Module SimplExprproof

Require Import FunInd.
Require Import Coqlib Maps Errors Integers.
Require Import AST Linking.
Require Import Values Memory Events Globalenvs Smallstep.
Require Import Ctypes Cop Csyntax Csem Cstrategy Clight.
Require Import SimplExpr SimplExprspec.

Relational specification of the translation.

Definition match_prog (p: Csyntax.program) (tp: Clight.program) :=
    match_program_gen tr_fundef eq p p tp
 /\ prog_types tp = prog_types p.

Lemma transf_program_match:
  forall p tp, transl_program p = OK tp -> match_prog p tp.

Semantic preservation

Section PRESERVATION.

Variable prog: Csyntax.program.
Variable tprog: Clight.program.
Hypothesis TRANSL: match_prog prog tprog.

Let ge := Csem.globalenv prog.
Let tge := Clight.globalenv tprog.


Lemma comp_env_preserved:
  Clight.genv_cenv tge = Csem.genv_cenv ge.

Lemma symbols_preserved:
  forall (s: qualident), Genv.find_symbol tge s = Genv.find_symbol ge s.
Proof (Genv.find_symbol_match (proj1 TRANSL)).

Lemma senv_preserved:
  Senv.equiv ge tge.
Proof (Genv.senv_match (proj1 TRANSL)).

Lemma function_ptr_translated:
  forall b f,
  Genv.find_funct_ptr ge b = Some f ->
  exists cu tf,
  Genv.find_funct_ptr tge b = Some tf /\ tr_fundef cu f tf /\ linkorder cu prog.
Proof (Genv.find_funct_ptr_match (proj1 TRANSL)).

Lemma functions_translated:
  forall v f,
  Genv.find_funct ge v = Some f ->
  exists cu tf,
  Genv.find_funct tge v = Some tf /\ tr_fundef cu f tf /\ linkorder cu prog.
Proof (Genv.find_funct_match (proj1 TRANSL)).

Lemma type_of_fundef_preserved:
  forall cu f tf, tr_fundef cu f tf ->
  type_of_fundef tf = Csyntax.type_of_fundef f.

Lemma function_return_preserved:
  forall ce f tf, tr_function ce f tf ->
  fn_return tf = Csyntax.fn_return f.


Section TRANSLATION.

Variable cunit: Csyntax.program.
Hypothesis LINKORDER: linkorder cunit prog.
Let ce := cunit.(prog_comp_env).

Lemma eval_Ederef':
  forall ge e le m a t l ofs,
  eval_expr ge e le m a (Vptr l ofs) ->
  eval_lvalue ge e le m (Ederef' a t) l ofs Full.

Lemma typeof_Ederef':
  forall a t, typeof (Ederef' a t) = t.

Lemma eval_Eaddrof':
  forall ge e le m a t l ofs,
  eval_lvalue ge e le m a l ofs Full ->
  eval_expr ge e le m (Eaddrof' a t) (Vptr l ofs).

Lemma typeof_Eaddrof':
  forall a t, typeof (Eaddrof' a t) = t.

Lemma eval_make_normalize:
  forall ge e le m a n sz sg sg1 attr width,
  0 < width -> width <= bitsize_intsize sz ->
  typeof a = Tint sz sg1 attr ->
  eval_expr ge e le m a (Vint n) ->
  eval_expr ge e le m (make_normalize sz sg width a) (Vint (bitfield_normalize sz sg width n)).


Lemma tr_simple_nil:
  (forall le dst r sl a tmps, tr_expr ce le dst r sl a tmps ->
   dst = For_val \/ dst = For_effects -> simple r = true -> sl = nil)
/\(forall le rl sl al tmps, tr_exprlist ce le rl sl al tmps ->
   simplelist rl = true -> sl = nil).

Lemma tr_simple_expr_nil:
  forall le dst r sl a tmps, tr_expr ce le dst r sl a tmps ->
  dst = For_val \/ dst = For_effects -> simple r = true -> sl = nil.
Proof (proj1 tr_simple_nil).

