Require Import Axioms.
Require Import Classical.
Require Import Coqlib.
Require Import Errors.
Require Import Maps.
Require Import Integers.
Require Import Floats.
Require Import Values.
Require Import AST.
Require Import Memory.
Require Import Events.
Require Import Globalenvs.
Require Import Smallstep.
Require Import Ctypes.
Require Import Cop.
Require Import Csyntax.
Require Import Csem.

Section STRATEGY.

Variable ge: genv.

Definition of the strategy

Fixpoint simple (a: expr) : bool :=
  match a with
  | Eloc _ _ _ => true
  | Evar _ _ => true
  | Ederef r _ => simple r
  | Efield r _ _ => simple r
  | Eval _ _ => true
  | Evalof l _ => simple l && negb(type_is_volatile (typeof l))
  | Eaddrof l _ => simple l
  | Eunop _ r1 _ => simple r1
  | Ebinop _ r1 r2 _ => simple r1 && simple r2
  | Ecast r1 _ => simple r1
  | Eseqand _ _ _ => false
  | Eseqor _ _ _ => false
  | Econdition _ _ _ _ => false
  | Esizeof _ _ => true
  | Ealignof _ _ => true
  | Eassign _ _ _ => false
  | Eassignop _ _ _ _ _ => false
  | Epostincr _ _ _ => false
  | Ecomma _ _ _ => false
  | Ecall _ _ _ => false
  | Ebuiltin _ _ _ _ => false
  | Eparen _ _ _ => false
  end.

Fixpoint simplelist (rl: exprlist) : bool :=
  match rl with Enil => true | Econs r rl' => simple r && simplelist rl' end.


Section SIMPLE_EXPRS.

Variable e: env.
Variable m: mem.

Inductive eval_simple_lvalue: expr -> block -> ptrofs -> Prop :=
  | esl_loc: forall b ofs ty,
      eval_simple_lvalue (Eloc b ofs ty) b ofs
  | esl_var_local: forall x ty b,
      e!x = Some(b, ty) ->
      eval_simple_lvalue (Evar x ty) b Ptrofs.zero
  | esl_var_global: forall x ty b,
      e!x = None ->
      Genv.find_symbol ge x = Some b ->
      eval_simple_lvalue (Evar x ty) b Ptrofs.zero
  | esl_deref: forall r ty b ofs,
      eval_simple_rvalue r (Vptr b ofs) ->
      eval_simple_lvalue (Ederef r ty) b ofs
  | esl_field_struct: forall r f ty b ofs id co a delta,
      eval_simple_rvalue r (Vptr b ofs) ->
      typeof r = Tstruct id a ->
      ge.(genv_cenv)!id = Some co ->
      field_offset ge f (co_members co) = OK delta ->
      eval_simple_lvalue (Efield r f ty) b (Ptrofs.add ofs (Ptrofs.repr delta))
  | esl_field_union: forall r f ty b ofs id co a,
      eval_simple_rvalue r (Vptr b ofs) ->
      typeof r = Tunion id a ->
      ge.(genv_cenv)!id = Some co ->
      eval_simple_lvalue (Efield r f ty) b ofs

