Module SimplExprspec


Relational specification of expression simplification.

Require Import Coqlib Maps Errors Integers Floats.
Require Import AST Linking Memory.
Require Import Ctypes Cop Csyntax Clight SimplExpr.

Section SPEC.

Variable ce: composite_env.

Local Open Scope gensym_monad_scope.

Relational specification of the translation.


Translation of expressions


This specification covers:

Definition final (dst: destination) (a: expr) : list statement :=
  match dst with
  | For_val => nil
  | For_effects => nil
  | For_set sd => do_set sd a
  end.

Definition tr_is_bitfield_access (l: expr) (bf: bitfield) : Prop :=
  match l with
  | Efield r f _ =>
      exists co ofs,
      match typeof r with
      | Tstruct id _ =>
          ce!id = Some co /\ field_offset ce f (co_members co) = OK (ofs, bf)
      | Tunion id _ =>
          ce!id = Some co /\ union_field_offset ce f (co_members co) = OK (ofs, bf)
      | _ => False
      end
  | _ => bf = Full
  end.

Inductive tr_rvalof: type -> expr -> list statement -> expr -> list ident -> Prop :=
  | tr_rvalof_nonvol: forall ty a tmp,
      type_is_volatile ty = false ->
      tr_rvalof ty a nil a tmp
  | tr_rvalof_vol: forall ty a t bf tmp,
      type_is_volatile ty = true -> In t tmp ->
      tr_is_bitfield_access a bf ->
      tr_rvalof ty a (make_set bf t a :: nil) (Etempvar t ty) tmp.

