. intros [v2 A].
  exists v2; exists m1; econstructor; eauto.
(* determ *)
  rewrite volatile_load_global_charact in *. 
  destruct H as [b1 [A1 B1]]. destruct H0 as [b2 [A2 B2]].
  rewrite A1 in A2; inv A2.
  eapply ec_determ. eapply volatile_load_ok. eauto. eauto. 
Qed.

(** ** Semantics of volatile stores *)

Inductive volatile_store_sem (chunk: memory_chunk) (F V: Type) (ge: Genv.t F V):
              list val -> mem -> trace -> val -> mem -> Prop :=
  | volatile_store_sem_intro: forall b ofs m1 v t m2,
      volatile_store ge chunk m1 b ofs v t m2 ->
      volatile_store_sem chunk ge (Vptr b ofs :: v :: nil) m1 t Vundef m2.

Lemma volatile_store_preserved:
  forall F1 V1 (ge1: Genv.t F1 V1) F2 V2 (ge2: Genv.t F2 V2) chunk m1 b ofs v t m2,
  (forall id, Genv.find_symbol ge2 id = Genv.find_symbol ge1 id) ->
  (forall b, block_is_volatile ge2 b = block_is_volatile ge1 b) ->
  volatile_store ge1 chunk m1 b ofs v t m2 ->
  volatile_store ge2 chunk m1 b ofs v t m2.
Proof.
  intros. inv H1; constructor; auto.
  rewrite H0; auto.
  rewrite H; auto.
  eapply eventval_match_preserved; eauto.
  rewrite H0; auto.
Qed.

Lemma volatile_store_extends:
  forall F V (ge: Genv.t F V) chunk m1 b ofs v t m2 m1' v',
  volatile_store ge chunk m1 b ofs v t m2 ->
  Mem.extends m1 m1' ->
  Val.lessdef v v' ->
  exists m2', 
     volatile_store ge chunk m1' b ofs v' t m2'
  /\ Mem.extends m2 m2'
  /\ mem_unchanged_on (loc_out_of_bounds m1) m1' m2'.
Proof.
  intros. inv H.
  econstructor; split. econstructor; eauto. eapply eventval_match_lessdef; eauto.
  split. auto. red; auto.
  exploit Mem.store_within_extends; eauto. intros [m2' [A B]].
  exists m2'; intuition. econstructor; eauto. 
  red; split; intros.
  eapply Mem.perm_store_1; eauto.
  rewrite <- H4. eapply Mem.load_store_other; eauto.
  destruct (eq_block b0 b); auto. subst b0; right. 
  exploit Mem.valid_access_in_bounds. 
  eapply Mem.store_valid_access_3. eexact H3.
  intros [C D].
  generalize (size_chunk_pos chunk0). intro E.
  generalize (size_chunk_pos chunk). intro G.
  apply (Intv.range_disjoint' (ofs0, ofs0 + size_chunk chunk0)
                              (Int.unsigned ofs, Int.unsigned ofs + size_chunk chunk)).
  red; intros. generalize (H x H5). unfold loc_out_of_bounds, Intv.In; simpl. omega.
  simpl; omega. simpl; omega.
Qed.

Lemma volatile_store_inject:
  forall F V (ge: Genv.t F V) f chunk m1 b ofs v t m2 m1' b' ofs' v',
  meminj_preserves_globals ge f ->
  volatile_store ge chunk m1 b ofs v t m2 ->
  val_inject f (Vptr b ofs) (Vptr b' ofs') ->
  val_inject f v v' ->
  Mem.inject f m1 m1' ->
  exists m2',
       volatile_store ge chunk m1' b' ofs' v' t m2'
    /\ Mem.inject f m2 m2'
    /\ mem_unchanged_on (loc_unmapped f) m1 m2
    /\ mem_unchanged_on (loc_out_of_reach f m1) m1' m2'.
Proof.
  intros. inv H0.
  inv H1. exploit (proj1 H); eauto. intros EQ; rewrite H9 in EQ; inv EQ.
  rewrite Int.add_zero. exists m1'.
  split. constructor; auto.  eapply eventval_match_inject; eauto.
  split. auto. split. red; auto. red; auto. 
  assert (Mem.storev chunk m1 (Vptr b ofs) v = Some m2). simpl; auto.
  exploit Mem.storev_mapped_inject; eauto. intros [m2' [A B]].
  inv H1. exists m2'; intuition. 
  constructor; auto. rewrite <- H4. eapply meminj_preserves_block_is_volatile; eauto.  
  split; intros. eapply Mem.perm_store_1; eauto.
  rewrite <- H6. eapply Mem.load_store_other; eauto. 
  left. exploit (H1 ofs0). generalize (size_chunk_pos chunk0). omega. 
  unfold loc_unmapped. congruence.
  split; intros. eapply Mem.perm_store_1; eauto.
  rewrite <- H6. eapply Mem.load_store_other; eauto.
  destruct (eq_block b0 b'); auto. subst b0; right.
  assert (EQ: Int.unsigned (Int.add ofs (Int.repr delta)) = Int.unsigned ofs + delta).
    eapply Mem.address_inject; eauto with mem.
  unfold Mem.storev in A. rewrite EQ in A. rewrite EQ.
  exploit Mem.valid_access_in_bounds. 
  eapply Mem.store_valid_access_3. eexact H0.
  intros [C D].
  generalize (size_chunk_pos chunk0). intro E.
  generalize (size_chunk_pos chunk). intro G.
  apply (Intv.range_disjoint' (ofs0, ofs0 + size_chunk chunk0)
                              (Int.unsigned ofs + delta, Int.unsigned ofs + delta + size_chunk chunk)).
  red; intros. exploit (H1 x H7). eauto. unfold Intv.In; simpl. omega.
  simpl; omega. simpl; omega.
Qed.