with eval_simple_rvalue: expr -> val -> Prop :=
  | esr_val: forall v ty,
      eval_simple_rvalue (Eval v ty) v
  | esr_rvalof: forall b ofs l ty v,
      eval_simple_lvalue l b ofs ->
      ty = typeof l -> type_is_volatile ty = false ->
      deref_loc ge ty m b ofs E0 v ->
      eval_simple_rvalue (Evalof l ty) v
  | esr_addrof: forall b ofs l ty,
      eval_simple_lvalue l b ofs ->
      eval_simple_rvalue (Eaddrof l ty) (Vptr b ofs)
  | esr_unop: forall op r1 ty v1 v,
      eval_simple_rvalue r1 v1 ->
      sem_unary_operation op v1 (typeof r1) m = Some v ->
      eval_simple_rvalue (Eunop op r1 ty) v
  | esr_binop: forall op r1 r2 ty v1 v2 v,
      eval_simple_rvalue r1 v1 -> eval_simple_rvalue r2 v2 ->
      sem_binary_operation ge op v1 (typeof r1) v2 (typeof r2) m = Some v ->
      eval_simple_rvalue (Ebinop op r1 r2 ty) v
  | esr_cast: forall ty r1 v1 v,
      eval_simple_rvalue r1 v1 ->
      sem_cast v1 (typeof r1) ty m = Some v ->
      eval_simple_rvalue (Ecast r1 ty) v
  | esr_sizeof: forall ty1 ty,
      eval_simple_rvalue (Esizeof ty1 ty) (Vptrofs (Ptrofs.repr (sizeof ge ty1)))
  | esr_alignof: forall ty1 ty,
      eval_simple_rvalue (Ealignof ty1 ty) (Vptrofs (Ptrofs.repr (alignof ge ty1))).

Inductive eval_simple_list: exprlist -> typelist -> list val -> Prop :=
  | esrl_nil:
      eval_simple_list Enil Tnil nil
  | esrl_cons: forall r rl ty tyl v vl v',
      eval_simple_rvalue r v' -> sem_cast v' (typeof r) ty m = Some v ->
      eval_simple_list rl tyl vl ->
      eval_simple_list (Econs r rl) (Tcons ty tyl) (v :: vl).

Scheme eval_simple_rvalue_ind2 := Minimality for eval_simple_rvalue Sort Prop
  with eval_simple_lvalue_ind2 := Minimality for eval_simple_lvalue Sort Prop.
Combined Scheme eval_simple_rvalue_lvalue_ind from eval_simple_rvalue_ind2, eval_simple_lvalue_ind2.

End SIMPLE_EXPRS.


Inductive leftcontext: kind -> kind -> (expr -> expr) -> Prop :=
  | lctx_top: forall k,
      leftcontext k k (fun x => x)
  | lctx_deref: forall k C ty,
      leftcontext k RV C -> leftcontext k LV (fun x => Ederef (C x) ty)
  | lctx_field: forall k C f ty,
      leftcontext k RV C -> leftcontext k LV (fun x => Efield (C x) f ty)
  | lctx_rvalof: forall k C ty,
      leftcontext k LV C -> leftcontext k RV (fun x => Evalof (C x) ty)
  | lctx_addrof: forall k C ty,
      leftcontext k LV C -> leftcontext k RV (fun x => Eaddrof (C x) ty)
  | lctx_unop: forall k C op ty,
      leftcontext k RV C -> leftcontext k RV (fun x => Eunop op (C x) ty)
  | lctx_binop_left: forall k C op e2 ty,
      leftcontext k RV C -> leftcontext k RV (fun x => Ebinop op (C x) e2 ty)
  | lctx_binop_right: forall k C op e1 ty,
      simple e1 = true -> leftcontext k RV C ->
      leftcontext k RV (fun x => Ebinop op e1 (C x) ty)
  | lctx_cast: forall k C ty,
      leftcontext k RV C -> leftcontext k RV (fun x => Ecast (C x) ty)
  | lctx_seqand: forall k C r2 ty,
      leftcontext k RV C -> leftcontext k RV (fun x => Eseqand (C x) r2 ty)
  | lctx_seqor: forall k C r2 ty,
      leftcontext k RV C -> leftcontext k RV (fun x => Eseqor (C x) r2 ty)
  | lctx_condition: forall k C r2 r3 ty,
      leftcontext k RV C -> leftcontext k RV (fun x => Econdition (C x) r2 r3 ty)
  | lctx_assign_left: forall k C e2 ty,
      leftcontext k LV C -> leftcontext k RV (fun x => Eassign (C x) e2 ty)
  | lctx_assign_right: forall k C e1 ty,
      simple e1 = true -> leftcontext k RV C ->
      leftcontext k RV (fun x => Eassign e1 (C x) ty)
  | lctx_assignop_left: forall k C op e2 tyres ty,
      leftcontext k LV C -> leftcontext k RV (fun x => Eassignop op (C x) e2 tyres ty)
  | lctx_assignop_right: forall k C op e1 tyres ty,
      simple e1 = true -> leftcontext k RV C ->
      leftcontext k RV (fun x => Eassignop op e1 (C x) tyres ty)
  | lctx_postincr: forall k C id ty,
      leftcontext k LV C -> leftcontext k RV (fun x => Epostincr id (C x) ty)
  | lctx_call_left: forall k C el ty,
      leftcontext k RV C -> leftcontext k RV (fun x => Ecall (C x) el ty)
  | lctx_call_right: forall k C e1 ty,
      simple e1 = true -> leftcontextlist k C ->
      leftcontext k RV (fun x => Ecall e1 (C x) ty)
  | lctx_builtin: forall k C ef tyargs ty,
      leftcontextlist k C ->
      leftcontext k RV (fun x => Ebuiltin ef tyargs (C x) ty)
  | lctx_comma: forall k C e2 ty,
      leftcontext k RV C -> leftcontext k RV (fun x => Ecomma (C x) e2 ty)
  | lctx_paren: forall k C tycast ty,
      leftcontext k RV C -> leftcontext k RV (fun x => Eparen (C x) tycast ty)