Inductive tr_expr: temp_env -> destination -> Csyntax.expr -> list statement -> expr -> list ident -> Prop :=
  | tr_var: forall le dst id ty tmp,
      tr_expr le dst (Csyntax.Evar id ty)
              (final dst (Evar id ty)) (Evar id ty) tmp
  | tr_deref: forall le dst e1 ty sl1 a1 tmp,
      tr_expr le For_val e1 sl1 a1 tmp ->
      tr_expr le dst (Csyntax.Ederef e1 ty)
              (sl1 ++ final dst (Ederef' a1 ty)) (Ederef' a1 ty) tmp
  | tr_field: forall le dst e1 f ty sl1 a1 tmp,
      tr_expr le For_val e1 sl1 a1 tmp ->
      tr_expr le dst (Csyntax.Efield e1 f ty)
              (sl1 ++ final dst (Efield a1 f ty)) (Efield a1 f ty) tmp
  | tr_val_effect: forall le v ty any tmp,
      tr_expr le For_effects (Csyntax.Eval v ty) nil any tmp
  | tr_val_value: forall le 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 le For_val (Csyntax.Eval v ty)
                           nil a tmp
  | tr_val_set: forall le sd v ty a any 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 le (For_set sd) (Csyntax.Eval v ty)
                   (do_set sd a) any tmp
  | tr_sizeof: forall le dst ty' ty tmp,
      tr_expr le dst (Csyntax.Esizeof ty' ty)
                   (final dst (Esizeof ty' ty))
                   (Esizeof ty' ty) tmp
  | tr_alignof: forall le dst ty' ty tmp,
      tr_expr le dst (Csyntax.Ealignof ty' ty)
                   (final dst (Ealignof ty' ty))
                   (Ealignof ty' ty) tmp
  | tr_valof: forall le dst e1 ty tmp sl1 a1 tmp1 sl2 a2 tmp2,
      tr_expr le For_val e1 sl1 a1 tmp1 ->
      tr_rvalof (Csyntax.typeof e1) a1 sl2 a2 tmp2 ->
      list_disjoint tmp1 tmp2 -> incl tmp1 tmp -> incl tmp2 tmp ->
      tr_expr le dst (Csyntax.Evalof e1 ty)
                    (sl1 ++ sl2 ++ final dst a2)
                    a2 tmp
  | tr_addrof: forall le dst e1 ty tmp sl1 a1,
      tr_expr le For_val e1 sl1 a1 tmp ->
      tr_expr le dst (Csyntax.Eaddrof e1 ty)
                   (sl1 ++ final dst (Eaddrof' a1 ty))
                   (Eaddrof' a1 ty) tmp
  | tr_unop: forall le dst op e1 ty tmp sl1 a1,
      tr_expr le For_val e1 sl1 a1 tmp ->
      tr_expr le dst (Csyntax.Eunop op e1 ty)
                   (sl1 ++ final dst (Eunop op a1 ty))
                   (Eunop op a1 ty) tmp
  | tr_binop: forall le dst op e1 e2 ty sl1 a1 tmp1 sl2 a2 tmp2 tmp,
      tr_expr le For_val e1 sl1 a1 tmp1 ->
      tr_expr le For_val e2 sl2 a2 tmp2 ->
      list_disjoint tmp1 tmp2 -> incl tmp1 tmp -> incl tmp2 tmp ->
      tr_expr le dst (Csyntax.Ebinop op e1 e2 ty)
                   (sl1 ++ sl2 ++ final dst (Ebinop op a1 a2 ty))
                   (Ebinop op a1 a2 ty) tmp
  | tr_cast_effects: forall le e1 ty sl1 a1 any tmp,
      tr_expr le For_effects e1 sl1 a1 tmp ->
      tr_expr le For_effects (Csyntax.Ecast e1 ty)
                   sl1
                   any tmp
  | tr_cast_val: forall le dst e1 ty sl1 a1 tmp,
      tr_expr le For_val e1 sl1 a1 tmp ->
      tr_expr le dst (Csyntax.Ecast e1 ty)
                   (sl1 ++ final dst (Ecast a1 ty))
                   (Ecast a1 ty) tmp
  | tr_seqand_val: forall le e1 e2 ty sl1 a1 tmp1 t sl2 a2 tmp2 tmp,
      tr_expr le For_val e1 sl1 a1 tmp1 ->
      tr_expr le (For_set (sd_seqbool_val t ty)) e2 sl2 a2 tmp2 ->
      list_disjoint tmp1 tmp2 ->
      incl tmp1 tmp -> incl tmp2 tmp -> In t tmp ->
      tr_expr le For_val (Csyntax.Eseqand e1 e2 ty)
                    (sl1 ++ makeif a1 (makeseq sl2)
                                      (Sset t (Econst_int Int.zero ty)) :: nil)
                    (Etempvar t ty) tmp
  | tr_seqand_effects: forall le e1 e2 ty sl1 a1 tmp1 sl2 a2 tmp2 any tmp,
      tr_expr le For_val e1 sl1 a1 tmp1 ->
      tr_expr le For_effects e2 sl2 a2 tmp2 ->
      list_disjoint tmp1 tmp2 ->
      incl tmp1 tmp -> incl tmp2 tmp ->
      tr_expr le For_effects (Csyntax.Eseqand e1 e2 ty)
                    (sl1 ++ makeif a1 (makeseq sl2) Sskip :: nil)
                    any tmp
  | tr_seqand_set: forall le sd e1 e2 ty sl1 a1 tmp1 sl2 a2 tmp2 any tmp,
      tr_expr le For_val e1 sl1 a1 tmp1 ->
      tr_expr le (For_set (sd_seqbool_set ty sd)) e2 sl2 a2 tmp2 ->
      list_disjoint tmp1 tmp2 ->
      incl tmp1 tmp -> incl tmp2 tmp -> In (sd_temp sd) tmp ->
      tr_expr le (For_set sd) (Csyntax.Eseqand e1 e2 ty)
                    (sl1 ++ makeif a1 (makeseq sl2)
                                      (makeseq (do_set sd (Econst_int Int.zero ty))) :: nil)
                    any tmp
  | tr_seqor_val: forall le e1 e2 ty sl1 a1 tmp1 t sl2 a2 tmp2 tmp,
      tr_expr le For_val e1 sl1 a1 tmp1 ->
      tr_expr le (For_set (sd_seqbool_val t ty)) e2 sl2 a2 tmp2 ->
      list_disjoint tmp1 tmp2 ->
      incl tmp1 tmp -> incl tmp2 tmp -> In t tmp ->
      tr_expr le For_val (Csyntax.Eseqor e1 e2 ty)
                    (sl1 ++ makeif a1 (Sset t (Econst_int Int.one ty))
                                      (makeseq sl2) :: nil)
                    (Etempvar t ty) tmp
  | tr_seqor_effects: forall le e1 e2 ty sl1 a1 tmp1 sl2 a2 tmp2 any tmp,
      tr_expr le For_val e1 sl1 a1 tmp1 ->
      tr_expr le For_effects e2 sl2 a2 tmp2 ->
      list_disjoint tmp1 tmp2 ->
      incl tmp1 tmp -> incl tmp2 tmp ->
      tr_expr le For_effects (Csyntax.Eseqor e1 e2 ty)
                    (sl1 ++ makeif a1 Sskip (makeseq sl2) :: nil)
                    any tmp
  | tr_seqor_set: forall le sd e1 e2 ty sl1 a1 tmp1 sl2 a2 tmp2 any tmp,
      tr_expr le For_val e1 sl1 a1 tmp1 ->
      tr_expr le (For_set (sd_seqbool_set ty sd)) e2 sl2 a2 tmp2 ->
      list_disjoint tmp1 tmp2 ->
      incl tmp1 tmp -> incl tmp2 tmp -> In (sd_temp sd) tmp ->
      tr_expr le (For_set sd) (Csyntax.Eseqor e1 e2 ty)
                    (sl1 ++ makeif a1 (makeseq (do_set sd (Econst_int Int.one ty)))
                                      (makeseq sl2) :: nil)
                    any tmp
  | tr_condition_val: forall le e1 e2 e3 ty sl1 a1 tmp1 sl2 a2 tmp2 sl3 a3 tmp3 t tmp,
      tr_expr le For_val e1 sl1 a1 tmp1 ->
      tr_expr le (For_set (SDbase ty ty t)) e2 sl2 a2 tmp2 ->
      tr_expr le (For_set (SDbase ty ty t)) e3 sl3 a3 tmp3 ->
      list_disjoint tmp1 tmp2 ->
      list_disjoint tmp1 tmp3 ->
      incl tmp1 tmp -> incl tmp2 tmp -> incl tmp3 tmp -> In t tmp ->
      tr_expr le For_val (Csyntax.Econdition e1 e2 e3 ty)
                      (sl1 ++ makeif a1 (makeseq sl2) (makeseq sl3) :: nil)
                      (Etempvar t ty) tmp
  | tr_condition_effects: forall le e1 e2 e3 ty sl1 a1 tmp1 sl2 a2 tmp2 sl3 a3 tmp3 any tmp,
      tr_expr le For_val e1 sl1 a1 tmp1 ->
      tr_expr le For_effects e2 sl2 a2 tmp2 ->
      tr_expr le For_effects e3 sl3 a3 tmp3 ->
      list_disjoint tmp1 tmp2 ->
      list_disjoint tmp1 tmp3 ->
      incl tmp1 tmp -> incl tmp2 tmp -> incl tmp3 tmp ->