Lemma tr_simple_exprlist_nil:
  forall le rl sl al tmps, tr_exprlist ce le rl sl al tmps ->
  simplelist rl = true -> sl = nil.
Proof (proj2 tr_simple_nil).


Remark deref_loc_translated:
  forall ty m b ofs bf t v,
  Csem.deref_loc ge ty m b ofs bf t v ->
  match chunk_for_volatile_type ty bf with
  | None => t = E0 /\ Clight.deref_loc ty m b ofs bf v
  | Some chunk => bf = Full /\ volatile_load tge chunk m b ofs t v
  end.

Remark assign_loc_translated:
  forall ty m b ofs bf v t m' v',
  Csem.assign_loc ge ty m b ofs bf v t m' v' ->
  match chunk_for_volatile_type ty bf with
  | None => t = E0 /\ Clight.assign_loc tge ty m b ofs bf v m'
  | Some chunk => bf = Full /\ volatile_store tge chunk m b ofs v t m'
  end.


Lemma is_bitfield_access_sound: forall e le m a b ofs bf bf',
  eval_lvalue tge e le m a b ofs bf ->
  tr_is_bitfield_access ce a bf' ->
  bf' = bf.

Lemma make_assign_value_sound:
  forall ty m b ofs bf v t m' v',
  Csem.assign_loc ge ty m b ofs bf v t m' v' ->
  forall tge e le m'' r,
  typeof r = ty ->
  eval_expr tge e le m'' r v ->
  eval_expr tge e le m'' (make_assign_value bf r) v'.

Lemma typeof_make_assign_value: forall bf r,
  typeof (make_assign_value bf r) = typeof r.


Lemma tr_simple:
 forall e m,
 (forall r v,
  eval_simple_rvalue ge e m r v ->
  forall le dst sl a tmps,
  tr_expr ce le dst r sl a tmps ->
  match dst with
  | For_val => sl = nil /\ Csyntax.typeof r = typeof a /\ eval_expr tge e le m a v
  | For_effects => sl = nil
  | For_set sd =>
      exists b, sl = do_set sd b
             /\ Csyntax.typeof r = typeof b
             /\ eval_expr tge e le m b v
  end)
/\
 (forall l b ofs bf,
  eval_simple_lvalue ge e m l b ofs bf ->
  forall le sl a tmps,
  tr_expr ce le For_val l sl a tmps ->
  sl = nil /\ Csyntax.typeof l = typeof a /\ eval_lvalue tge e le m a b ofs bf).

Lemma tr_simple_rvalue:
  forall e m r v,
  eval_simple_rvalue ge e m r v ->
  forall le dst sl a tmps,
  tr_expr ce le dst r sl a tmps ->
  match dst with
  | For_val => sl = nil /\ Csyntax.typeof r = typeof a /\ eval_expr tge e le m a v
  | For_effects => sl = nil
  | For_set sd =>
      exists b, sl = do_set sd b
             /\ Csyntax.typeof r = typeof b
             /\ eval_expr tge e le m b v
  end.

Lemma tr_simple_lvalue:
  forall e m l b ofs bf,
  eval_simple_lvalue ge e m l b ofs bf ->
  forall le sl a tmps,
  tr_expr ce le For_val l sl a tmps ->
  sl = nil /\ Csyntax.typeof l = typeof a /\ eval_lvalue tge e le m a b ofs bf.

Lemma tr_simple_exprlist:
  forall le rl sl al tmps,
  tr_exprlist ce le rl sl al tmps ->
  forall e m tyl vl,
  eval_simple_list ge e m rl tyl vl ->
  sl = nil /\ eval_exprlist tge e le m al tyl vl.