with leftcontextlist: kind -> (expr -> exprlist) -> Prop :=
  | lctx_list_head: forall k C el,
      leftcontext k RV C -> leftcontextlist k (fun x => Econs (C x) el)
  | lctx_list_tail: forall k C e1,
      simple e1 = true -> leftcontextlist k C ->
      leftcontextlist k (fun x => Econs e1 (C x)).

Lemma leftcontext_context:
  forall k1 k2 C, leftcontext k1 k2 C -> context k1 k2 C
with leftcontextlist_contextlist:
  forall k C, leftcontextlist k C -> contextlist k C.

Local Hint Resolve leftcontext_context : core.


Inductive estep: state -> trace -> state -> Prop :=

  | step_expr: forall f r k e m v ty,
      eval_simple_rvalue e m r v ->
      match r with Eval _ _ => False | _ => True end ->
      ty = typeof r ->
      estep (ExprState f r k e m)
         E0 (ExprState f (Eval v ty) k e m)

  | step_rvalof_volatile: forall f C l ty k e m b ofs t v,
      leftcontext RV RV C ->
      eval_simple_lvalue e m l b ofs ->
      deref_loc ge ty m b ofs t v ->
      ty = typeof l -> type_is_volatile ty = true ->
      estep (ExprState f (C (Evalof l ty)) k e m)
          t (ExprState f (C (Eval v ty)) k e m)

  | step_seqand_true: forall f C r1 r2 ty k e m v,
      leftcontext RV RV C ->
      eval_simple_rvalue e m r1 v ->
      bool_val v (typeof r1) m = Some true ->
      estep (ExprState f (C (Eseqand r1 r2 ty)) k e m)
         E0 (ExprState f (C (Eparen r2 type_bool ty)) k e m)
  | step_seqand_false: forall f C r1 r2 ty k e m v,
      leftcontext RV RV C ->
      eval_simple_rvalue e m r1 v ->
      bool_val v (typeof r1) m = Some false ->
      estep (ExprState f (C (Eseqand r1 r2 ty)) k e m)
         E0 (ExprState f (C (Eval (Vint Int.zero) ty)) k e m)