      tr_expr le For_effects (Csyntax.Econdition e1 e2 e3 ty)
                       (sl1 ++ makeif a1 (makeseq sl2) (makeseq sl3) :: nil)
                       any tmp
  | tr_condition_set: forall le sd t e1 e2 e3 ty sl1 a1 tmp1 sl2 a2 tmp2 sl3 a3 tmp3 any tmp,
      tr_expr le For_val e1 sl1 a1 tmp1 ->
      tr_expr le (For_set (SDcons ty ty t sd)) e2 sl2 a2 tmp2 ->
      tr_expr le (For_set (SDcons ty ty t sd)) e3 sl3 a3 tmp3 ->
      list_disjoint tmp1 tmp2 ->
      list_disjoint tmp1 tmp3 ->
      incl tmp1 tmp -> incl tmp2 tmp -> incl tmp3 tmp -> In t tmp ->
      tr_expr le (For_set sd) (Csyntax.Econdition e1 e2 e3 ty)
                       (sl1 ++ makeif a1 (makeseq sl2) (makeseq sl3) :: nil)
                       any tmp
  | tr_assign_effects: forall le e1 e2 ty sl1 a1 tmp1 sl2 a2 tmp2 bf any tmp,
      tr_expr le For_val e1 sl1 a1 tmp1 ->
      tr_expr le For_val e2 sl2 a2 tmp2 ->
      list_disjoint tmp1 tmp2 ->
      incl tmp1 tmp -> incl tmp2 tmp ->
      tr_is_bitfield_access a1 bf ->
      tr_expr le For_effects (Csyntax.Eassign e1 e2 ty)
                      (sl1 ++ sl2 ++ make_assign bf a1 a2 :: nil)
                      any tmp
  | tr_assign_val: forall le dst e1 e2 ty sl1 a1 tmp1 sl2 a2 tmp2 t tmp ty1 ty2 bf,
      tr_expr le For_val e1 sl1 a1 tmp1 ->
      tr_expr le For_val e2 sl2 a2 tmp2 ->
      incl tmp1 tmp -> incl tmp2 tmp ->
      list_disjoint tmp1 tmp2 ->
      In t tmp -> ~In t tmp1 -> ~In t tmp2 ->
      ty1 = Csyntax.typeof e1 ->
      ty2 = Csyntax.typeof e2 ->
      tr_is_bitfield_access a1 bf ->
      tr_expr le dst (Csyntax.Eassign e1 e2 ty)
                   (sl1 ++ sl2 ++
                    Sset t (Ecast a2 ty1) ::
                    make_assign bf a1 (Etempvar t ty1) ::
                    final dst (make_assign_value bf (Etempvar t ty1)))
                   (make_assign_value bf (Etempvar t ty1)) tmp
  | tr_assignop_effects: forall le op e1 e2 tyres ty ty1 sl1 a1 tmp1 sl2 a2 tmp2 bf sl3 a3 tmp3 any tmp,
      tr_expr le For_val e1 sl1 a1 tmp1 ->
      tr_expr le For_val e2 sl2 a2 tmp2 ->
      ty1 = Csyntax.typeof e1 ->
      tr_rvalof ty1 a1 sl3 a3 tmp3 ->
      list_disjoint tmp1 tmp2 -> list_disjoint tmp1 tmp3 -> list_disjoint tmp2 tmp3 ->
      incl tmp1 tmp -> incl tmp2 tmp -> incl tmp3 tmp ->
      tr_is_bitfield_access a1 bf ->
      tr_expr le For_effects (Csyntax.Eassignop op e1 e2 tyres ty)
                      (sl1 ++ sl2 ++ sl3 ++ make_assign bf a1 (Ebinop op a3 a2 tyres) :: nil)
                      any tmp
  | tr_assignop_val: forall le dst op e1 e2 tyres ty sl1 a1 tmp1 sl2 a2 tmp2 sl3 a3 tmp3 t bf tmp ty1,
      tr_expr le For_val e1 sl1 a1 tmp1 ->
      tr_expr le For_val e2 sl2 a2 tmp2 ->
      tr_rvalof ty1 a1 sl3 a3 tmp3 ->
      list_disjoint tmp1 tmp2 -> list_disjoint tmp1 tmp3 -> list_disjoint tmp2 tmp3 ->
      incl tmp1 tmp -> incl tmp2 tmp -> incl tmp3 tmp ->
      In t tmp -> ~In t tmp1 -> ~In t tmp2 -> ~In t tmp3 ->
      ty1 = Csyntax.typeof e1 ->
      tr_is_bitfield_access a1 bf ->
      tr_expr le dst (Csyntax.Eassignop op e1 e2 tyres ty)
                   (sl1 ++ sl2 ++ sl3 ++
                    Sset t (Ecast (Ebinop op a3 a2 tyres) ty1) ::
                    make_assign bf a1 (Etempvar t ty1) ::
                    final dst (make_assign_value bf (Etempvar t ty1)))
                   (make_assign_value bf (Etempvar t ty1)) tmp
  | tr_postincr_effects: forall le id e1 ty ty1 sl1 a1 tmp1 sl2 a2 tmp2 bf any tmp,
      tr_expr le For_val e1 sl1 a1 tmp1 ->
      tr_rvalof ty1 a1 sl2 a2 tmp2 ->
      ty1 = Csyntax.typeof e1 ->
      incl tmp1 tmp -> incl tmp2 tmp ->
      list_disjoint tmp1 tmp2 ->
      tr_is_bitfield_access a1 bf ->
      tr_expr le For_effects (Csyntax.Epostincr id e1 ty)
                      (sl1 ++ sl2 ++ make_assign bf a1 (transl_incrdecr id a2 ty1) :: nil)
                      any tmp
  | tr_postincr_val: forall le dst id e1 ty sl1 a1 tmp1 bf t ty1 tmp,
      tr_expr le For_val e1 sl1 a1 tmp1 ->
      incl tmp1 tmp -> In t tmp -> ~In t tmp1 ->
      ty1 = Csyntax.typeof e1 ->
      tr_is_bitfield_access a1 bf ->
      tr_expr le dst (Csyntax.Epostincr id e1 ty)
                   (sl1 ++ make_set bf t a1 ::
                    make_assign bf a1 (transl_incrdecr id (Etempvar t ty1) ty1) ::
                    final dst (Etempvar t ty1))
                   (Etempvar t ty1) tmp
  | tr_comma: forall le dst e1 e2 ty sl1 a1 tmp1 sl2 a2 tmp2 tmp,
      tr_expr le For_effects e1 sl1 a1 tmp1 ->
      tr_expr le dst e2 sl2 a2 tmp2 ->
      list_disjoint tmp1 tmp2 ->
      incl tmp1 tmp -> incl tmp2 tmp ->
      tr_expr le dst (Csyntax.Ecomma e1 e2 ty) (sl1 ++ sl2) a2 tmp
  | tr_call_effects: forall le e1 el2 ty sl1 a1 tmp1 sl2 al2 tmp2 any tmp,
      tr_expr le For_val e1 sl1 a1 tmp1 ->
      tr_exprlist le el2 sl2 al2 tmp2 ->
      list_disjoint tmp1 tmp2 ->
      incl tmp1 tmp -> incl tmp2 tmp ->
      tr_expr le For_effects (Csyntax.Ecall e1 el2 ty)
                   (sl1 ++ sl2 ++ Scall None a1 al2 :: nil)
                   any tmp
  | tr_call_val: forall le dst e1 el2 ty sl1 a1 tmp1 sl2 al2 tmp2 t tmp,
      dst <> For_effects ->
      tr_expr le For_val e1 sl1 a1 tmp1 ->
      tr_exprlist le el2 sl2 al2 tmp2 ->
      list_disjoint tmp1 tmp2 -> In t tmp ->
      incl tmp1 tmp -> incl tmp2 tmp ->
      tr_expr le dst (Csyntax.Ecall e1 el2 ty)
                   (sl1 ++ sl2 ++ Scall (Some t) a1 al2 :: final dst (Etempvar t ty))
                   (Etempvar t ty) tmp
  | tr_builtin_effects: forall le ef tyargs el ty sl al tmp1 any tmp,
      tr_exprlist le el sl al tmp1 ->
      incl tmp1 tmp ->
      tr_expr le For_effects (Csyntax.Ebuiltin ef tyargs el ty)
                   (sl ++ Sbuiltin None ef tyargs al :: nil)
                   any tmp
  | tr_builtin_val: forall le dst ef tyargs el ty sl al tmp1 t tmp,
      dst <> For_effects ->
      tr_exprlist le el sl al tmp1 ->
      In t tmp -> incl tmp1 tmp ->
      tr_expr le dst (Csyntax.Ebuiltin ef tyargs el ty)
                   (sl ++ Sbuiltin (Some t) ef tyargs al :: final dst (Etempvar t ty))
                   (Etempvar t ty) tmp
  | tr_paren_val: forall le e1 tycast ty sl1 a1 t tmp,
      tr_expr le (For_set (SDbase tycast ty t)) e1 sl1 a1 tmp ->
      In t tmp ->
      tr_expr le For_val (Csyntax.Eparen e1 tycast ty)
                       sl1
                       (Etempvar t ty) tmp
  | tr_paren_effects: forall le e1 tycast ty sl1 a1 tmp any,
      tr_expr le For_effects e1 sl1 a1 tmp ->
      tr_expr le For_effects (Csyntax.Eparen e1 tycast ty) sl1 any tmp
  | tr_paren_set: forall le t sd e1 tycast ty sl1 a1 tmp any,
      tr_expr le (For_set (SDcons tycast ty t sd)) e1 sl1 a1 tmp ->
      In t tmp ->
      tr_expr le (For_set sd) (Csyntax.Eparen e1 tycast ty) sl1 any tmp

