Require Import String.
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

Inductive eventval: Type :=
  | EVint: int -> eventval
  | EVlong: int64 -> eventval
  | EVfloat: float -> eventval
  | EVsingle: float32 -> eventval
  | EVptr_global: ident -> int -> eventval.

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


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.


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

Lemma E0_left: forall t, E0 ** t = t.

Lemma E0_right: forall t, t ** E0 = t.

Lemma Eapp_assoc: forall t1 t2 t3, (t1 ** t2) ** t3 = t1 ** (t2 ** t3).

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.

Lemma Eappinf_assoc: forall t1 t2 T, (t1 ** t2) *** T = t1 *** (t2 *** T).

Hint Rewrite E0_left E0_right Eapp_assoc
             E0_left_inf Eappinf_assoc: trace_rewrite.

Opaque trace E0 Eapp Eappinf.


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.


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.

Lemma traceinf_sim_sym:
  forall T1 T2, traceinf_sim T1 T2 -> traceinf_sim T2 T1.

Lemma traceinf_sim_trans:
  forall T1 T2 T3,
  traceinf_sim T1 T2 -> traceinf_sim T2 T3 -> traceinf_sim T1 T3.

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.


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.

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.

Remark unroll_traceinf:
  forall T, T = match T with Econsinf t T' => Econsinf t T' end.

Lemma traceinf_traceinf'_app:
  forall t T NE,
  traceinf_of_traceinf' (Econsinf' t T NE) = t *** traceinf_of_traceinf' T.


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).

Lemma traceinf_prefix_app:
  forall t1 T2 t,
  traceinf_prefix t1 T2 ->
  traceinf_prefix (t ** t1) (t *** T2).

Relating values and event values

Set Implicit Arguments.

Section EVENTVAL.

Variable ge: Senv.t.


Inductive eventval_match: eventval -> typ -> val -> Prop :=
  | ev_match_int: forall i,
      eventval_match (EVint i) Tint (Vint i)
  | ev_match_long: forall i,
      eventval_match (EVlong i) Tlong (Vlong i)
  | ev_match_float: forall f,
      eventval_match (EVfloat f) Tfloat (Vfloat f)
  | ev_match_single: forall f,
      eventval_match (EVsingle f) Tsingle (Vsingle f)
  | ev_match_ptr: forall id b ofs,
      Senv.public_symbol ge id = true ->
      Senv.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).


Lemma eventval_match_type:
  forall ev ty v,
  eventval_match ev ty v -> Val.has_type v ty.

Lemma eventval_list_match_length:
  forall evl tyl vl, eventval_list_match evl tyl vl -> List.length vl = List.length tyl.

Lemma eventval_match_lessdef:
  forall ev ty v1 v2,
  eventval_match ev ty v1 -> Val.lessdef v1 v2 -> eventval_match ev ty v2.

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.


Lemma eventval_match_determ_1:
  forall ev ty v1 v2, eventval_match ev ty v1 -> eventval_match ev ty v2 -> v1 = v2.

Lemma eventval_match_determ_2:
  forall ev1 ev2 ty v, eventval_match ev1 ty v -> eventval_match ev2 ty v -> ev1 = ev2.

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.


Definition eventval_valid (ev: eventval) : Prop :=
  match ev with
  | EVint _ => True
  | EVlong _ => True
  | EVfloat _ => True
  | EVsingle _ => True
  | EVptr_global id ofs => Senv.public_symbol ge id = true
  end.

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

Lemma eventval_match_receptive:
  forall ev1 ty v1 ev2,
  eventval_match ev1 ty v1 ->
  eventval_valid ev1 -> eventval_valid ev2 -> eventval_type ev1 = eventval_type ev2 ->
  exists v2, eventval_match ev2 ty v2.

Lemma eventval_match_valid:
  forall ev ty v, eventval_match ev ty v -> eventval_valid ev.

Lemma eventval_match_same_type:
  forall ev1 ty v1 ev2 v2,
  eventval_match ev1 ty v1 -> eventval_match ev2 ty v2 -> eventval_type ev1 = eventval_type ev2.

End EVENTVAL.


Section EVENTVAL_INV.

Variables ge1 ge2: Senv.t.