  | step_seqor_true: forall f C r1 r2 ty k e m v,
      leftcontext RV RV C ->
      eval_simple_rvalue e m r1 v ->
      bool_val v (typeof r1) m = Some true ->
      estep (ExprState f (C (Eseqor r1 r2 ty)) k e m)
         E0 (ExprState f (C (Eval (Vint Int.one) ty)) k e m)
  | step_seqor_false: forall f C r1 r2 ty k e m v,
      leftcontext RV RV C ->
      eval_simple_rvalue e m r1 v ->
      bool_val v (typeof r1) m = Some false ->
      estep (ExprState f (C (Eseqor r1 r2 ty)) k e m)
         E0 (ExprState f (C (Eparen r2 type_bool ty)) k e m)

  | step_condition: forall f C r1 r2 r3 ty k e m v b,
      leftcontext RV RV C ->
      eval_simple_rvalue e m r1 v ->
      bool_val v (typeof r1) m = Some b ->
      estep (ExprState f (C (Econdition r1 r2 r3 ty)) k e m)
         E0 (ExprState f (C (Eparen (if b then r2 else r3) ty ty)) k e m)

  | step_assign: forall f C l r ty k e m b ofs v v' t m',
      leftcontext RV RV C ->
      eval_simple_lvalue e m l b ofs ->
      eval_simple_rvalue e m r v ->
      sem_cast v (typeof r) (typeof l) m = Some v' ->
      assign_loc ge (typeof l) m b ofs v' t m' ->
      ty = typeof l ->
      estep (ExprState f (C (Eassign l r ty)) k e m)
          t (ExprState f (C (Eval v' ty)) k e m')

  | step_assignop: forall f C op l r tyres ty k e m b ofs v1 v2 v3 v4 t1 t2 m' t,
      leftcontext RV RV C ->
      eval_simple_lvalue e m l b ofs ->
      deref_loc ge (typeof l) m b ofs t1 v1 ->
      eval_simple_rvalue e m r v2 ->
      sem_binary_operation ge op v1 (typeof l) v2 (typeof r) m = Some v3 ->
      sem_cast v3 tyres (typeof l) m = Some v4 ->
      assign_loc ge (typeof l) m b ofs v4 t2 m' ->
      ty = typeof l ->
      t = t1 ** t2 ->
      estep (ExprState f (C (Eassignop op l r tyres ty)) k e m)
          t (ExprState f (C (Eval v4 ty)) k e m')

  | step_assignop_stuck: forall f C op l r tyres ty k e m b ofs v1 v2 t,
      leftcontext RV RV C ->
      eval_simple_lvalue e m l b ofs ->
      deref_loc ge (typeof l) m b ofs t v1 ->
      eval_simple_rvalue e m r v2 ->
      match sem_binary_operation ge op v1 (typeof l) v2 (typeof r) m with
      | None => True
      | Some v3 =>
          match sem_cast v3 tyres (typeof l) m with
          | None => True
          | Some v4 => forall t2 m', ~(assign_loc ge (typeof l) m b ofs v4 t2 m')
          end
      end ->
      ty = typeof l ->
      estep (ExprState f (C (Eassignop op l r tyres ty)) k e m)
          t Stuckstate

  | step_postincr: forall f C id l ty k e m b ofs v1 v2 v3 t1 t2 m' t,
      leftcontext RV RV C ->
      eval_simple_lvalue e m l b ofs ->
      deref_loc ge ty m b ofs t1 v1 ->
      sem_incrdecr ge id v1 ty m = Some v2 ->
      sem_cast v2 (incrdecr_type ty) ty m = Some v3 ->
      assign_loc ge ty m b ofs v3 t2 m' ->
      ty = typeof l ->
      t = t1 ** t2 ->
      estep (ExprState f (C (Epostincr id l ty)) k e m)
          t (ExprState f (C (Eval v1 ty)) k e m')