Lemma volatile_store_receptive:
  forall F V (ge: Genv.t F V) chunk m b ofs v t1 m1 t2,
  volatile_store ge chunk m b ofs v t1 m1 -> match_traces ge t1 t2 -> t1 = t2.
Proof.
  intros. inv H; inv H0; auto.  
Qed.

Lemma volatile_store_ok:
  forall chunk,
  extcall_properties (volatile_store_sem chunk) 
                     (mksignature (Tint :: type_of_chunk chunk :: nil) None).
Proof.
  intros; constructor; intros.
(* well typed *)
  unfold proj_sig_res; simpl. inv H; constructor.
(* arity *)
  inv H; simpl; auto.
(* symbols preserved *)
  inv H1. constructor. eapply volatile_store_preserved; eauto. 
(* valid block *)
  inv H. inv H1. auto. eauto with mem.
(* bounds *)
  inv H. inv H1. auto. eapply Mem.bounds_store; eauto.
(* mem extends*)
  inv H. inv H1. inv H6. inv H7. inv H4.
  exploit volatile_store_extends; eauto. intros [m2' [A [B C]]]. 
  exists Vundef; exists m2'; intuition. constructor; auto. 
(* mem inject *)
  inv H0. inv H2. inv H7. inv H8. inversion H5; subst.
  exploit volatile_store_inject; eauto. intros [m2' [A [B [C D]]]].
  exists f; exists Vundef; exists m2'; intuition. constructor; auto. red; intros; congruence.
(* trace length *)
  inv H; inv H0; simpl; omega.
(* receptive *)
  assert (t1 = t2). inv H. eapply volatile_store_receptive; eauto.
  subst t2; exists vres1; exists m1; auto.
(* determ *)
  inv H; inv H0. inv H1; inv H8; try congruence.
  assert (id = id0) by (eapply Genv.genv_vars_inj; eauto). subst id0.
  assert (ev = ev0) by (eapply eventval_match_determ_2; eauto). subst ev0.
  split. constructor. auto.
  split. constructor. intuition congruence.
Qed.

Inductive volatile_store_global_sem (chunk: memory_chunk) (id: ident) (ofs: int)
                                   (F V: Type) (ge: Genv.t F V):
              list val -> mem -> trace -> val -> mem -> Prop :=
  | volatile_store_global_sem_intro: forall b m1 v t m2,
      Genv.find_symbol ge id = Some b ->
      volatile_store ge chunk m1 b ofs v t m2 ->
      volatile_store_global_sem chunk id ofs ge (v :: nil) m1 t Vundef m2.

Remark volatile_store_global_charact:
  forall chunk id ofs (F V: Type) (ge: Genv.t F V) vargs m t vres m',
  volatile_store_global_sem chunk id ofs ge vargs m t vres m' <->
  exists b, Genv.find_symbol ge id = Some b /\ volatile_store_sem chunk ge (Vptr b ofs :: vargs) m t vres m'.
Proof.
  intros; split. 
  intros. inv H; exists b; split; auto; econstructor; eauto.
  intros [b [P Q]]. inv Q. econstructor; eauto. 
Qed.