with tr_exprlist: temp_env -> Csyntax.exprlist -> list statement -> list expr -> list ident -> Prop :=
  | tr_nil: forall le tmp,
      tr_exprlist le Csyntax.Enil nil nil tmp
  | tr_cons: forall le e1 el2 sl1 a1 tmp1 sl2 al2 tmp2 tmp,
      tr_expr le For_val e1 sl1 a1 tmp1 ->
      tr_exprlist le el2 sl2 al2 tmp2 ->
      list_disjoint tmp1 tmp2 ->
      incl tmp1 tmp -> incl tmp2 tmp ->
      tr_exprlist le (Csyntax.Econs e1 el2) (sl1 ++ sl2) (a1 :: al2) tmp.

Scheme tr_expr_ind2 := Minimality for tr_expr Sort Prop
  with tr_exprlist_ind2 := Minimality for tr_exprlist Sort Prop.
Combined Scheme tr_expr_exprlist from tr_expr_ind2, tr_exprlist_ind2.

Useful invariance properties.

Lemma tr_expr_invariant:
  forall le dst r sl a tmps, tr_expr le dst r sl a tmps ->
  forall le', (forall x, In x tmps -> le'!x = le!x) ->
  tr_expr le' dst r sl a tmps
with tr_exprlist_invariant:
  forall le rl sl al tmps, tr_exprlist le rl sl al tmps ->
  forall le', (forall x, In x tmps -> le'!x = le!x) ->
  tr_exprlist le' rl sl al tmps.
Proof.
  induction 1; intros; econstructor; eauto.
    intros. apply H0. intros. transitivity (le'!id); auto.
    intros. apply H0. auto. intros. transitivity (le'!id); auto.
  induction 1; intros; econstructor; eauto.
Qed.

Lemma tr_rvalof_monotone:
  forall ty a sl b tmps, tr_rvalof ty a sl b tmps ->
  forall tmps', incl tmps tmps' -> tr_rvalof ty a sl b tmps'.
Proof.
  induction 1; intros; econstructor; unfold incl in *; eauto.
Qed.

Lemma tr_expr_monotone:
  forall le dst r sl a tmps, tr_expr le dst r sl a tmps ->
  forall tmps', incl tmps tmps' -> tr_expr le dst r sl a tmps'
with tr_exprlist_monotone:
  forall le rl sl al tmps, tr_exprlist le rl sl al tmps ->
  forall tmps', incl tmps tmps' -> tr_exprlist le rl sl al tmps'.
Proof.
  specialize tr_rvalof_monotone. intros RVALOF.
  induction 1; intros; econstructor; unfold incl in *; eauto.
  induction 1; intros; econstructor; unfold incl in *; eauto.
Qed.

Top-level translation


The "top-level" translation is equivalent to tr_expr above for source terms. It brings additional flexibility in the matching between Csyntax values and Cminor expressions: in the case of tr_expr, the Cminor expression must not depend on memory, while in the case of tr_top it can depend on the current memory state.

Section TR_TOP.

Variable ge: genv.
Variable e: env.
Variable le: temp_env.
Variable m: mem.

Inductive tr_top: destination -> Csyntax.expr -> list statement -> expr -> list ident -> Prop :=
  | tr_top_val_val: forall v ty a tmp,
      typeof a = ty -> eval_expr ge e le m a v ->
      tr_top For_val (Csyntax.Eval v ty) nil a tmp
  | tr_top_base: forall dst r sl a tmp,
      tr_expr le dst r sl a tmp ->
      tr_top dst r sl a tmp.

End TR_TOP.

Translation of statements


Inductive tr_expression: Csyntax.expr -> statement -> expr -> Prop :=
  | tr_expression_intro: forall r sl a tmps,
      (forall ge e le m, tr_top ge e le m For_val r sl a tmps) ->
      tr_expression r (makeseq sl) a.

Inductive tr_expr_stmt: Csyntax.expr -> statement -> Prop :=
  | tr_expr_stmt_intro: forall r sl a tmps,
      (forall ge e le m, tr_top ge e le m For_effects r sl a tmps) ->
      tr_expr_stmt r (makeseq sl).

Inductive tr_if: Csyntax.expr -> statement -> statement -> statement -> Prop :=
  | tr_if_intro: forall r s1 s2 sl a tmps,
      (forall ge e le m, tr_top ge e le m For_val r sl a tmps) ->
      tr_if r s1 s2 (makeseq (sl ++ makeif a s1 s2 :: nil)).

Inductive tr_stmt: Csyntax.statement -> statement -> Prop :=
  | tr_skip:
      tr_stmt Csyntax.Sskip Sskip
  | tr_do: forall r s,
      tr_expr_stmt r s ->
      tr_stmt (Csyntax.Sdo r) s
  | tr_seq: forall s1 s2 ts1 ts2,
      tr_stmt s1 ts1 -> tr_stmt s2 ts2 ->
      tr_stmt (Csyntax.Ssequence s1 s2) (Ssequence ts1 ts2)
  | tr_ifthenelse_empty: forall r s' a,
      tr_expression r s' a ->
      tr_stmt (Csyntax.Sifthenelse r Csyntax.Sskip Csyntax.Sskip) (Ssequence s' Sskip)
  | tr_ifthenelse: forall r s1 s2 s' a ts1 ts2,
      tr_expression r s' a ->
      tr_stmt s1 ts1 -> tr_stmt s2 ts2 ->
      tr_stmt (Csyntax.Sifthenelse r s1 s2) (Ssequence s' (Sifthenelse a ts1 ts2))
  | tr_while: forall r s1 s' ts1,
      tr_if r Sskip Sbreak s' ->
      tr_stmt s1 ts1 ->
      tr_stmt (Csyntax.Swhile r s1)
              (Sloop (Ssequence s' ts1) Sskip)
  | tr_dowhile: forall r s1 s' ts1,
      tr_if r Sskip Sbreak s' ->
      tr_stmt s1 ts1 ->
      tr_stmt (Csyntax.Sdowhile r s1)
              (Sloop ts1 s')
  | tr_for_1: forall r s3 s4 s' ts3 ts4,
      tr_if r Sskip Sbreak s' ->
      tr_stmt s3 ts3 ->
      tr_stmt s4 ts4 ->
      tr_stmt (Csyntax.Sfor Csyntax.Sskip r s3 s4)
              (Sloop (Ssequence s' ts4) ts3)
  | tr_for_2: forall s1 r s3 s4 s' ts1 ts3 ts4,
      tr_if r Sskip Sbreak s' ->
      s1 <> Csyntax.Sskip ->
      tr_stmt s1 ts1 ->
      tr_stmt s3 ts3 ->
      tr_stmt s4 ts4 ->
      tr_stmt (Csyntax.Sfor s1 r s3 s4)
              (Ssequence ts1 (Sloop (Ssequence s' ts4) ts3))
  | tr_break:
      tr_stmt Csyntax.Sbreak Sbreak
  | tr_continue:
      tr_stmt Csyntax.Scontinue Scontinue
  | tr_return_none:
      tr_stmt (Csyntax.Sreturn None) (Sreturn None)
  | tr_return_some: forall r s' a,
      tr_expression r s' a ->
      tr_stmt (Csyntax.Sreturn (Some r)) (Ssequence s' (Sreturn (Some a)))
  | tr_switch: forall r ls s' a tls,
      tr_expression r s' a ->
      tr_lblstmts ls tls ->
      tr_stmt (Csyntax.Sswitch r ls) (Ssequence s' (Sswitch a tls))
  | tr_label: forall lbl s ts,
      tr_stmt s ts ->
      tr_stmt (Csyntax.Slabel lbl s) (Slabel lbl ts)
  | tr_goto: forall lbl,
      tr_stmt (Csyntax.Sgoto lbl) (Sgoto lbl)

with tr_lblstmts: Csyntax.labeled_statements -> labeled_statements -> Prop :=
  | tr_ls_nil:
      tr_lblstmts Csyntax.LSnil LSnil
  | tr_ls_cons: forall c s ls ts tls,
      tr_stmt s ts ->
      tr_lblstmts ls tls ->
      tr_lblstmts (Csyntax.LScons c s ls) (LScons c ts tls).

Correctness proof with respect to the specification.


Properties of the monad


Remark bind_inversion:
  forall (A B: Type) (f: mon A) (g: A -> mon B) (y: B) (z1 z3: generator) I,
  bind f g z1 = Res y z3 I ->
  exists x, exists z2, exists I1, exists I2,
  f z1 = Res x z2 I1 /\ g x z2 = Res y z3 I2.
Proof.
  intros until I. unfold bind. destruct (f z1).
  congruence.
  caseEq (g a g'); intros; inv H0.
  econstructor; econstructor; econstructor; econstructor; eauto.
Qed.