Lemma typeof_context:
  forall k1 k2 C, leftcontext k1 k2 C ->
  forall e1 e2, Csyntax.typeof e1 = Csyntax.typeof e2 ->
  Csyntax.typeof (C e1) = Csyntax.typeof (C e2).

Scheme leftcontext_ind2 := Minimality for leftcontext Sort Prop
  with leftcontextlist_ind2 := Minimality for leftcontextlist Sort Prop.
Combined Scheme leftcontext_leftcontextlist_ind from leftcontext_ind2, leftcontextlist_ind2.

Lemma tr_expr_leftcontext_rec:
 (
  forall from to C, leftcontext from to C ->
  forall le e dst sl a tmps,
  tr_expr ce le dst (C e) sl a tmps ->
  exists dst', exists sl1, exists sl2, exists a', exists tmp',
  tr_expr ce le dst' e sl1 a' tmp'
  /\ sl = sl1 ++ sl2
  /\ incl tmp' tmps
  /\ (forall le' e' sl3,
        tr_expr ce le' dst' e' sl3 a' tmp' ->
        (forall id, ~In id tmp' -> le'!id = le!id) ->
        Csyntax.typeof e' = Csyntax.typeof e ->
        tr_expr ce le' dst (C e') (sl3 ++ sl2) a tmps)
 ) /\ (
  forall from C, leftcontextlist from C ->
  forall le e sl a tmps,
  tr_exprlist ce le (C e) sl a tmps ->
  exists dst', exists sl1, exists sl2, exists a', exists tmp',
  tr_expr ce le dst' e sl1 a' tmp'
  /\ sl = sl1 ++ sl2
  /\ incl tmp' tmps
  /\ (forall le' e' sl3,
        tr_expr ce le' dst' e' sl3 a' tmp' ->
        (forall id, ~In id tmp' -> le'!id = le!id) ->
        Csyntax.typeof e' = Csyntax.typeof e ->
        tr_exprlist ce le' (C e') (sl3 ++ sl2) a tmps)
).

Theorem tr_expr_leftcontext:
  forall C le r dst sl a tmps,
  leftcontext RV RV C ->
  tr_expr ce le dst (C r) sl a tmps ->
  exists dst', exists sl1, exists sl2, exists a', exists tmp',
  tr_expr ce le dst' r sl1 a' tmp'
  /\ sl = sl1 ++ sl2
  /\ incl tmp' tmps
  /\ (forall le' r' sl3,
        tr_expr ce le' dst' r' sl3 a' tmp' ->
        (forall id, ~In id tmp' -> le'!id = le!id) ->
        Csyntax.typeof r' = Csyntax.typeof r ->
        tr_expr ce le' dst (C r') (sl3 ++ sl2) a tmps).

Theorem tr_top_leftcontext:
  forall e le m dst rtop sl a tmps,
  tr_top ce tge e le m dst rtop sl a tmps ->
  forall r C,
  rtop = C r ->
  leftcontext RV RV C ->
  exists dst', exists sl1, exists sl2, exists a', exists tmp',
  tr_top ce tge e le m dst' r sl1 a' tmp'
  /\ sl = sl1 ++ sl2
  /\ incl tmp' tmps
  /\ (forall le' m' r' sl3,
        tr_expr ce le' dst' r' sl3 a' tmp' ->
        (forall id, ~In id tmp' -> le'!id = le!id) ->
        Csyntax.typeof r' = Csyntax.typeof r ->
        tr_top ce tge e le' m' dst (C r') (sl3 ++ sl2) a tmps).


Remark sem_cast_deterministic:
  forall v ty ty' m1 v1 m2 v2,
  sem_cast v ty ty' m1 = Some v1 ->
  sem_cast v ty ty' m2 = Some v2 ->
  v1 = v2.

Lemma eval_simpl_expr_sound:
  forall e le m a v, eval_expr tge e le m a v ->
  match eval_simpl_expr a with Some v' => v' = v | None => True end.