Lemma volatile_store_global_ok:
  forall chunk id ofs,
  extcall_properties (volatile_store_global_sem chunk id ofs) 
                     (mksignature (type_of_chunk chunk :: nil) None).
Proof.
  intros; constructor; intros.
(* well typed *)
  unfold proj_sig_res; simpl. inv H; constructor.
(* arity *)
  inv H; simpl; auto.
(* symbols preserved *)
  inv H1. econstructor. rewrite H; eauto. eapply volatile_store_preserved; eauto. 
(* valid block *)
  inv H. inv H2. auto. eauto with mem.
(* bounds *)
  inv H. inv H2. auto. eapply Mem.bounds_store; eauto.
(* mem extends*)
  rewrite volatile_store_global_charact in H. destruct H as [b [P Q]]. 
  exploit ec_mem_extends. eapply volatile_store_ok. eexact Q. eauto. eauto. 
  intros [vres' [m2' [A [B [C D]]]]]. 
  exists vres'; exists m2'; intuition. rewrite volatile_store_global_charact. exists b; auto. 
(* mem inject *)
  rewrite volatile_store_global_charact in H0. destruct H0 as [b [P Q]]. 
  exploit (proj1 H). eauto. intros EQ. 
  assert (val_inject f (Vptr b ofs) (Vptr b ofs)). econstructor; eauto. rewrite Int.add_zero; auto.
  exploit ec_mem_inject. eapply volatile_store_ok. eauto. eexact Q. eauto. eauto. 
  intros [f' [vres' [m2' [A [B [C [D [E G]]]]]]]].
  exists f'; exists vres'; exists m2'; intuition.
  rewrite volatile_store_global_charact. exists b; auto. 
(* trace length *)
  inv H. inv H1; simpl; omega.
(* receptive *)
  assert (t1 = t2). inv H. eapply volatile_store_receptive; eauto. subst t2. 
  exists vres1; exists m1; congruence.
(* determ *)
  rewrite volatile_store_global_charact in *. 
  destruct H as [b1 [A1 B1]]. destruct H0 as [b2 [A2 B2]]. rewrite A1 in A2; inv A2. 
  eapply ec_determ. eapply volatile_store_ok. eauto. eauto. 
Qed.

(** ** Semantics of dynamic memory allocation (malloc) *)

Inductive extcall_malloc_sem (F V: Type) (ge: Genv.t F V):
              list val -> mem -> trace -> val -> mem -> Prop :=
  | extcall_malloc_sem_intro: forall n m m' b m'',
      Mem.alloc m (-4) (Int.unsigned n) = (m', b) ->
      Mem.store Mint32 m' b (-4) (Vint n) = Some m'' ->
      extcall_malloc_sem ge (Vint n :: nil) m E0 (Vptr b Int.zero) m''.

Lemma extcall_malloc_ok:
  extcall_properties extcall_malloc_sem 
                     (mksignature (Tint :: nil) (Some Tint)).
Proof.
  assert (UNCHANGED:
    forall (P: block -> Z -> Prop) m n m' b m'',
    Mem.alloc m (-4) (Int.unsigned n) = (m', b) ->
    Mem.store Mint32 m' b (-4) (Vint n) = Some m'' ->
    mem_unchanged_on P m m'').
  intros; split; intros.
  eauto with mem.
  transitivity (Mem.load chunk m' b0 ofs). 
  eapply Mem.load_store_other; eauto. left. 
  apply Mem.valid_not_valid_diff with m; eauto with mem.
  eapply Mem.load_alloc_other; eauto. 

  constructor; intros.
(* well typed *)
  inv H. unfold proj_sig_res; simpl. auto.
(* arity *)
  inv H. auto.
(* symbols preserved *)
  inv H1; econstructor; eauto.
(* valid block *)
  inv H. eauto with mem.
(* bounds *)
  inv H. transitivity (Mem.bounds m' b).
  eapply Mem.bounds_store; eauto.
  eapply Mem.bounds_alloc_other; eauto.
  apply Mem.valid_not_valid_diff with m1; eauto with mem.
(* mem extends *)
  inv H. inv H1. inv H5. inv H7. 
  exploit Mem.alloc_extends; eauto. apply Zle_refl. apply Zle_refl.
  intros [m3' [A B]].
  exploit Mem.store_within_extends. eexact B. eauto. 
  instantiate (1 := Vint n). auto. 
  intros [m2' [C D]].
  exists (Vptr b Int.zero); exists m2'; intuition.
  econstructor; eauto.
  eapply UNCHANGED; eauto.
(* mem injects *)
  inv H0. inv H2. inv H6. inv H8.
  exploit Mem.alloc_parallel_inject; eauto. apply Zle_refl. apply Zle_refl. 
  intros [f' [m3' [b' [ALLOC [A [B [C D]]]]]]].
  exploit Mem.store_mapped_inject. eexact A. eauto. eauto. 
  instantiate (1 := Vint n). auto. 
  intros [m2' [E G]].
  exists f'; exists (Vptr b' Int.zero); exists m2'; intuition.
  econstructor; eauto.
  econstructor. eauto. auto.
  eapply UNCHANGED; eauto.
  eapply UNCHANGED; eauto.
  red; intros. destruct (eq_block b1 b). 
  subst b1. rewrite C in H2. inv H2. eauto with mem. 
  rewrite D in H2. congruence. auto. 
(* trace length *)
  inv H; simpl; omega.
(* receptive *)
  assert (t1 = t2). inv H; inv H0; auto. subst t2.
  exists vres1; exists m1; auto.
(* determ *)
  inv H; inv H0. split. constructor. intuition congruence. 
Qed.