Hypothesis public_preserved:
  forall id, Senv.public_symbol ge2 id = Senv.public_symbol ge1 id.

Lemma eventval_valid_preserved:
  forall ev, eventval_valid ge1 ev -> eventval_valid ge2 ev.

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

Lemma eventval_match_preserved:
  forall ev ty v,
  eventval_match ge1 ev ty v -> eventval_match ge2 ev ty v.

Lemma eventval_list_match_preserved:
  forall evl tyl vl,
  eventval_list_match ge1 evl tyl vl -> eventval_list_match ge2 evl tyl vl.

End EVENTVAL_INV.


Section EVENTVAL_INJECT.

Variable f: block -> option (block * Z).
Variable ge1 ge2: Senv.t.

Definition symbols_inject : Prop :=
   (forall id, Senv.public_symbol ge2 id = Senv.public_symbol ge1 id)
/\ (forall id b1 b2 delta,
     f b1 = Some(b2, delta) -> Senv.find_symbol ge1 id = Some b1 ->
     delta = 0 /\ Senv.find_symbol ge2 id = Some b2)
/\ (forall id b1,
     Senv.public_symbol ge1 id = true -> Senv.find_symbol ge1 id = Some b1 ->
     exists b2, f b1 = Some(b2, 0) /\ Senv.find_symbol ge2 id = Some b2)
/\ (forall b1 b2 delta,
     f b1 = Some(b2, delta) ->
     Senv.block_is_volatile ge2 b2 = Senv.block_is_volatile ge1 b1).

Hypothesis symb_inj: symbols_inject.

Lemma eventval_match_inject:
  forall ev ty v1 v2,
  eventval_match ge1 ev ty v1 -> Val.inject f v1 v2 -> eventval_match ge2 ev ty v2.

Lemma eventval_match_inject_2:
  forall ev ty v1,
  eventval_match ge1 ev ty v1 ->
  exists v2, eventval_match ge2 ev ty v2 /\ Val.inject f v1 v2.

Lemma eventval_list_match_inject:
  forall evl tyl vl1, eventval_list_match ge1 evl tyl vl1 ->
  forall vl2, Val.inject_list f vl1 vl2 -> eventval_list_match ge2 evl tyl vl2.

End EVENTVAL_INJECT.

Matching traces.

Section MATCH_TRACES.

Variable ge: Senv.t.


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.


Section MATCH_TRACES_INV.

Variables ge1 ge2: Senv.t.

Hypothesis public_preserved:
  forall id, Senv.public_symbol ge2 id = Senv.public_symbol ge1 id.

Lemma match_traces_preserved:
  forall t1 t2, match_traces ge1 t1 t2 -> match_traces ge2 t1 t2.

End MATCH_TRACES_INV.


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

Inductive volatile_load (ge: Senv.t):
                   memory_chunk -> mem -> block -> int -> trace -> val -> Prop :=
  | volatile_load_vol: forall chunk m b ofs id ev v,
      Senv.block_is_volatile ge b = true ->
      Senv.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,
      Senv.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 (ge: Senv.t):
                  memory_chunk -> mem -> block -> int -> val -> trace -> mem -> Prop :=
  | volatile_store_vol: forall chunk m b ofs id ev v,
      Senv.block_is_volatile ge b = true ->
      Senv.find_symbol ge id = Some b ->
      eventval_match ge ev (type_of_chunk chunk) (Val.load_result 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',
      Senv.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

Definition extcall_sem : Type :=
  Senv.t -> list val -> mem -> trace -> val -> mem -> Prop.


Definition loc_out_of_bounds (m: mem) (b: block) (ofs: Z) : Prop :=
  ~Mem.perm m b ofs Max Nonempty.

Definition loc_not_writable (m: mem) (b: block) (ofs: Z) : Prop :=
  ~Mem.perm m b ofs Max Writable.

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) -> ~Mem.perm m b0 (ofs - delta) Max Nonempty.