Lemma static_bool_val_sound:
  forall v t m b, bool_val v t Mem.empty = Some b -> bool_val v t m = Some b.

Lemma step_makeif:
  forall f a s1 s2 k e le m v1 b,
  eval_expr tge e le m a v1 ->
  bool_val v1 (typeof a) m = Some b ->
  star step1 tge (State f (makeif a s1 s2) k e le m)
             E0 (State f (if b then s1 else s2) k e le m).

Lemma step_make_set:
  forall id a ty m b ofs bf t v e le f k,
  Csem.deref_loc ge ty m b ofs bf t v ->
  eval_lvalue tge e le m a b ofs bf ->
  typeof a = ty ->
  step1 tge (State f (make_set bf id a) k e le m)
          t (State f Sskip k e (PTree.set id v le) m).

Lemma step_make_assign:
  forall a1 a2 ty m b ofs bf t v m' v' v2 e le f k,
  Csem.assign_loc ge ty m b ofs bf v t m' v' ->
  eval_lvalue tge e le m a1 b ofs bf ->
  eval_expr tge e le m a2 v2 ->
  sem_cast v2 (typeof a2) ty m = Some v ->
  typeof a1 = ty ->
  step1 tge (State f (make_assign bf a1 a2) k e le m)
          t (State f Sskip k e le m').

Fixpoint Kseqlist (sl: list statement) (k: cont) :=
  match sl with
  | nil => k
  | s :: l => Kseq s (Kseqlist l k)
  end.

Remark Kseqlist_app:
  forall sl1 sl2 k,
  Kseqlist (sl1 ++ sl2) k = Kseqlist sl1 (Kseqlist sl2 k).

Lemma push_seq:
  forall f sl k e le m,
  star step1 tge (State f (makeseq sl) k e le m)
              E0 (State f Sskip (Kseqlist sl k) e le m).

Lemma step_tr_rvalof:
  forall ty m b ofs bf t v e le a sl a' tmp f k,
  Csem.deref_loc ge ty m b ofs bf t v ->
  eval_lvalue tge e le m a b ofs bf ->
  tr_rvalof ce ty a sl a' tmp ->
  typeof a = ty ->
  exists le',
    star step1 tge (State f Sskip (Kseqlist sl k) e le m)
                 t (State f Sskip k e le' m)
  /\ eval_expr tge e le' m a' v
  /\ typeof a' = typeof a
  /\ forall x, ~In x tmp -> le'!x = le!x.

End TRANSLATION.



Inductive match_cont : composite_env -> Csem.cont -> cont -> Prop :=
  | match_Kstop: forall ce,
      match_cont ce Csem.Kstop Kstop
  | match_Kseq: forall ce s k ts tk,
      tr_stmt ce s ts ->
      match_cont ce k tk ->
      match_cont ce (Csem.Kseq s k) (Kseq ts tk)
  | match_Kwhile2: forall ce r s k s' ts tk,
      tr_if ce r Sskip Sbreak s' ->
      tr_stmt ce s ts ->
      match_cont ce k tk ->
      match_cont ce (Csem.Kwhile2 r s k)
                    (Kloop1 (Ssequence s' ts) Sskip tk)
  | match_Kdowhile1: forall ce r s k s' ts tk,
      tr_if ce r Sskip Sbreak s' ->
      tr_stmt ce s ts ->
      match_cont ce k tk ->
      match_cont ce (Csem.Kdowhile1 r s k)
                    (Kloop1 ts s' tk)
  | match_Kfor3: forall ce r s3 s k ts3 s' ts tk,
      tr_if ce r Sskip Sbreak s' ->
      tr_stmt ce s3 ts3 ->
      tr_stmt ce s ts ->
      match_cont ce k tk ->
      match_cont ce (Csem.Kfor3 r s3 s k)
                    (Kloop1 (Ssequence s' ts) ts3 tk)
  | match_Kfor4: forall ce r s3 s k ts3 s' ts tk,
      tr_if ce r Sskip Sbreak s' ->
      tr_stmt ce s3 ts3 ->
      tr_stmt ce s ts ->
      match_cont ce k tk ->
      match_cont ce (Csem.Kfor4 r s3 s k)
                    (Kloop2 (Ssequence s' ts) ts3 tk)
  | match_Kswitch2: forall ce k tk,
      match_cont ce k tk ->
      match_cont ce (Csem.Kswitch2 k) (Kswitch tk)
  | match_Kcall: forall f e C ty k optid tf le sl tk a dest tmps cu ce,
      linkorder cu prog ->
      tr_function cu.(prog_comp_env) f tf ->
      leftcontext RV RV C ->
      (forall v m, tr_top cu.(prog_comp_env) tge e (set_opttemp optid v le) m dest (C (Csyntax.Eval v ty)) sl a tmps) ->
      match_cont_exp cu.(prog_comp_env) dest a k tk ->
      match_cont ce (Csem.Kcall f e C ty k)
                    (Kcall optid tf e le (Kseqlist sl tk))