(** ** Semantics of dynamic memory deallocation (free) *)

Inductive extcall_free_sem  (F V: Type) (ge: Genv.t F V):
              list val -> mem -> trace -> val -> mem -> Prop :=
  | extcall_free_sem_intro: forall b lo sz m m',
      Mem.load Mint32 m b (Int.unsigned lo - 4) = Some (Vint sz) ->
      Int.unsigned sz > 0 ->
      Mem.free m b (Int.unsigned lo - 4) (Int.unsigned lo + Int.unsigned sz) = Some m' ->
      extcall_free_sem ge (Vptr b lo :: nil) m E0 Vundef m'.

Lemma extcall_free_ok:
  extcall_properties extcall_free_sem 
                     (mksignature (Tint :: nil) None).
Proof.
  assert (UNCHANGED:
    forall (P: block -> Z -> Prop) m b lo hi m',
    Mem.free m b lo hi = Some m' ->
    lo < hi ->
    (forall b' ofs, P b' ofs -> b' <> b \/ ofs < lo \/ hi <= ofs) ->
    mem_unchanged_on P m m').
  intros; split; intros.
  eapply Mem.perm_free_1; eauto.
  rewrite <- H3. eapply Mem.load_free; eauto. 
  destruct (eq_block b0 b); auto. right. right. 
  apply (Intv.range_disjoint' (ofs, ofs + size_chunk chunk) (lo, hi)).
  red; intros. apply Intv.notin_range. simpl. exploit H1; eauto. intuition. 
  simpl; generalize (size_chunk_pos chunk); omega.
  simpl; omega.

  constructor; intros.
(* well typed *)
  inv H. unfold proj_sig_res. simpl. auto.
(* arity *)
  inv H. auto.
(* symbols preserved *)
  inv H1; econstructor; eauto.
(* valid block *)
  inv H. eauto with mem.
(* bounds *)
  inv H. eapply Mem.bounds_free; eauto.
(* mem extends *)
  inv H. inv H1. inv H8. inv H6. 
  exploit Mem.load_extends; eauto. intros [vsz [A B]]. inv B. 
  exploit Mem.free_parallel_extends; eauto. intros [m2' [C D]].
  exists Vundef; exists m2'; intuition.
  econstructor; eauto.
  eapply UNCHANGED; eauto. omega. 
  intros. destruct (eq_block b' b); auto. subst b; right.
  red in H.
  exploit Mem.range_perm_in_bounds. 
  eapply Mem.free_range_perm. eexact H4. omega. omega.
(* mem inject *)
  inv H0. inv H2. inv H7. inv H9.
  exploit Mem.load_inject; eauto. intros [vsz [A B]]. inv B. 
  assert (Mem.range_perm m1 b (Int.unsigned lo - 4) (Int.unsigned lo + Int.unsigned sz) Freeable).
    eapply Mem.free_range_perm; eauto.
  exploit Mem.address_inject; eauto. 
    apply Mem.perm_implies with Freeable; auto with mem.
    apply H0. instantiate (1 := lo). omega. 
  intro EQ.
  assert (Mem.range_perm m1' b2 (Int.unsigned lo + delta - 4) (Int.unsigned lo + delta + Int.unsigned sz) Freeable).
    red; intros. 
    replace ofs with ((ofs - delta) + delta) by omega.
    eapply Mem.perm_inject; eauto. apply H0. omega. 
  destruct (Mem.range_perm_free _ _ _ _ H2) as [m2' FREE].
  exists f; exists Vundef; exists m2'; intuition.

  econstructor.
  rewrite EQ. replace (Int.unsigned lo + delta - 4) with (Int.unsigned lo - 4 + delta) by omega.
  eauto. auto. 
  rewrite EQ. auto.
  
  assert (Mem.free_list m1 ((b, Int.unsigned lo - 4, Int.unsigned lo + Int.unsigned sz) :: nil) = Some m2).
    simpl. rewrite H5. auto.
  eapply Mem.free_inject; eauto. 
  intros. destruct (eq_block b b1).
  subst b. assert (delta0 = delta) by congruence. subst delta0. 
  exists (Int.unsigned lo - 4); exists (Int.unsigned lo + Int.unsigned sz); split.
  simpl; auto. omega.
  elimtype False.
  exploit Mem.inject_no_overlap. eauto. eauto. eauto. eauto. 
  instantiate (1 := ofs + delta0 - delta). 
  apply Mem.perm_implies with Freeable; auto with mem.
  apply H0. omega. eauto with mem.
  unfold block; omega.