Remark bind2_inversion:
  forall (A B Csyntax: Type) (f: mon (A*B)) (g: A -> B -> mon Csyntax) (y: Csyntax) (z1 z3: generator) I,
  bind2 f g z1 = Res y z3 I ->
  exists x1, exists x2, exists z2, exists I1, exists I2,
  f z1 = Res (x1,x2) z2 I1 /\ g x1 x2 z2 = Res y z3 I2.
Proof.
  unfold bind2. intros.
  exploit bind_inversion; eauto.
  intros [[x1 x2] [z2 [I1 [I2 [P Q]]]]]. simpl in Q.
  exists x1; exists x2; exists z2; exists I1; exists I2; auto.
Qed.

Ltac monadInv1 H :=
  match type of H with
  | (Res _ _ _ = Res _ _ _) =>
      inversion H; clear H; try subst
  | (@ret _ _ _ = Res _ _ _) =>
      inversion H; clear H; try subst
  | (@error _ _ _ = Res _ _ _) =>
      inversion H
  | (bind ?F ?G ?Z = Res ?X ?Z' ?I) =>
      let x := fresh "x" in (
      let z := fresh "z" in (
      let I1 := fresh "I" in (
      let I2 := fresh "I" in (
      let EQ1 := fresh "EQ" in (
      let EQ2 := fresh "EQ" in (
      destruct (bind_inversion _ _ F G X Z Z' I H) as [x [z [I1 [I2 [EQ1 EQ2]]]]];
      clear H;
      try (monadInv1 EQ2)))))))
   | (bind2 ?F ?G ?Z = Res ?X ?Z' ?I) =>
      let x := fresh "x" in (
      let y := fresh "y" in (
      let z := fresh "z" in (
      let I1 := fresh "I" in (
      let I2 := fresh "I" in (
      let EQ1 := fresh "EQ" in (
      let EQ2 := fresh "EQ" in (
      destruct (bind2_inversion _ _ _ F G X Z Z' I H) as [x [y [z [I1 [I2 [EQ1 EQ2]]]]]];
      clear H;
      try (monadInv1 EQ2))))))))
 end.

Ltac monadInv H :=
  match type of H with
  | (@ret _ _ _ = Res _ _ _) => monadInv1 H
  | (@error _ _ _ = Res _ _ _) => monadInv1 H
  | (bind ?F ?G ?Z = Res ?X ?Z' ?I) => monadInv1 H
  | (bind2 ?F ?G ?Z = Res ?X ?Z' ?I) => monadInv1 H
  | (?F _ _ _ _ _ _ _ _ = Res _ _ _) =>
      ((progress simpl in H) || unfold F in H); monadInv1 H
  | (?F _ _ _ _ _ _ _ = Res _ _ _) =>
      ((progress simpl in H) || unfold F in H); monadInv1 H
  | (?F _ _ _ _ _ _ = Res _ _ _) =>
      ((progress simpl in H) || unfold F in H); monadInv1 H
  | (?F _ _ _ _ _ = Res _ _ _) =>
      ((progress simpl in H) || unfold F in H); monadInv1 H
  | (?F _ _ _ _ = Res _ _ _) =>
      ((progress simpl in H) || unfold F in H); monadInv1 H
  | (?F _ _ _ = Res _ _ _) =>
      ((progress simpl in H) || unfold F in H); monadInv1 H
  | (?F _ _ = Res _ _ _) =>
      ((progress simpl in H) || unfold F in H); monadInv1 H
  | (?F _ = Res _ _ _) =>
      ((progress simpl in H) || unfold F in H); monadInv1 H
  end.

Freshness and separation properties.


Definition within (id: ident) (g1 g2: generator) : Prop :=
  Ple (gen_next g1) id /\ Plt id (gen_next g2).

Lemma gensym_within:
  forall ty g1 id g2 I,
  gensym ty g1 = Res id g2 I -> within id g1 g2.
Proof.
  intros. monadInv H. split. apply Ple_refl. apply Plt_succ.
Qed.

Lemma within_widen:
  forall id g1 g2 g1' g2',
  within id g1 g2 ->
  Ple (gen_next g1') (gen_next g1) ->
  Ple (gen_next g2) (gen_next g2') ->
  within id g1' g2'.
Proof.
  intros. destruct H. split.
  eapply Ple_trans; eauto.
  eapply Plt_Ple_trans; eauto.
Qed.

Definition contained (l: list ident) (g1 g2: generator) : Prop :=
  forall id, In id l -> within id g1 g2.

Lemma contained_nil:
  forall g1 g2, contained nil g1 g2.
Proof.
  intros; red; intros; contradiction.
Qed.

Lemma contained_widen:
  forall l g1 g2 g1' g2',
  contained l g1 g2 ->
  Ple (gen_next g1') (gen_next g1) ->
  Ple (gen_next g2) (gen_next g2') ->
  contained l g1' g2'.
Proof.
  intros; red; intros. eapply within_widen; eauto.
Qed.

Lemma contained_cons:
  forall id l g1 g2,
  within id g1 g2 -> contained l g1 g2 -> contained (id :: l) g1 g2.
Proof.
  intros; red; intros. simpl in H1; destruct H1. subst id0. auto. auto.
Qed.

Lemma contained_app:
  forall l1 l2 g1 g2,
  contained l1 g1 g2 -> contained l2 g1 g2 -> contained (l1 ++ l2) g1 g2.
Proof.
  intros; red; intros. destruct (in_app_or _ _ _ H1); auto.
Qed.

Lemma contained_disjoint:
  forall g1 l1 g2 l2 g3,
  contained l1 g1 g2 -> contained l2 g2 g3 -> list_disjoint l1 l2.
Proof.
  intros; red; intros. red; intro; subst y.
  exploit H; eauto. intros [A B]. exploit H0; eauto. intros [Csyntax D].
  elim (Plt_strict x). apply Plt_Ple_trans with (gen_next g2); auto.
Qed.

Lemma contained_notin:
  forall g1 l g2 id g3,
  contained l g1 g2 -> within id g2 g3 -> ~In id l.
Proof.
  intros; red; intros. exploit H; eauto. intros [Csyntax D]. destruct H0 as [A B].
  elim (Plt_strict id). apply Plt_Ple_trans with (gen_next g2); auto.
Qed.

Definition dest_below (dst: destination) (g: generator) : Prop :=
  match dst with
  | For_set sd => Plt (sd_temp sd) g.(gen_next)
  | _ => True
  end.

Remark dest_for_val_below: forall g, dest_below For_val g.
Proof.
intros; simpl; auto. Qed.

Remark dest_for_effect_below: forall g, dest_below For_effects g.
Proof.
intros; simpl; auto. Qed.

Lemma dest_for_set_seqbool_val:
  forall tmp ty g1 g2,
  within tmp g1 g2 -> dest_below (For_set (sd_seqbool_val tmp ty)) g2.
Proof.
  intros. destruct H. simpl. auto.
Qed.

Lemma dest_for_set_seqbool_set:
  forall sd ty g, dest_below (For_set sd) g -> dest_below (For_set (sd_seqbool_set ty sd)) g.
Proof.
  intros. assumption.
Qed.

Lemma dest_for_set_condition_val:
  forall tmp tycast ty g1 g2, within tmp g1 g2 -> dest_below (For_set (SDbase tycast ty tmp)) g2.
Proof.
  intros. destruct H. simpl. auto.
Qed.