with match_cont_exp : composite_env -> destination -> expr -> Csem.cont -> cont -> Prop :=
  | match_Kdo: forall ce k a tk,
      match_cont ce k tk ->
      match_cont_exp ce For_effects a (Csem.Kdo k) tk
  | match_Kifthenelse_empty: forall ce a k tk,
      match_cont ce k tk ->
      match_cont_exp ce For_val a (Csem.Kifthenelse Csyntax.Sskip Csyntax.Sskip k) (Kseq Sskip tk)
  | match_Kifthenelse_1: forall ce a s1 s2 k ts1 ts2 tk,
      tr_stmt ce s1 ts1 -> tr_stmt ce s2 ts2 ->
      match_cont ce k tk ->
      match_cont_exp ce For_val a (Csem.Kifthenelse s1 s2 k) (Kseq (Sifthenelse a ts1 ts2) tk)
  | match_Kwhile1: forall ce r s k s' a ts tk,
      tr_if ce r Sskip Sbreak s' ->
      tr_stmt ce s ts ->
      match_cont ce k tk ->
      match_cont_exp ce For_val a
         (Csem.Kwhile1 r s k)
         (Kseq (makeif a Sskip Sbreak)
           (Kseq ts (Kloop1 (Ssequence s' ts) Sskip tk)))
  | match_Kdowhile2: forall ce r s k s' a ts tk,
      tr_if ce r Sskip Sbreak s' ->
      tr_stmt ce s ts ->
      match_cont ce k tk ->
      match_cont_exp ce For_val a
        (Csem.Kdowhile2 r s k)
        (Kseq (makeif a Sskip Sbreak) (Kloop2 ts s' tk))
  | match_Kfor2: forall ce r s3 s k s' a ts3 ts tk,
      tr_if ce r Sskip Sbreak s' ->
      tr_stmt ce s3 ts3 ->
      tr_stmt ce s ts ->
      match_cont ce k tk ->
      match_cont_exp ce For_val a
        (Csem.Kfor2 r s3 s k)
        (Kseq (makeif a Sskip Sbreak)
          (Kseq ts (Kloop1 (Ssequence s' ts) ts3 tk)))
  | match_Kswitch1: forall ce ls k a tls tk,
      tr_lblstmts ce ls tls ->
      match_cont ce k tk ->
      match_cont_exp ce For_val a (Csem.Kswitch1 ls k) (Kseq (Sswitch a tls) tk)
  | match_Kreturn: forall ce k a tk,
      match_cont ce k tk ->
      match_cont_exp ce For_val a (Csem.Kreturn k) (Kseq (Sreturn (Some a)) tk).

Lemma match_cont_is_call_cont:
  forall ce k tk,
  match_cont ce k tk -> Csem.is_call_cont k ->
  forall ce', match_cont ce' k tk.