  eapply UNCHANGED; eauto. omega. intros.  
  red in H7. left. congruence. 

  eapply UNCHANGED; eauto. omega. intros.
  destruct (eq_block b' b2); auto. subst b'. right. 
  red in H7. generalize (H7 _ _ H6). intros. 
  exploit Mem.range_perm_in_bounds. eexact H0. omega. intros. omega.

  red; intros. congruence.
(* trace length *)
  inv H; simpl; omega.
(* receptive *)
  assert (t1 = t2). inv H; inv H0; auto. subst t2.
  exists vres1; exists m1; auto.
(* determ *)
  inv H; inv H0. split. constructor. intuition congruence.
Qed.

(** ** Semantics of [memcpy] operations. *)

Inductive extcall_memcpy_sem (sz al: Z) (F V: Type) (ge: Genv.t F V): list val -> mem -> trace -> val -> mem -> Prop :=
  | extcall_memcpy_sem_intro: forall bdst odst bsrc osrc m bytes m',
      al = 1 \/ al = 2 \/ al = 4 -> sz > 0 ->
      (al | sz) -> (al | Int.unsigned osrc) -> (al | Int.unsigned odst) ->
      bsrc <> bdst \/ Int.unsigned osrc = Int.unsigned odst
                   \/ Int.unsigned osrc + sz <= Int.unsigned odst
                   \/ Int.unsigned odst + sz <= Int.unsigned osrc ->
      Mem.loadbytes m bsrc (Int.unsigned osrc) sz = Some bytes ->
      Mem.storebytes m bdst (Int.unsigned odst) bytes = Some m' ->
      extcall_memcpy_sem sz al ge (Vptr bdst odst :: Vptr bsrc osrc :: nil) m E0 Vundef m'.

Lemma extcall_memcpy_ok:
  forall sz al,
  extcall_properties (extcall_memcpy_sem sz al) (mksignature (Tint :: Tint :: nil) None).
Proof.
  intros. constructor.
(* return type *)
  intros. inv H. constructor. 
(* arity *)
  intros. inv H. auto.
(* change of globalenv *)
  intros. inv H1. econstructor; eauto.
(* valid blocks *)
  intros. inv H. eauto with mem. 
(* bounds *)
  intros. inv H. eapply Mem.bounds_storebytes; eauto. 
(* extensions *)
  intros. inv H. 
  inv H1. inv H13. inv H14. inv H10. inv H11.
  exploit Mem.loadbytes_length; eauto. intros LEN.
  exploit Mem.loadbytes_extends; eauto. intros [bytes2 [A B]].
  exploit Mem.storebytes_within_extends; eauto. intros [m2' [C D]].
  exists Vundef; exists m2'.
  split. econstructor; eauto.
  split. constructor.
  split. auto.
  red; split; intros.
  eauto with mem.  
  exploit Mem.loadbytes_length. eexact H8. intros.
  rewrite <- H1. eapply Mem.load_storebytes_other; eauto. 
  destruct (eq_block b bdst); auto. subst b; right.
  exploit Mem.range_perm_in_bounds. eapply Mem.storebytes_range_perm. eexact H9.
  rewrite H10. rewrite nat_of_Z_eq. omega. omega.
  intros [P Q].
  exploit list_forall2_length; eauto. intros R. rewrite R in Q.
  apply (Intv.range_disjoint' (ofs, ofs + size_chunk chunk)
                              (Int.unsigned odst, Int.unsigned odst + Z_of_nat (length bytes2))); simpl.
  red; intros. generalize (H x H11). unfold loc_out_of_bounds, Intv.In; simpl. omega.
  generalize (size_chunk_pos chunk); omega.
  rewrite <- R; rewrite H10. rewrite nat_of_Z_eq. omega. omega.
(* injections *)
  intros. inv H0. inv H2. inv H14. inv H15. inv H11. inv H12.
  exploit Mem.loadbytes_length; eauto. intros LEN.
  assert (RPSRC: Mem.range_perm m1 bsrc (Int.unsigned osrc) (Int.unsigned osrc + sz) Nonempty).
    eapply Mem.range_perm_implies. eapply Mem.loadbytes_range_perm; eauto. auto with mem.
  assert (RPDST: Mem.range_perm m1 bdst (Int.unsigned odst) (Int.unsigned odst + sz) Nonempty).
    replace sz with (Z_of_nat (length bytes)).
    eapply Mem.range_perm_implies. eapply Mem.storebytes_range_perm; eauto. auto with mem.
    rewrite LEN. apply nat_of_Z_eq. omega.
  assert (PSRC: Mem.perm m1 bsrc (Int.unsigned osrc) Nonempty).
    apply RPSRC. omega.
  assert (PDST: Mem.perm m1 bdst (Int.unsigned odst) Nonempty).
    apply RPDST. omega.
  exploit Mem.address_inject.  eauto. eexact PSRC. eauto. intros EQ1.
  exploit Mem.address_inject.  eauto. eexact PDST. eauto. intros EQ2.
  exploit Mem.loadbytes_inject; eauto. intros [bytes2 [A B]].
  exploit Mem.storebytes_mapped_inject; eauto. intros [m2' [C D]].
  exists f; exists Vundef; exists m2'.
  split. econstructor; try rewrite EQ1; try rewrite EQ2; eauto. 
  eapply Mem.aligned_area_inject with (m := m1); eauto.
  eapply Mem.aligned_area_inject with (m := m1); eauto.
  eapply Mem.disjoint_or_equal_inject with (m := m1); eauto.
  split. constructor.
  split. auto.
  split. red; split; intros. eauto with mem. 
  rewrite <- H2. eapply Mem.load_storebytes_other; eauto. 
  destruct (eq_block b bdst); auto. subst b. 
  assert (loc_unmapped f bdst ofs). apply H0. generalize (size_chunk_pos chunk); omega. 
  red in H12. congruence.
  split. red; split; intros. eauto with mem.
  rewrite <- H2. eapply Mem.load_storebytes_other; eauto. 
  destruct (eq_block b b0); auto. subst b0; right.
  rewrite <- (list_forall2_length B). rewrite LEN. rewrite nat_of_Z_eq; try omega.
  apply (Intv.range_disjoint' (ofs, ofs + size_chunk chunk)
                              (Int.unsigned odst + delta0, Int.unsigned odst + delta0 + sz)); simpl.
  red; intros. generalize (H0 x H12). unfold loc_out_of_reach, Intv.In; simpl. 
  intros. exploit H14; eauto. 
  exploit Mem.range_perm_in_bounds. eexact RPDST. omega. 
  omega.
  generalize (size_chunk_pos chunk); omega.
  omega.
  split. apply inject_incr_refl.
  red; intros; congruence.
(* trace length *)
  intros; inv H. simpl; omega.
(* receptive *)
  intros. 
  assert (t1 = t2). inv H; inv H0; auto. subst t2.
  exists vres1; exists m1; auto.
(* determ *)
  intros; inv H; inv H0. split. constructor. intros; split; congruence.
Qed.