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 {

  ec_well_typed:
    forall ge vargs m1 t vres m2,
    sem ge vargs m1 t vres m2 ->
    Val.has_type vres (proj_sig_res sg);

  ec_symbols_preserved:
    forall ge1 ge2 vargs m1 t vres m2,
    (forall id, Senv.find_symbol ge2 id = Senv.find_symbol ge1 id) ->
    (forall id, Senv.public_symbol ge2 id = Senv.public_symbol ge1 id) ->
    (forall b, Senv.block_is_volatile ge2 b = Senv.block_is_volatile ge1 b) ->
    sem ge1 vargs m1 t vres m2 ->
    sem ge2 vargs m1 t vres m2;

  ec_valid_block:
    forall ge vargs m1 t vres m2 b,
    sem ge vargs m1 t vres m2 ->
    Mem.valid_block m1 b -> Mem.valid_block m2 b;

  ec_max_perm:
    forall ge vargs m1 t vres m2 b ofs p,
    sem ge vargs m1 t vres m2 ->
    Mem.valid_block m1 b -> Mem.perm m2 b ofs Max p -> Mem.perm m1 b ofs Max p;

  ec_readonly:
    forall ge vargs m1 t vres m2,
    sem ge vargs m1 t vres m2 ->
    Mem.unchanged_on (loc_not_writable m1) m1 m2;

  ec_mem_extends:
    forall ge vargs m1 t vres m2 m1' vargs',
    sem ge vargs m1 t vres m2 ->
    Mem.extends m1 m1' ->
    Val.lessdef_list vargs vargs' ->
    exists vres', exists m2',
       sem ge vargs' m1' t vres' m2'
    /\ Val.lessdef vres vres'
    /\ Mem.extends m2 m2'
    /\ Mem.unchanged_on (loc_out_of_bounds m1) m1' m2';

  ec_mem_inject:
    forall ge1 ge2 vargs m1 t vres m2 f m1' vargs',
    symbols_inject f ge1 ge2 ->
    sem ge1 vargs m1 t vres m2 ->
    Mem.inject f m1 m1' ->
    Val.inject_list f vargs vargs' ->
    exists f', exists vres', exists m2',
       sem ge2 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';

  ec_trace_length:
    forall ge vargs m t vres m',
    sem ge vargs m t vres m' -> (length t <= 1)%nat;

  ec_receptive:
    forall ge vargs m t1 vres1 m1 t2,
    sem ge vargs m t1 vres1 m1 -> match_traces ge t1 t2 ->
    exists vres2, exists m2, sem ge vargs m t2 vres2 m2;

  ec_determ:
    forall ge vargs m t1 vres1 m1 t2 vres2 m2,
    sem ge vargs m t1 vres1 m1 -> sem 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) (ge: Senv.t):
              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 ge1 ge2 chunk m b ofs t v,
  (forall id, Senv.find_symbol ge2 id = Senv.find_symbol ge1 id) ->
  (forall id, Senv.public_symbol ge2 id = Senv.public_symbol ge1 id) ->
  (forall b, Senv.block_is_volatile ge2 b = Senv.block_is_volatile ge1 b) ->
  volatile_load ge1 chunk m b ofs t v ->
  volatile_load ge2 chunk m b ofs t v.

Lemma volatile_load_extends:
  forall ge 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'.

Lemma volatile_load_inject:
  forall ge1 ge2 f chunk m b ofs t v b' ofs' m',
  symbols_inject f ge1 ge2 ->
  volatile_load ge1 chunk m b ofs t v ->
  Val.inject f (Vptr b ofs) (Vptr b' ofs') ->
  Mem.inject f m m' ->
  exists v', volatile_load ge2 chunk m' b' ofs' t v' /\ Val.inject f v v'.

Lemma volatile_load_receptive:
  forall ge 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.

Lemma volatile_load_ok:
  forall chunk,
  extcall_properties (volatile_load_sem chunk)
                     (mksignature (Tint :: nil) (Some (type_of_chunk chunk)) cc_default).

Semantics of volatile stores

Inductive volatile_store_sem (chunk: memory_chunk) (ge: Senv.t):
              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 ge1 ge2 chunk m1 b ofs v t m2,
  (forall id, Senv.find_symbol ge2 id = Senv.find_symbol ge1 id) ->
  (forall id, Senv.public_symbol ge2 id = Senv.public_symbol ge1 id) ->
  (forall b, Senv.block_is_volatile ge2 b = Senv.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.