Lemma dest_for_set_condition_set:
  forall sd tmp tycast ty g1 g2, dest_below (For_set sd) g2 -> within tmp g1 g2 -> dest_below (For_set (SDcons tycast ty tmp sd)) g2.
Proof.
  intros. destruct H0. simpl. auto.
Qed.

Lemma sd_temp_notin:
  forall sd g1 g2 l, dest_below (For_set sd) g1 -> contained l g1 g2 -> ~In (sd_temp sd) l.
Proof.
  intros. simpl in H. red; intros. exploit H0; eauto. intros [A B].
  elim (Plt_strict (sd_temp sd)). apply Plt_Ple_trans with (gen_next g1); auto.
Qed.

Lemma dest_below_le:
  forall dst g1 g2,
  dest_below dst g1 -> Ple g1.(gen_next) g2.(gen_next) -> dest_below dst g2.
Proof.
  intros. destruct dst; simpl in *; auto. eapply Plt_Ple_trans; eauto.
Qed.

Hint Resolve gensym_within within_widen contained_widen
             contained_cons contained_app contained_disjoint
             contained_notin contained_nil
             dest_for_set_seqbool_val dest_for_set_seqbool_set
             dest_for_set_condition_val dest_for_set_condition_set
             sd_temp_notin dest_below_le
             incl_refl incl_tl incl_app incl_appl incl_appr incl_same_head
             in_eq in_cons
             Ple_trans Ple_refl: gensym.

Local Hint Resolve dest_for_val_below dest_for_effect_below : core.

Correctness of the translation functions


Lemma finish_meets_spec_1:
  forall dst sl a sl' a',
  finish dst sl a = (sl', a') -> sl' = sl ++ final dst a.
Proof.
  intros. destruct dst; simpl in *; inv H. apply app_nil_end. apply app_nil_end. auto.
Qed.

Lemma finish_meets_spec_2:
  forall dst sl a sl' a',
  finish dst sl a = (sl', a') -> a' = a.
Proof.
  intros. destruct dst; simpl in *; inv H; auto.
Qed.

Ltac UseFinish :=
  match goal with
  | [ H: finish _ _ _ = (_, _) |- _ ] =>
      try (rewrite (finish_meets_spec_2 _ _ _ _ _ H));
      try (rewrite (finish_meets_spec_1 _ _ _ _ _ H));
      repeat rewrite app_ass
  end.


Definition add_dest (dst: destination) (tmps: list ident) :=
  match dst with
  | For_set sd => sd_temp sd :: tmps
  | _ => tmps
  end.

Lemma add_dest_incl:
  forall dst tmps, incl tmps (add_dest dst tmps).
Proof.
  intros. destruct dst; simpl; eauto with coqlib.
Qed.

Lemma tr_expr_add_dest:
  forall le dst r sl a tmps,
  tr_expr le dst r sl a tmps ->
  tr_expr le dst r sl a (add_dest dst tmps).
Proof.
  intros. apply tr_expr_monotone with tmps; auto. apply add_dest_incl.
Qed.

Lemma is_bitfield_access_meets_spec: forall l g bf g' I,
  is_bitfield_access ce l g = Res bf g' I ->
  tr_is_bitfield_access l bf.
Proof.
  unfold is_bitfield_access; intros; red. destruct l; try (monadInv H; auto).
  assert (AUX: forall fn id,
               is_bitfield_access_aux ce fn id i g = Res bf g' I ->
               exists co ofs,
               ce!id = Some co /\ fn ce i (co_members co) = OK (ofs, bf)).
  { unfold is_bitfield_access_aux; intros.
    destruct ce!id as [co|]; try discriminate.
    destruct (fn ce i (co_members co)) as [[ofs1 bf1]|] eqn:FN; inv H0.
    exists co, ofs1; auto. }
  destruct (typeof l); try discriminate; apply AUX; auto.
Qed.

Lemma transl_valof_meets_spec:
  forall ty a g sl b g' I,
  transl_valof ce ty a g = Res (sl, b) g' I ->
  exists tmps, tr_rvalof ty a sl b tmps /\ contained tmps g g'.
Proof.
  unfold transl_valof; intros.
  destruct (type_is_volatile ty) eqn:?; monadInv H.
  exists (x :: nil); split; eauto with gensym.
  econstructor; eauto using is_bitfield_access_meets_spec with coqlib.
  exists (@nil ident); split; eauto with gensym. constructor; auto.
Qed.

Scheme expr_ind2 := Induction for Csyntax.expr Sort Prop
  with exprlist_ind2 := Induction for Csyntax.exprlist Sort Prop.
Combined Scheme expr_exprlist_ind from expr_ind2, exprlist_ind2.