(** ** Semantics of system calls. *)

Inductive extcall_io_sem (name: ident) (sg: signature) (F V: Type) (ge: Genv.t F V):
              list val -> mem -> trace -> val -> mem -> Prop :=
  | extcall_io_sem_intro: forall vargs m args res vres,
      eventval_list_match ge args (sig_args sg) vargs ->
      eventval_match ge res (proj_sig_res sg) vres ->
      extcall_io_sem name sg ge vargs m (Event_syscall name args res :: E0) vres m.

Lemma extcall_io_ok:
  forall name sg,
  extcall_properties (extcall_io_sem name sg) sg.
Proof.
  intros; constructor; intros.
(* well typed *)
  inv H. eapply eventval_match_type; eauto.
(* arity *)
  inv H. eapply eventval_list_match_length; eauto.
(* symbols preserved *)
  inv H1. econstructor; eauto. 
  eapply eventval_list_match_preserved; eauto.
  eapply eventval_match_preserved; eauto. 
(* valid block *)
  inv H; auto.
(* bounds *)
  inv H; auto.
(* mem extends *)
  inv H.
  exists vres; exists m1'; intuition.
  econstructor; eauto.
  eapply eventval_list_match_lessdef; eauto.
  red; auto.
(* mem injects *)
  inv H0.
  exists f; exists vres; exists m1'; intuition.
  econstructor; eauto.
  eapply eventval_list_match_inject; eauto.
  eapply eventval_match_inject_2; eauto.
  red; auto.
  red; auto.
  red; intros; congruence.
(* trace length *)
  inv H; simpl; omega.
(* receptive *)
  inv H; inv H0.
  exploit eventval_match_valid; eauto. intros [A B]. 
  exploit eventval_valid_match. eexact H7. rewrite <- H8; eauto. 
  intros [v' EVM]. exists v'; exists m1. econstructor; eauto. 
(* determ *)
  inv H; inv H0.
  assert (args = args0). eapply eventval_list_match_determ_2; eauto. subst args0.
  exploit eventval_match_valid. eexact H2. intros [V1 T1].
  exploit eventval_match_valid. eexact H3. intros [V2 T2].
  split. constructor; auto. congruence.
  intros EQ; inv EQ. split; auto. eapply eventval_match_determ_1; eauto.
Qed.