Lemma volatile_store_readonly:
  forall ge chunk1 m1 b1 ofs1 v t m2,
  volatile_store ge chunk1 m1 b1 ofs1 v t m2 ->
  Mem.unchanged_on (loc_not_writable m1) m1 m2.

Lemma volatile_store_extends:
  forall ge 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'.

Lemma volatile_store_inject:
  forall ge1 ge2 f chunk m1 b ofs v t m2 m1' b' ofs' v',
  symbols_inject f ge1 ge2 ->
  volatile_store ge1 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 ge2 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'.

Lemma volatile_store_receptive:
  forall ge 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.

Lemma volatile_store_ok:
  forall chunk,
  extcall_properties (volatile_store_sem chunk)
                     (mksignature (Tint :: type_of_chunk chunk :: nil) None cc_default).

Semantics of dynamic memory allocation (malloc)

Inductive extcall_malloc_sem (ge: Senv.t):
              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) cc_default).

Semantics of dynamic memory deallocation (free)

Inductive extcall_free_sem (ge: Senv.t):
              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 cc_default).

Semantics of memcpy operations.

Inductive extcall_memcpy_sem (sz al: Z) (ge: Senv.t):
                        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 \/ al = 8 -> sz >= 0 -> (al | sz) ->
      (sz > 0 -> (al | Int.unsigned osrc)) ->
      (sz > 0 -> (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 cc_default).

Semantics of annotations.

Inductive extcall_annot_sem (text: string) (targs: list typ) (ge: Senv.t):
              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 cc_default).

Inductive extcall_annot_val_sem (text: string) (targ: typ) (ge: Senv.t):
              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) cc_default).

Inductive extcall_debug_sem (ge: Senv.t):
              list val -> mem -> trace -> val -> mem -> Prop :=
  | extcall_debug_sem_intro: forall vargs m,
      extcall_debug_sem ge vargs m E0 Vundef m.

Lemma extcall_debug_ok:
  forall targs,
  extcall_properties extcall_debug_sem
                     (mksignature targs None cc_default).

Semantics of external functions.

Parameter external_functions_sem: String.string -> signature -> extcall_sem.

Axiom external_functions_properties:
  forall id sg, extcall_properties (external_functions_sem id sg) sg.


Parameter inline_assembly_sem: String.string -> signature -> extcall_sem.

Axiom inline_assembly_properties:
  forall id sg, extcall_properties (inline_assembly_sem id sg) sg.

Combined semantics of external calls

Definition external_call (ef: external_function): extcall_sem :=
  match ef with
  | EF_external name sg => external_functions_sem name sg
  | EF_builtin name sg => external_functions_sem name sg
  | EF_vload chunk => volatile_load_sem chunk
  | EF_vstore chunk => volatile_store_sem chunk
  | 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
  | EF_inline_asm txt sg clb => inline_assembly_sem txt sg
  | EF_debug kind txt targs => extcall_debug_sem
  end.

Theorem external_call_spec:
  forall ef,
  extcall_properties (external_call ef) (ef_sig ef).

Definition external_call_well_typed ef := ec_well_typed (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_max_perm ef := ec_max_perm (external_call_spec ef).
Definition external_call_readonly ef := ec_readonly (external_call_spec ef).
Definition external_call_mem_extends ef := ec_mem_extends (external_call_spec ef).
Definition external_call_mem_inject_gen 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).


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 id, Genv.public_symbol ge2 id = Genv.public_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.


Lemma external_call_nextblock:
  forall ef ge vargs m1 t vres m2,
  external_call ef ge vargs m1 t vres m2 ->
  Ple (Mem.nextblock m1) (Mem.nextblock m2).


Definition meminj_preserves_globals (F V: Type) (ge: Genv.t F V) (f: block -> option (block * Z)) : 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).

Lemma external_call_mem_inject:
  forall ef F V (ge: Genv.t F V) vargs m1 t vres m2 f m1' vargs',
  meminj_preserves_globals ge f ->
  external_call ef ge vargs m1 t vres m2 ->
  Mem.inject f m1 m1' ->
  Val.inject_list f vargs vargs' ->
  exists f', exists vres', exists m2',
     external_call ef 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'.