  | step_postincr_stuck: forall f C id l ty k e m b ofs v1 t,
      leftcontext RV RV C ->
      eval_simple_lvalue e m l b ofs ->
      deref_loc ge ty m b ofs t v1 ->
      match sem_incrdecr ge id v1 ty m with
      | None => True
      | Some v2 =>
          match sem_cast v2 (incrdecr_type ty) ty m with
          | None => True
          | Some v3 => forall t2 m', ~(assign_loc ge (typeof l) m b ofs v3 t2 m')
          end
      end ->
      ty = typeof l ->
      estep (ExprState f (C (Epostincr id l ty)) k e m)
          t Stuckstate

  | step_comma: forall f C r1 r2 ty k e m v,
      leftcontext RV RV C ->
      eval_simple_rvalue e m r1 v ->
      ty = typeof r2 ->
      estep (ExprState f (C (Ecomma r1 r2 ty)) k e m)
         E0 (ExprState f (C r2) k e m)

  | step_paren: forall f C r tycast ty k e m v1 v,
      leftcontext RV RV C ->
      eval_simple_rvalue e m r v1 ->
      sem_cast v1 (typeof r) tycast m = Some v ->
      estep (ExprState f (C (Eparen r tycast ty)) k e m)
         E0 (ExprState f (C (Eval v ty)) k e m)

  | step_call: forall f C rf rargs ty k e m targs tres cconv vf vargs fd,
      leftcontext RV RV C ->
      classify_fun (typeof rf) = fun_case_f targs tres cconv ->
      eval_simple_rvalue e m rf vf ->
      eval_simple_list e m rargs targs vargs ->
      Genv.find_funct ge vf = Some fd ->
      type_of_fundef fd = Tfunction targs tres cconv ->
      estep (ExprState f (C (Ecall rf rargs ty)) k e m)
         E0 (Callstate fd vargs (Kcall f e C ty k) m)

  | step_builtin: forall f C ef tyargs rargs ty k e m vargs t vres m',
      leftcontext RV RV C ->
      eval_simple_list e m rargs tyargs vargs ->
      external_call ef ge vargs m t vres m' ->
      estep (ExprState f (C (Ebuiltin ef tyargs rargs ty)) k e m)
          t (ExprState f (C (Eval vres ty)) k e m').

Definition step (S: state) (t: trace) (S': state) : Prop :=
  estep S t S' \/ sstep ge S t S'.


Lemma context_compose:
  forall k2 k3 C2, context k2 k3 C2 ->
  forall k1 C1, context k1 k2 C1 ->
  context k1 k3 (fun x => C2(C1 x))
with contextlist_compose:
  forall k2 C2, contextlist k2 C2 ->
  forall k1 C1, context k1 k2 C1 ->
  contextlist k1 (fun x => C2(C1 x)).

Local Hint Constructors context contextlist : core.
Local Hint Resolve context_compose contextlist_compose : core.

Safe executions.

Definition safe (s: Csem.state) : Prop :=
  forall s', star Csem.step ge s E0 s' ->
  (exists r, final_state s' r) \/ (exists t, exists s'', Csem.step ge s' t s'').

Lemma safe_steps:
  forall s s',
  safe s -> star Csem.step ge s E0 s' -> safe s'.

Lemma star_safe:
  forall s1 s2 t s3,
  safe s1 -> star Csem.step ge s1 E0 s2 -> (safe s2 -> star Csem.step ge s2 t s3) ->
  star Csem.step ge s1 t s3.

Lemma plus_safe:
  forall s1 s2 t s3,
  safe s1 -> star Csem.step ge s1 E0 s2 -> (safe s2 -> plus Csem.step ge s2 t s3) ->
  plus Csem.step ge s1 t s3.

Lemma safe_imm_safe:
  forall f C a k e m K,
  safe (ExprState f (C a) k e m) ->
  context K RV C ->
  imm_safe ge e K a m.


Definition expr_kind (a: expr) : kind :=
  match a with
  | Eloc _ _ _ => LV
  | Evar _ _ => LV
  | Ederef _ _ => LV
  | Efield _ _ _ => LV
  | _ => RV
  end.