(** ** Semantics of annotation. *)

Inductive extcall_annot_sem (text: ident) (targs: list typ) (F V: Type) (ge: Genv.t F V):
              list val -> mem -> trace -> val -> mem -> Prop :=
  | extcall_annot_sem_intro: forall vargs m args,
      eventval_list_match ge args targs vargs ->
      extcall_annot_sem text targs ge vargs m (Event_annot text args :: E0) Vundef m.

Lemma extcall_annot_ok:
  forall text targs,
  extcall_properties (extcall_annot_sem text targs) (mksignature targs None).
Proof.
  intros; constructor; intros.
(* well typed *)
  inv H. simpl. auto.
(* arity *)
  inv H. eapply eventval_list_match_length; eauto.
(* symbols *)
  inv H1. econstructor; eauto. 
  eapply eventval_list_match_preserved; eauto.
(* valid blocks *)
  inv H; auto.
(* bounds *)
  inv H; auto.
(* mem extends *)
  inv H.
  exists Vundef; exists m1'; intuition.
  econstructor; eauto.
  eapply eventval_list_match_lessdef; eauto.
  red; auto.
(* mem injects *)
  inv H0.
  exists f; exists Vundef; exists m1'; intuition.
  econstructor; eauto.
  eapply eventval_list_match_inject; eauto.
  red; auto.
  red; auto.
  red; intros; congruence.
(* trace length *)
  inv H; simpl; omega.
(* receptive *)
  assert (t1 = t2). inv H; inv H0; auto. 
  exists vres1; exists m1; congruence.
(* determ *)
  inv H; inv H0.
  assert (args = args0). eapply eventval_list_match_determ_2; eauto. subst args0.
  split. constructor. auto.
Qed.

Inductive extcall_annot_val_sem (text: ident) (targ: typ) (F V: Type) (ge: Genv.t F V):
              list val -> mem -> trace -> val -> mem -> Prop :=
  | extcall_annot_val_sem_intro: forall varg m arg,
      eventval_match ge arg targ varg ->
      extcall_annot_val_sem text targ ge (varg :: nil) m (Event_annot text (arg :: nil) :: E0) varg m.

Lemma extcall_annot_val_ok:
  forall text targ,
  extcall_properties (extcall_annot_val_sem text targ) (mksignature (targ :: nil) (Some targ)).
Proof.
  intros; constructor; intros.

  inv H. unfold proj_sig_res; simpl. eapply eventval_match_type; eauto.

  inv H. auto.

  inv H1. econstructor; eauto. 
  eapply eventval_match_preserved; eauto.

  inv H; auto.

  inv H; auto.

  inv H. inv H1. inv H6. 
  exists v2; exists m1'; intuition.
  econstructor; eauto.
  eapply eventval_match_lessdef; eauto.
  red; auto.

  inv H0. inv H2. inv H7.
  exists f; exists v'; exists m1'; intuition.
  econstructor; eauto.
  eapply eventval_match_inject; eauto.
  red; auto.
  red; auto.
  red; intros; congruence.

  inv H; simpl; omega.

  assert (t1 = t2). inv H; inv H0; auto. subst t2.
  exists vres1; exists m1; auto.

  inv H; inv H0.
  assert (arg = arg0). eapply eventval_match_determ_2; eauto. subst arg0.
  split. constructor. auto.
Qed.

(** ** Combined semantics of external calls *)

(** Combining the semantics given above for the various kinds of external calls,
  we define the predicate [external_call] that relates:
- the external function being invoked
- the values of the arguments passed to this function
- the memory state before the call
- the result value of the call
- the memory state after the call
- the trace generated by the call (can be empty).

This predicate is used in the semantics of all CompCert languages. *)

Definition external_call (ef: external_function): extcall_sem :=
  match ef with
  | EF_external name sg  => extcall_io_sem name sg
  | EF_builtin name sg   => extcall_io_sem name sg
  | EF_vload chunk       => volatile_load_sem chunk
  | EF_vstore chunk      => volatile_store_sem chunk
  | EF_vload_global chunk id ofs => volatile_load_global_sem chunk id ofs
  | EF_vstore_global chunk id ofs => volatile_store_global_sem chunk id ofs
  | EF_malloc            => extcall_malloc_sem 
  | EF_free              => extcall_free_sem
  | EF_memcpy sz al      => extcall_memcpy_sem sz al
  | EF_annot txt targs   => extcall_annot_sem txt targs
  | EF_annot_val txt targ=> extcall_annot_val_sem txt targ
  end.