Lemma external_call_match_traces:
  forall ef ge 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.

Lemma external_call_deterministic:
  forall ef ge 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.


Fixpoint decode_longs (tyl: list typ) (vl: list val) : list val :=
  match tyl with
  | nil => nil
  | Tlong :: tys =>
      match vl with
      | v1 :: v2 :: vs => Val.longofwords v1 v2 :: decode_longs tys vs
      | _ => nil
      end
  | ty :: tys =>
      match vl with
      | v1 :: vs => v1 :: decode_longs tys vs
      | _ => nil
      end
  end.

Definition encode_long (oty: option typ) (v: val) : list val :=
  match oty with
  | Some Tlong => Val.hiword v :: Val.loword v :: nil
  | _ => v :: nil
  end.

Definition proj_sig_res' (s: signature) : list typ :=
  match s.(sig_res) with
  | Some Tlong => Tint :: Tint :: nil
  | Some ty => ty :: nil
  | None => Tint :: nil
  end.

Inductive external_call'
      (ef: external_function) (ge: Senv.t)
      (vargs: list val) (m1: mem) (t: trace) (vres: list val) (m2: mem) : Prop :=
  external_call'_intro: forall v,
    external_call ef ge (decode_longs (sig_args (ef_sig ef)) vargs) m1 t v m2 ->
    vres = encode_long (sig_res (ef_sig ef)) v ->
    external_call' ef ge vargs m1 t vres m2.

Lemma decode_longs_lessdef:
  forall tyl vl1 vl2, Val.lessdef_list vl1 vl2 -> Val.lessdef_list (decode_longs tyl vl1) (decode_longs tyl vl2).

Lemma decode_longs_inject:
  forall f tyl vl1 vl2, Val.inject_list f vl1 vl2 -> Val.inject_list f (decode_longs tyl vl1) (decode_longs tyl vl2).

Lemma encode_long_lessdef:
  forall oty v1 v2, Val.lessdef v1 v2 -> Val.lessdef_list (encode_long oty v1) (encode_long oty v2).

Lemma encode_long_inject:
  forall f oty v1 v2, Val.inject f v1 v2 -> Val.inject_list f (encode_long oty v1) (encode_long oty v2).

Lemma encode_long_has_type:
  forall v sg,
  Val.has_type v (proj_sig_res sg) ->
  Val.has_type_list (encode_long (sig_res sg) v) (proj_sig_res' sg).

Lemma external_call_well_typed':
  forall ef ge vargs m1 t vres m2,
  external_call' ef ge vargs m1 t vres m2 ->
  Val.has_type_list vres (proj_sig_res' (ef_sig ef)).

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 id, Genv.public_symbol ge2 id = Genv.public_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.

Lemma external_call_valid_block':
  forall ef ge vargs m1 t vres m2 b,
  external_call' ef ge vargs m1 t vres m2 ->
  Mem.valid_block m1 b -> Mem.valid_block m2 b.

Lemma external_call_nextblock':
  forall ef ge vargs m1 t vres m2,
  external_call' ef ge vargs m1 t vres m2 ->
  Ple (Mem.nextblock m1) (Mem.nextblock m2).

Lemma external_call_mem_extends':
  forall ef ge vargs m1 t vres m2 m1' vargs',
  external_call' ef ge vargs m1 t vres m2 ->
  Mem.extends m1 m1' ->
  Val.lessdef_list vargs vargs' ->
  exists vres' m2',
     external_call' ef ge vargs' m1' t vres' m2'
  /\ Val.lessdef_list vres vres'
  /\ Mem.extends m2 m2'
  /\ Mem.unchanged_on (loc_out_of_bounds m1) m1' m2'.

Lemma external_call_mem_inject':
  forall ef F V (ge: Genv.t F V) vargs m1 t vres m2 f m1' vargs',
  meminj_preserves_globals ge f ->
  external_call' ef ge vargs m1 t vres m2 ->
  Mem.inject f m1 m1' ->
  Val.inject_list f vargs vargs' ->
  exists f' vres' m2',
     external_call' ef ge vargs' m1' t vres' m2'
  /\ Val.inject_list 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'.