Lemma match_cont_call_cont:
  forall ce k tk,
  match_cont ce k tk ->
  forall ce', match_cont ce' (Csem.call_cont k) (call_cont tk).


Inductive match_states: Csem.state -> state -> Prop :=
  | match_exprstates: forall f r k e m tf sl tk le dest a tmps cu
      (LINK: linkorder cu prog)
      (TRF: tr_function cu.(prog_comp_env) f tf)
      (TR: tr_top cu.(prog_comp_env) tge e le m dest r sl a tmps)
      (MK: match_cont_exp cu.(prog_comp_env) dest a k tk),
      match_states (Csem.ExprState f r k e m)
                   (State tf Sskip (Kseqlist sl tk) e le m)
  | match_regularstates: forall f s k e m tf ts tk le cu
      (LINK: linkorder cu prog)
      (TRF: tr_function cu.(prog_comp_env) f tf)
      (TR: tr_stmt cu.(prog_comp_env) s ts)
      (MK: match_cont cu.(prog_comp_env) k tk),
      match_states (Csem.State f s k e m)
                   (State tf ts tk e le m)
  | match_callstates: forall fd args k m tfd tk cu
      (LINK: linkorder cu prog)
      (TR: tr_fundef cu fd tfd)
      (MK: forall ce, match_cont ce k tk),
      match_states (Csem.Callstate fd args k m)
                   (Callstate tfd args tk m)
  | match_returnstates: forall res k m tk
      (MK: forall ce, match_cont ce k tk),
      match_states (Csem.Returnstate res k m)
                   (Returnstate res tk m)
  | match_stuckstate: forall S,
      match_states Csem.Stuckstate S.


Lemma tr_select_switch:
  forall ce n ls tls,
  tr_lblstmts ce ls tls ->
  tr_lblstmts ce (Csem.select_switch n ls) (select_switch n tls).

Lemma tr_seq_of_labeled_statement:
  forall ce ls tls,
  tr_lblstmts ce ls tls ->
  tr_stmt ce (Csem.seq_of_labeled_statement ls) (seq_of_labeled_statement tls).


Section FIND_LABEL.

Variable ce: composite_env.
Variable lbl: label.

Definition nolabel (s: statement) : Prop :=
  forall k, find_label lbl s k = None.

Fixpoint nolabel_list (sl: list statement) : Prop :=
  match sl with
  | nil => True
  | s1 :: sl' => nolabel s1 /\ nolabel_list sl'
  end.

Lemma nolabel_list_app:
  forall sl2 sl1, nolabel_list sl1 -> nolabel_list sl2 -> nolabel_list (sl1 ++ sl2).

Lemma makeseq_nolabel:
  forall sl, nolabel_list sl -> nolabel (makeseq sl).

Lemma makeif_nolabel:
  forall a s1 s2, nolabel s1 -> nolabel s2 -> nolabel (makeif a s1 s2).

Lemma make_set_nolabel:
  forall bf t a, nolabel (make_set bf t a).

Lemma make_assign_nolabel:
  forall bf l r, nolabel (make_assign bf l r).

Lemma tr_rvalof_nolabel:
  forall ce ty a sl a' tmp, tr_rvalof ce ty a sl a' tmp -> nolabel_list sl.

Lemma nolabel_do_set:
  forall sd a, nolabel_list (do_set sd a).

Lemma nolabel_final:
  forall dst a, nolabel_list (final dst a).

Ltac NoLabelTac :=
  match goal with
  | [ |- nolabel_list nil ] => exact I
  | [ |- nolabel_list (final _ _) ] => apply nolabel_final
  | [ |- nolabel_list (_ :: _) ] => simpl; split; NoLabelTac
  | [ |- nolabel_list (_ ++ _) ] => apply nolabel_list_app; NoLabelTac
  | [ H: _ -> nolabel_list ?x |- nolabel_list ?x ] => apply H; NoLabelTac
  | [ |- nolabel (makeseq _) ] => apply makeseq_nolabel; NoLabelTac
  | [ |- nolabel (makeif _ _ _) ] => apply makeif_nolabel; NoLabelTac
  | [ |- nolabel (make_set _ _ _) ] => apply make_set_nolabel
  | [ |- nolabel (make_assign _ _ _) ] => apply make_assign_nolabel
  | [ |- nolabel _ ] => red; intros; simpl; auto
  | [ |- _ /\ _ ] => split; NoLabelTac
  | _ => auto
  end.