Theorem external_call_spec:
  forall ef, 
  extcall_properties (external_call ef) (ef_sig ef).
Proof.
  intros. unfold external_call, ef_sig. destruct ef.
  apply extcall_io_ok.
  apply extcall_io_ok.
  apply volatile_load_ok.
  apply volatile_store_ok.
  apply volatile_load_global_ok.
  apply volatile_store_global_ok.
  apply extcall_malloc_ok.
  apply extcall_free_ok.
  apply extcall_memcpy_ok.
  apply extcall_annot_ok.
  apply extcall_annot_val_ok.
Qed.

Definition external_call_well_typed ef := ec_well_typed (external_call_spec ef).
Definition external_call_arity ef := ec_arity (external_call_spec ef).
Definition external_call_symbols_preserved_gen ef := ec_symbols_preserved (external_call_spec ef).
Definition external_call_valid_block ef := ec_valid_block (external_call_spec ef).
Definition external_call_bounds ef := ec_bounds (external_call_spec ef).
Definition external_call_mem_extends ef := ec_mem_extends (external_call_spec ef).
Definition external_call_mem_inject ef := ec_mem_inject (external_call_spec ef).
Definition external_call_trace_length ef := ec_trace_length (external_call_spec ef).
Definition external_call_receptive ef := ec_receptive (external_call_spec ef).
Definition external_call_determ ef := ec_determ (external_call_spec ef).

(** Special cases of [external_call_symbols_preserved_gen]. *)

Lemma external_call_symbols_preserved:
  forall ef F1 F2 V (ge1: Genv.t F1 V) (ge2: Genv.t F2 V) vargs m1 t vres m2,
  external_call ef ge1 vargs m1 t vres m2 ->
  (forall id, Genv.find_symbol ge2 id = Genv.find_symbol ge1 id) ->
  (forall b, Genv.find_var_info ge2 b = Genv.find_var_info ge1 b) ->
  external_call ef ge2 vargs m1 t vres m2.
Proof.
  intros. eapply external_call_symbols_preserved_gen; eauto.
  intros. unfold block_is_volatile. rewrite H1. auto.
Qed.

Require Import Errors.

Lemma external_call_symbols_preserved_2:
  forall ef F1 V1 F2 V2 (tvar: V1 -> res V2)
         (ge1: Genv.t F1 V1) (ge2: Genv.t F2 V2) vargs m1 t vres m2,
  external_call ef ge1 vargs m1 t vres m2 ->
  (forall id, Genv.find_symbol ge2 id = Genv.find_symbol ge1 id) ->
  (forall b gv1, Genv.find_var_info ge1 b = Some gv1 ->
     exists gv2, Genv.find_var_info ge2 b = Some gv2 /\ transf_globvar tvar gv1 = OK gv2) ->
  (forall b gv2, Genv.find_var_info ge2 b = Some gv2 ->
     exists gv1, Genv.find_var_info ge1 b = Some gv1 /\ transf_globvar tvar gv1 = OK gv2) ->
  external_call ef ge2 vargs m1 t vres m2.
Proof.
  intros. eapply external_call_symbols_preserved_gen; eauto.
  intros. unfold block_is_volatile.
  case_eq (Genv.find_var_info ge1 b); intros.
  exploit H1; eauto. intros [g2 [A B]]. rewrite A. monadInv B. destruct g; auto.
  case_eq (Genv.find_var_info ge2 b); intros.
  exploit H2; eauto. intros [g1 [A B]]. congruence.
  auto.
Qed.

(** Corollary of [external_call_valid_block]. *)

Lemma external_call_nextblock:
  forall ef (F V : Type) (ge : Genv.t F V) vargs m1 t vres m2,
  external_call ef ge vargs m1 t vres m2 ->
  Mem.nextblock m1 <= Mem.nextblock m2.
Proof.
  intros. 
  exploit external_call_valid_block; eauto. 
  instantiate (1 := Mem.nextblock m1 - 1). red; omega.
  unfold Mem.valid_block. omega.
Qed.

(** Corollaries of [external_call_determ]. *)

Lemma external_call_match_traces:
  forall ef (F V : Type) (ge : Genv.t F V) vargs m t1 vres1 m1 t2 vres2 m2,
  external_call ef ge vargs m t1 vres1 m1 ->
  external_call ef ge vargs m t2 vres2 m2 ->
  match_traces ge t1 t2.
Proof.
  intros. exploit external_call_determ. eexact H. eexact H0. tauto.
Qed.

Lemma external_call_deterministic:
  forall ef (F V : Type) (ge : Genv.t F V) vargs m t vres1 m1 vres2 m2,
  external_call ef ge vargs m t vres1 m1 ->
  external_call ef ge vargs m t vres2 m2 ->
  vres1 = vres2 /\ m1 = m2.
Proof.
  intros. exploit external_call_determ. eexact H. eexact H0. intuition.
Qed.