Lemma lred_kind:
  forall e a m a' m', lred ge e a m a' m' -> expr_kind a = LV.

Lemma rred_kind:
  forall a m t a' m', rred ge a m t a' m' -> expr_kind a = RV.

Lemma callred_kind:
  forall a m fd args ty, callred ge a m fd args ty -> expr_kind a = RV.

Lemma context_kind:
  forall a from to C, context from to C -> expr_kind a = from -> expr_kind (C a) = to.

Lemma imm_safe_kind:
  forall e k a m, imm_safe ge e k a m -> expr_kind a = k.

Lemma safe_expr_kind:
  forall from C f a k e m,
  context from RV C ->
  safe (ExprState f (C a) k e m) ->
  expr_kind a = from.


Section INVERSION_LEMMAS.

Variable e: env.

Fixpoint exprlist_all_values (rl: exprlist) : Prop :=
  match rl with
  | Enil => True
  | Econs (Eval v ty) rl' => exprlist_all_values rl'
  | Econs _ _ => False
  end.

Definition invert_expr_prop (a: expr) (m: mem) : Prop :=
  match a with
  | Eloc b ofs ty => False
  | Evar x ty =>
      exists b,
      e!x = Some(b, ty)
      \/ (e!x = None /\ Genv.find_symbol ge x = Some b)
  | Ederef (Eval v ty1) ty =>
      exists b, exists ofs, v = Vptr b ofs
  | Efield (Eval v ty1) f ty =>
      exists b, exists ofs, v = Vptr b ofs /\
      match ty1 with
      | Tstruct id _ => exists co delta, ge.(genv_cenv)!id = Some co /\ field_offset ge f (co_members co) = Errors.OK delta
      | Tunion id _ => exists co, ge.(genv_cenv)!id = Some co
      | _ => False
      end
  | Eval v ty => False
  | Evalof (Eloc b ofs ty') ty =>
      ty' = ty /\ exists t, exists v, deref_loc ge ty m b ofs t v
  | Eunop op (Eval v1 ty1) ty =>
      exists v, sem_unary_operation op v1 ty1 m = Some v
  | Ebinop op (Eval v1 ty1) (Eval v2 ty2) ty =>
      exists v, sem_binary_operation ge op v1 ty1 v2 ty2 m = Some v
  | Ecast (Eval v1 ty1) ty =>
      exists v, sem_cast v1 ty1 ty m = Some v
  | Eseqand (Eval v1 ty1) r2 ty =>
      exists b, bool_val v1 ty1 m = Some b
  | Eseqor (Eval v1 ty1) r2 ty =>
      exists b, bool_val v1 ty1 m = Some b
  | Econdition (Eval v1 ty1) r1 r2 ty =>
      exists b, bool_val v1 ty1 m = Some b
  | Eassign (Eloc b ofs ty1) (Eval v2 ty2) ty =>
      exists v, exists m', exists t,
      ty = ty1 /\ sem_cast v2 ty2 ty1 m = Some v /\ assign_loc ge ty1 m b ofs v t m'
  | Eassignop op (Eloc b ofs ty1) (Eval v2 ty2) tyres ty =>
      exists t, exists v1,
      ty = ty1
      /\ deref_loc ge ty1 m b ofs t v1
  | Epostincr id (Eloc b ofs ty1) ty =>
      exists t, exists v1,
      ty = ty1
      /\ deref_loc ge ty m b ofs t v1
  | Ecomma (Eval v ty1) r2 ty =>
      typeof r2 = ty
  | Eparen (Eval v1 ty1) ty2 ty =>
      exists v, sem_cast v1 ty1 ty2 m = Some v
  | Ecall (Eval vf tyf) rargs ty =>
      exprlist_all_values rargs ->
      exists tyargs tyres cconv fd vl,
         classify_fun tyf = fun_case_f tyargs tyres cconv
      /\ Genv.find_funct ge vf = Some fd
      /\ cast_arguments m rargs tyargs vl
      /\ type_of_fundef fd = Tfunction tyargs tyres cconv
  | Ebuiltin ef tyargs rargs ty =>
      exprlist_all_values rargs ->
      exists vargs, exists t, exists vres, exists m',
         cast_arguments m rargs tyargs vargs
      /\ external_call ef ge vargs m t vres m'
  | _ => True
  end.