Lemma tr_find_label_expr:
  (forall le dst r sl a tmps, tr_expr ce le dst r sl a tmps -> nolabel_list sl)
/\(forall le rl sl al tmps, tr_exprlist ce le rl sl al tmps -> nolabel_list sl).

Lemma tr_find_label_top:
  forall e le m dst r sl a tmps,
  tr_top ce tge e le m dst r sl a tmps -> nolabel_list sl.

Lemma tr_find_label_expression:
  forall r s a, tr_expression ce r s a -> forall k, find_label lbl s k = None.

Lemma tr_find_label_expr_stmt:
  forall r s, tr_expr_stmt ce r s -> forall k, find_label lbl s k = None.

Lemma tr_find_label_if:
  forall r s,
  tr_if ce r Sskip Sbreak s ->
  forall k, find_label lbl s k = None.

Lemma tr_find_label:
  forall s k ts tk
    (TR: tr_stmt ce s ts)
    (MC: match_cont ce k tk),
  match Csem.find_label lbl s k with
  | None =>
      find_label lbl ts tk = None
  | Some (s', k') =>
      exists ts', exists tk',
          find_label lbl ts tk = Some (ts', tk')
       /\ tr_stmt ce s' ts'
       /\ match_cont ce k' tk'
  end
with tr_find_label_ls:
  forall s k ts tk
    (TR: tr_lblstmts ce s ts)
    (MC: match_cont ce k tk),
  match Csem.find_label_ls lbl s k with
  | None =>
      find_label_ls lbl ts tk = None
  | Some (s', k') =>
      exists ts', exists tk',
          find_label_ls lbl ts tk = Some (ts', tk')
       /\ tr_stmt ce s' ts'
       /\ match_cont ce k' tk'
  end.

End FIND_LABEL.



Fixpoint esize (a: Csyntax.expr) : nat :=
  match a with
  | Csyntax.Eloc _ _ _ _ => 1%nat
  | Csyntax.Evar _ _ => 1%nat
  | Csyntax.Ederef r1 _ => S(esize r1)
  | Csyntax.Efield l1 _ _ => S(esize l1)
  | Csyntax.Eval _ _ => O
  | Csyntax.Evalof l1 _ => S(esize l1)
  | Csyntax.Eaddrof l1 _ => S(esize l1)
  | Csyntax.Eunop _ r1 _ => S(esize r1)
  | Csyntax.Ebinop _ r1 r2 _ => S(esize r1 + esize r2)%nat
  | Csyntax.Ecast r1 _ => S(esize r1)
  | Csyntax.Eseqand r1 _ _ => S(esize r1)
  | Csyntax.Eseqor r1 _ _ => S(esize r1)
  | Csyntax.Econdition r1 _ _ _ => S(esize r1)
  | Csyntax.Esizeof _ _ => 1%nat
  | Csyntax.Ealignof _ _ => 1%nat
  | Csyntax.Eassign l1 r2 _ => S(esize l1 + esize r2)%nat
  | Csyntax.Eassignop _ l1 r2 _ _ => S(esize l1 + esize r2)%nat
  | Csyntax.Epostincr _ l1 _ => S(esize l1)
  | Csyntax.Ecomma r1 r2 _ => S(esize r1 + esize r2)%nat
  | Csyntax.Ecall r1 rl2 _ => S(esize r1 + esizelist rl2)%nat
  | Csyntax.Ebuiltin ef _ rl _ => S(esizelist rl)%nat
  | Csyntax.Eparen r1 _ _ => S(esize r1)
  end

with esizelist (el: Csyntax.exprlist) : nat :=
  match el with
  | Csyntax.Enil => O
  | Csyntax.Econs r1 rl2 => (esize r1 + esizelist rl2)%nat
  end.