Lemma transl_meets_spec:
   (forall r dst g sl a g' I,
    transl_expr ce dst r g = Res (sl, a) g' I ->
    dest_below dst g ->
    exists tmps, (forall le, tr_expr le dst r sl a (add_dest dst tmps)) /\ contained tmps g g')
  /\
   (forall rl g sl al g' I,
    transl_exprlist ce rl g = Res (sl, al) g' I ->
    exists tmps, (forall le, tr_exprlist le rl sl al tmps) /\ contained tmps g g').
Proof.
  apply expr_exprlist_ind; simpl add_dest; intros.
- (* val *)
  simpl in H. destruct v; monadInv H; exists (@nil ident); split; auto with gensym.
Opaque makeif.
+ intros. destruct dst; simpl in *; inv H2.
    constructor. auto. intros; constructor.
    constructor.
    constructor. auto. intros; constructor.
+ intros. destruct dst; simpl in *; inv H2.
    constructor. auto. intros; constructor.
    constructor.
    constructor. auto. intros; constructor.
+ intros. destruct dst; simpl in *; inv H2.
    constructor. auto. intros; constructor.
    constructor.
    constructor. auto. intros; constructor.
+ intros. destruct dst; simpl in *; inv H2.
    constructor. auto. intros; constructor.
    constructor.
    constructor. auto. intros; constructor.
- (* var *)
  monadInv H; econstructor; split; auto with gensym. UseFinish. constructor.
- (* field *)
  monadInv H0. exploit H; eauto. auto. intros [tmp [A B]]. UseFinish.
  econstructor; split; eauto. intros; apply tr_expr_add_dest. constructor; auto.
- (* valof *)
  monadInv H0. exploit H; eauto. intros [tmp1 [A B]].
  exploit transl_valof_meets_spec; eauto. intros [tmp2 [Csyntax D]]. UseFinish.
  exists (tmp1 ++ tmp2); split.
  intros; apply tr_expr_add_dest. econstructor; eauto with gensym.
  eauto with gensym.
- (* deref *)
  monadInv H0. exploit H; eauto. intros [tmp [A B]]. UseFinish.
  econstructor; split; eauto. intros; apply tr_expr_add_dest. constructor; auto.
- (* addrof *)
  monadInv H0. exploit H; eauto. intros [tmp [A B]]. UseFinish.
  econstructor; split; eauto. intros; apply tr_expr_add_dest. econstructor; eauto.
- (* unop *)
  monadInv H0. exploit H; eauto. intros [tmp [A B]]. UseFinish.
  econstructor; split; eauto. intros; apply tr_expr_add_dest. constructor; auto.
- (* binop *)
  monadInv H1. exploit H; eauto. intros [tmp1 [A B]].
  exploit H0; eauto. intros [tmp2 [Csyntax D]]. UseFinish.
  exists (tmp1 ++ tmp2); split.
  intros; apply tr_expr_add_dest. econstructor; eauto with gensym.
  eauto with gensym.
- (* cast *)
  destruct dst.
+ (* for value *)
  monadInv H0. exploit H; eauto. intros [tmp [A B]].
  econstructor; split; eauto. intros; apply tr_expr_add_dest.
  rewrite (app_nil_end sl).
  apply tr_cast_val with (dst := For_val); auto.
+ (* for effects *)
  exploit H; eauto. intros [tmp [A B]].
  econstructor; split; eauto. intros; eapply tr_cast_effects; eauto.
+ (* for set *)
  monadInv H0. exploit H; eauto. intros [tmp [A B]].
  econstructor; split; eauto. intros; apply tr_expr_add_dest.
  apply tr_cast_val with (dst := For_set sd); auto.
- (* seqand *)
  monadInv H1. exploit H; eauto. intros [tmp1 [A B]].
  destruct dst; monadInv EQ0.
+ (* for value *)
  exploit H0; eauto with gensym. intros [tmp2 [C D]].
  simpl add_dest in *.
  exists (x0 :: tmp1 ++ tmp2); split.
  intros; eapply tr_seqand_val; eauto with gensym.
  apply list_disjoint_cons_r; eauto with gensym.
  apply contained_cons. eauto with gensym.
  apply contained_app; eauto with gensym.
+ (* for effects *)
  exploit H0; eauto with gensym. intros [tmp2 [Csyntax D]].
  simpl add_dest in *.
  exists (tmp1 ++ tmp2); split.
  intros; eapply tr_seqand_effects; eauto with gensym.
  apply contained_app; eauto with gensym.
+ (* for set *)
  exploit H0; eauto with gensym. intros [tmp2 [C D]].
  simpl add_dest in *.
  exists (tmp1 ++ tmp2); split.
  intros; eapply tr_seqand_set; eauto with gensym.
  apply list_disjoint_cons_r; eauto with gensym.
  apply contained_app; eauto with gensym.
- (* seqor *)
  monadInv H1. exploit H; eauto. intros [tmp1 [A B]].
  destruct dst; monadInv EQ0.
+ (* for value *)
  exploit H0; eauto with gensym. intros [tmp2 [Csyntax D]].
  simpl add_dest in *.
  exists (x0 :: tmp1 ++ tmp2); split.
  intros; eapply tr_seqor_val; eauto with gensym.
  apply list_disjoint_cons_r; eauto with gensym.
  apply contained_cons. eauto with gensym.
  apply contained_app; eauto with gensym.
+ (* for effects *)
  exploit H0; eauto with gensym. intros [tmp2 [C D]].
  simpl add_dest in *.
  exists (tmp1 ++ tmp2); split.
  intros; eapply tr_seqor_effects; eauto with gensym.
  apply contained_app; eauto with gensym.
+ (* for set *)
  exploit H0; eauto with gensym. intros [tmp2 [C D]].
  simpl add_dest in *.
  exists (tmp1 ++ tmp2); split.
  intros; eapply tr_seqor_set; eauto with gensym.
  apply list_disjoint_cons_r; eauto with gensym.
  apply contained_app; eauto with gensym.
- (* condition *)
  monadInv H2. exploit H; eauto. intros [tmp1 [A B]].
  destruct dst; monadInv EQ0.
+ (* for value *)
  exploit H0; eauto with gensym. intros [tmp2 [C D]].
  exploit H1; eauto with gensym. intros [tmp3 [E F]].
  simpl add_dest in *.
  exists (x0 :: tmp1 ++ tmp2 ++ tmp3); split.
  simpl; intros; eapply tr_condition_val; eauto with gensym.
  apply list_disjoint_cons_r; eauto with gensym.
  apply list_disjoint_cons_r; eauto with gensym.
  apply contained_cons. eauto with gensym.
  apply contained_app. eauto with gensym.
  apply contained_app; eauto with gensym.
+ (* for effects *)
  exploit H0; eauto. intros [tmp2 [Csyntax D]].
  exploit H1; eauto. intros [tmp3 [E F]].
  simpl add_dest in *.
  exists (tmp1 ++ tmp2 ++ tmp3); split.
  intros; eapply tr_condition_effects; eauto with gensym.
  apply contained_app; eauto with gensym.
+ (* for test *)
  exploit H0; eauto with gensym. intros [tmp2 [C D]].
  exploit H1; eauto 10 with gensym. intros [tmp3 [E F]].
  simpl add_dest in *.
  exists (x0 :: tmp1 ++ tmp2 ++ tmp3); split.
  intros; eapply tr_condition_set; eauto with gensym.
  apply list_disjoint_cons_r; eauto with gensym.
  apply list_disjoint_cons_r; eauto with gensym.
  apply contained_cons; eauto with gensym.
  apply contained_app; eauto with gensym.
  apply contained_app; eauto with gensym.
- (* sizeof *)
  monadInv H. UseFinish.
  exists (@nil ident); split; auto with gensym. constructor.
- (* alignof *)
  monadInv H. UseFinish.
  exists (@nil ident); split; auto with gensym. constructor.
- (* assign *)
  monadInv H1. exploit H; eauto. intros [tmp1 [A B]].
  exploit H0; eauto. intros [tmp2 [Csyntax D]].
  apply is_bitfield_access_meets_spec in EQ0.
  destruct dst; monadInv EQ3; simpl add_dest in *.
+ (* for value *)
  exists (x2 :: tmp1 ++ tmp2); split.
  intros. eapply tr_assign_val with (dst := For_val); eauto with gensym.
  apply contained_cons. eauto with gensym.
  apply contained_app; eauto with gensym.
+ (* for effects *)
  exists (tmp1 ++ tmp2); split.
  econstructor; eauto with gensym.
  apply contained_app; eauto with gensym.
+ (* for set *)
  exists (x2 :: tmp1 ++ tmp2); split.
  repeat rewrite app_ass. simpl.
  intros. eapply tr_assign_val with (dst := For_set sd); eauto with gensym.
  apply contained_cons. eauto with gensym.
  apply contained_app; eauto with gensym.
- (* assignop *)
  monadInv H1. exploit H; eauto. intros [tmp1 [A B]].
  exploit H0; eauto. intros [tmp2 [Csyntax D]].
  exploit transl_valof_meets_spec; eauto. intros [tmp3 [E F]].
  apply is_bitfield_access_meets_spec in EQ2.
  destruct dst; monadInv EQ4; simpl add_dest in *.
+ (* for value *)
  exists (x3 :: tmp1 ++ tmp2 ++ tmp3); split.
  intros. eapply tr_assignop_val with (dst := For_val); eauto 6 with gensym.
  apply contained_cons. eauto 6 with gensym.
  apply contained_app; eauto 6 with gensym.
+ (* for effects *)
  exists (tmp1 ++ tmp2 ++ tmp3); split.
  econstructor; eauto with gensym.
  apply contained_app; eauto with gensym.
+ (* for set *)
  exists (x3 :: tmp1 ++ tmp2 ++ tmp3); split.
  repeat rewrite app_ass. simpl.
  intros. eapply tr_assignop_val with (dst := For_set sd); eauto 6 with gensym.
  apply contained_cons. eauto 6 with gensym.
  apply contained_app; eauto 6 with gensym.
- (* postincr *)
  monadInv H0. exploit H; eauto. intros [tmp1 [A B]].
  apply is_bitfield_access_meets_spec in EQ1.
  destruct dst; monadInv EQ2; simpl add_dest in *.
+ (* for value *)
  exists (x1 :: tmp1); split.
  econstructor; eauto with gensym.
  apply contained_cons; eauto with gensym.
+ (* for effects *)
  exploit transl_valof_meets_spec; eauto. intros [tmp2 [C D]].
  exists (tmp1 ++ tmp2); split.
  econstructor; eauto with gensym.
  eauto with gensym.
+ (* for set *)
  repeat rewrite app_ass; simpl.
  exists (x1 :: tmp1); split.
  econstructor; eauto with gensym.
  apply contained_cons; eauto with gensym.
- (* comma *)
  monadInv H1. exploit H; eauto. intros [tmp1 [A B]].
  exploit H0; eauto with gensym. intros [tmp2 [Csyntax D]].
  exists (tmp1 ++ tmp2); split.
  econstructor; eauto with gensym.
  destruct dst; simpl; eauto with gensym.
  apply list_disjoint_cons_r; eauto with gensym.
  simpl. eapply incl_tran. 2: apply add_dest_incl. auto with gensym.
  destruct dst; simpl; auto with gensym.
  apply contained_app; eauto with gensym.
- (* call *)
  monadInv H1. exploit H; eauto. intros [tmp1 [A B]].
  exploit H0; eauto. intros [tmp2 [Csyntax D]].
  destruct dst; monadInv EQ2; simpl add_dest in *.
+ (* for value *)
  exists (x1 :: tmp1 ++ tmp2); split.
  econstructor; eauto with gensym. congruence.
  apply contained_cons. eauto with gensym.
  apply contained_app; eauto with gensym.
+ (* for effects *)
  exists (tmp1 ++ tmp2); split.
  econstructor; eauto with gensym.
  apply contained_app; eauto with gensym.
+ (* for set *)
  exists (x1 :: tmp1 ++ tmp2); split.
  repeat rewrite app_ass. econstructor; eauto with gensym. congruence.
  apply contained_cons. eauto with gensym.
  apply contained_app; eauto with gensym.
- (* builtin *)
  monadInv H0. exploit H; eauto. intros [tmp1 [A B]].
  destruct dst; monadInv EQ0; simpl add_dest in *.
+ (* for value *)
  exists (x0 :: tmp1); split.
  econstructor; eauto with gensym. congruence.
  apply contained_cons; eauto with gensym.
+ (* for effects *)
  exists tmp1; split.
  econstructor; eauto with gensym.
  auto.
+ (* for set *)
  exists (x0 :: tmp1); split.
  repeat rewrite app_ass. econstructor; eauto with gensym. congruence.
  apply contained_cons; eauto with gensym.
- (* loc *)
  monadInv H.
- (* paren *)
  monadInv H0.
- (* nil *)
  monadInv H; exists (@nil ident); split; auto with gensym. constructor.
- (* cons *)
  monadInv H1. exploit H; eauto. intros [tmp1 [A B]].
  exploit H0; eauto. intros [tmp2 [Csyntax D]].
  exists (tmp1 ++ tmp2); split.
  econstructor; eauto with gensym.
  eauto with gensym.
Qed.