Lemma external_call_determ':
  forall ef ge 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 /\ (t1 = t2 -> vres1 = vres2 /\ m1 = m2).

Lemma external_call_match_traces':
  forall ef ge 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.

Lemma external_call_deterministic':
  forall ef ge 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.

Evaluation of builtin arguments

Section EVAL_BUILTIN_ARG.

Variable A: Type.
Variable ge: Senv.t.
Variable e: A -> val.
Variable sp: val.
Variable m: mem.

Inductive eval_builtin_arg: builtin_arg A -> val -> Prop :=
  | eval_BA: forall x,
      eval_builtin_arg (BA x) (e x)
  | eval_BA_int: forall n,
      eval_builtin_arg (BA_int n) (Vint n)
  | eval_BA_long: forall n,
      eval_builtin_arg (BA_long n) (Vlong n)
  | eval_BA_float: forall n,
      eval_builtin_arg (BA_float n) (Vfloat n)
  | eval_BA_single: forall n,
      eval_builtin_arg (BA_single n) (Vsingle n)
  | eval_BA_loadstack: forall chunk ofs v,
      Mem.loadv chunk m (Val.add sp (Vint ofs)) = Some v ->
      eval_builtin_arg (BA_loadstack chunk ofs) v
  | eval_BA_addrstack: forall ofs,
      eval_builtin_arg (BA_addrstack ofs) (Val.add sp (Vint ofs))
  | eval_BA_loadglobal: forall chunk id ofs v,
      Mem.loadv chunk m (Senv.symbol_address ge id ofs) = Some v ->
      eval_builtin_arg (BA_loadglobal chunk id ofs) v
  | eval_BA_addrglobal: forall id ofs,
      eval_builtin_arg (BA_addrglobal id ofs) (Senv.symbol_address ge id ofs)
  | eval_BA_splitlong: forall hi lo vhi vlo,
      eval_builtin_arg hi vhi -> eval_builtin_arg lo vlo ->
      eval_builtin_arg (BA_splitlong hi lo) (Val.longofwords vhi vlo).

Definition eval_builtin_args (al: list (builtin_arg A)) (vl: list val) : Prop :=
  list_forall2 eval_builtin_arg al vl.

Lemma eval_builtin_arg_determ:
  forall a v, eval_builtin_arg a v -> forall v', eval_builtin_arg a v' -> v' = v.

Lemma eval_builtin_args_determ:
  forall al vl, eval_builtin_args al vl -> forall vl', eval_builtin_args al vl' -> vl' = vl.

End EVAL_BUILTIN_ARG.

Hint Constructors eval_builtin_arg: barg.


Section EVAL_BUILTIN_ARG_PRESERVED.

Variables A F1 V1 F2 V2: Type.
Variable ge1: Genv.t F1 V1.
Variable ge2: Genv.t F2 V2.
Variable e: A -> val.
Variable sp: val.
Variable m: mem.

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

Lemma eval_builtin_arg_preserved:
  forall a v, eval_builtin_arg ge1 e sp m a v -> eval_builtin_arg ge2 e sp m a v.

Lemma eval_builtin_args_preserved:
  forall al vl, eval_builtin_args ge1 e sp m al vl -> eval_builtin_args ge2 e sp m al vl.

End EVAL_BUILTIN_ARG_PRESERVED.


Section EVAL_BUILTIN_ARG_LESSDEF.

Variable A: Type.
Variable ge: Senv.t.
Variables e1 e2: A -> val.
Variable sp: val.
Variables m1 m2: mem.

Hypothesis env_lessdef: forall x, Val.lessdef (e1 x) (e2 x).
Hypothesis mem_extends: Mem.extends m1 m2.

Lemma eval_builtin_arg_lessdef:
  forall a v1, eval_builtin_arg ge e1 sp m1 a v1 ->
  exists v2, eval_builtin_arg ge e2 sp m2 a v2 /\ Val.lessdef v1 v2.

Lemma eval_builtin_args_lessdef:
  forall al vl1, eval_builtin_args ge e1 sp m1 al vl1 ->
  exists vl2, eval_builtin_args ge e2 sp m2 al vl2 /\ Val.lessdef_list vl1 vl2.

End EVAL_BUILTIN_ARG_LESSDEF.