Definition measure (st: Csem.state) : nat :=
  match st with
  | Csem.ExprState _ r _ _ _ => (esize r + 1)%nat
  | Csem.State _ Csyntax.Sskip _ _ _ => 0%nat
  | Csem.State _ (Csyntax.Sdo r) _ _ _ => (esize r + 2)%nat
  | Csem.State _ (Csyntax.Sifthenelse r _ _) _ _ _ => (esize r + 2)%nat
  | _ => 0%nat
  end.

Lemma leftcontext_size:
  forall from to C,
  leftcontext from to C ->
  forall e1 e2,
  (esize e1 < esize e2)%nat ->
  (esize (C e1) < esize (C e2))%nat
with leftcontextlist_size:
  forall from C,
  leftcontextlist from C ->
  forall e1 e2,
  (esize e1 < esize e2)%nat ->
  (esizelist (C e1) < esizelist (C e2))%nat.


Lemma tr_val_gen:
  forall ce le dst v ty a tmp,
  typeof a = ty ->
  (forall tge e le' m,
      (forall id, In id tmp -> le'!id = le!id) ->
      eval_expr tge e le' m a v) ->
  tr_expr ce le dst (Csyntax.Eval v ty) (final dst a) a tmp.

Lemma estep_simulation:
  forall S1 t S2, Cstrategy.estep ge S1 t S2 ->
  forall S1' (MS: match_states S1 S1'),
  exists S2',
     (plus step1 tge S1' t S2' \/
       (star step1 tge S1' t S2' /\ measure S2 < measure S1)%nat)
  /\ match_states S2 S2'.


Lemma tr_top_val_for_val_inv:
  forall ce e le m v ty sl a tmps,
  tr_top ce tge e le m For_val (Csyntax.Eval v ty) sl a tmps ->
  sl = nil /\ typeof a = ty /\ eval_expr tge e le m a v.

Lemma alloc_variables_preserved:
  forall e m params e' m',
  Csem.alloc_variables ge e m params e' m' ->
  alloc_variables tge e m params e' m'.

Lemma bind_parameters_preserved:
  forall e m params args m',
  Csem.bind_parameters ge e m params args m' ->
  bind_parameters tge e m params args m'.

Lemma blocks_of_env_preserved:
  forall e, blocks_of_env tge e = Csem.blocks_of_env ge e.

Lemma sstep_simulation:
  forall S1 t S2, Csem.sstep ge S1 t S2 ->
  forall S1' (MS: match_states S1 S1'),
  exists S2',
     (plus step1 tge S1' t S2' \/
       (star step1 tge S1' t S2' /\ measure S2 < measure S1)%nat)
  /\ match_states S2 S2'.


Theorem simulation:
  forall S1 t S2, Cstrategy.step ge S1 t S2 ->
  forall S1' (MS: match_states S1 S1'),
  exists S2',
     (plus step1 tge S1' t S2' \/
       (star step1 tge S1' t S2' /\ measure S2 < measure S1)%nat)
  /\ match_states S2 S2'.

Lemma transl_initial_states:
  forall S,
  Csem.initial_state prog S ->
  exists S', Clight.initial_state tprog S' /\ match_states S S'.

Lemma transl_final_states:
  forall S S' r,
  match_states S S' -> Csem.final_state S r -> Clight.final_state S' r.

Theorem transl_program_correct:
  forward_simulation (Cstrategy.semantics prog) (Clight.semantics1 tprog).

End PRESERVATION.

Commutation with linking

Global Instance TransfSimplExprLink : TransfLink match_prog.