Lemma transl_expr_meets_spec:
   forall r dst g sl a g' I,
   transl_expr ce dst r g = Res (sl, a) g' I ->
   dest_below dst g ->
   exists tmps, forall ge e le m, tr_top ge e le m dst r sl a tmps.
Proof.
  intros. exploit (proj1 transl_meets_spec); eauto. intros [tmps [A B]].
  exists (add_dest dst tmps); intros. apply tr_top_base. auto.
Qed.

Lemma transl_expression_meets_spec:
  forall r g s a g' I,
  transl_expression ce r g = Res (s, a) g' I ->
  tr_expression r s a.
Proof.
  intros. monadInv H. exploit transl_expr_meets_spec; eauto.
  intros [tmps A]. econstructor; eauto.
Qed.

Lemma transl_expr_stmt_meets_spec:
  forall r g s g' I,
  transl_expr_stmt ce r g = Res s g' I ->
  tr_expr_stmt r s.
Proof.
  intros. monadInv H. exploit transl_expr_meets_spec; eauto.
  intros [tmps A]. econstructor; eauto.
Qed.

Lemma transl_if_meets_spec:
  forall r s1 s2 g s g' I,
  transl_if ce r s1 s2 g = Res s g' I ->
  tr_if r s1 s2 s.
Proof.
  intros. monadInv H. exploit transl_expr_meets_spec; eauto.
  intros [tmps A]. econstructor; eauto.
Qed.

Lemma transl_stmt_meets_spec:
  forall s g ts g' I, transl_stmt ce s g = Res ts g' I -> tr_stmt s ts
with transl_lblstmt_meets_spec:
  forall s g ts g' I, transl_lblstmt ce s g = Res ts g' I -> tr_lblstmts s ts.
Proof.
  generalize transl_expression_meets_spec transl_expr_stmt_meets_spec transl_if_meets_spec; intros T1 T2 T3.
Opaque transl_expression transl_expr_stmt.
  clear transl_stmt_meets_spec.
  induction s; simpl; intros until I; intros TR;
  try (monadInv TR); try (constructor; eauto).
  destruct (is_Sskip s1); destruct (is_Sskip s2) eqn:?; monadInv EQ3; try (constructor; eauto).
  eapply tr_ifthenelse_empty; eauto.
  destruct (is_Sskip s1); monadInv EQ4.
  apply tr_for_1; eauto.
  apply tr_for_2; eauto.
  destruct o; monadInv TR; constructor; eauto.
  clear transl_lblstmt_meets_spec.
  induction s; simpl; intros until I; intros TR;
  monadInv TR; constructor; eauto.
Qed.

Relational presentation for the transformation of functions, fundefs, and variables.

Inductive tr_function: Csyntax.function -> Clight.function -> Prop :=
  | tr_function_intro: forall f tf,
      tr_stmt f.(Csyntax.fn_body) tf.(fn_body) ->
      fn_return tf = Csyntax.fn_return f ->
      fn_callconv tf = Csyntax.fn_callconv f ->
      fn_params tf = Csyntax.fn_params f ->
      fn_vars tf = Csyntax.fn_vars f ->
      tr_function f tf.

Lemma transl_function_spec:
  forall f tf,
  transl_function ce f = OK tf ->
  tr_function f tf.
Proof.
  unfold transl_function; intros.
  destruct (transl_stmt ce (Csyntax.fn_body f) (initial_generator tt)) eqn:T; inv H.
  constructor; auto. simpl. eapply transl_stmt_meets_spec; eauto.
Qed.

End SPEC.

Inductive tr_fundef (p: Csyntax.program): Csyntax.fundef -> Clight.fundef -> Prop :=
  | tr_internal: forall f tf,
      tr_function p.(prog_comp_env) f tf ->
      tr_fundef p (Internal f) (Internal tf)
  | tr_external: forall ef targs tres cconv,
      tr_fundef p (External ef targs tres cconv) (External ef targs tres cconv).

Lemma transl_fundef_spec:
  forall p fd tfd,
  transl_fundef p.(prog_comp_env) fd = OK tfd ->
  tr_fundef p fd tfd.
Proof.
  unfold transl_fundef; intros.
  destruct fd; Errors.monadInv H.
+ constructor. eapply transl_function_spec; eauto.
+ constructor.
Qed.