Lemma lred_invert:
  forall l m l' m', lred ge e l m l' m' -> invert_expr_prop l m.

Lemma rred_invert:
  forall r m t r' m', rred ge r m t r' m' -> invert_expr_prop r m.

Lemma callred_invert:
  forall r fd args ty m,
  callred ge r m fd args ty ->
  invert_expr_prop r m.

Scheme context_ind2 := Minimality for context Sort Prop
  with contextlist_ind2 := Minimality for contextlist Sort Prop.
Combined Scheme context_contextlist_ind from context_ind2, contextlist_ind2.

Lemma invert_expr_context:
  (forall from to C, context from to C ->
   forall a m,
   invert_expr_prop a m ->
   invert_expr_prop (C a) m)
/\(forall from C, contextlist from C ->
  forall a m,
  invert_expr_prop a m ->
  ~exprlist_all_values (C a)).

Lemma imm_safe_inv:
  forall k a m,
  imm_safe ge e k a m ->
  match a with
  | Eloc _ _ _ => True
  | Eval _ _ => True
  | _ => invert_expr_prop a m
  end.

Lemma safe_inv:
  forall k C f a K m,
  safe (ExprState f (C a) K e m) ->
  context k RV C ->
  match a with
  | Eloc _ _ _ => True
  | Eval _ _ => True
  | _ => invert_expr_prop a m
  end.

End INVERSION_LEMMAS.

Correctness of the strategy.

Section SIMPLE_EVAL.

Variable f: function.
Variable k: cont.
Variable e: env.
Variable m: mem.

Lemma eval_simple_steps:
   (forall a v, eval_simple_rvalue e m a v ->
    forall C, context RV RV C ->
    star Csem.step ge (ExprState f (C a) k e m)
                   E0 (ExprState f (C (Eval v (typeof a))) k e m))
/\ (forall a b ofs, eval_simple_lvalue e m a b ofs ->
    forall C, context LV RV C ->
    star Csem.step ge (ExprState f (C a) k e m)
                   E0 (ExprState f (C (Eloc b ofs (typeof a))) k e m)).

Lemma eval_simple_rvalue_steps:
  forall a v, eval_simple_rvalue e m a v ->
  forall C, context RV RV C ->
  star Csem.step ge (ExprState f (C a) k e m)
                E0 (ExprState f (C (Eval v (typeof a))) k e m).
Proof (proj1 eval_simple_steps).

Lemma eval_simple_lvalue_steps:
  forall a b ofs, eval_simple_lvalue e m a b ofs ->
  forall C, context LV RV C ->
  star Csem.step ge (ExprState f (C a) k e m)
                E0 (ExprState f (C (Eloc b ofs (typeof a))) k e m).
Proof (proj2 eval_simple_steps).

Corollary eval_simple_rvalue_safe:
  forall C a v,
  eval_simple_rvalue e m a v ->
  context RV RV C -> safe (ExprState f (C a) k e m) ->
  safe (ExprState f (C (Eval v (typeof a))) k e m).

Corollary eval_simple_lvalue_safe:
  forall C a b ofs,
  eval_simple_lvalue e m a b ofs ->
  context LV RV C -> safe (ExprState f (C a) k e m) ->
  safe (ExprState f (C (Eloc b ofs (typeof a))) k e m).

Lemma simple_can_eval:
  forall a from C,
  simple a = true -> context from RV C -> safe (ExprState f (C a) k e m) ->
  match from with
  | LV => exists b, exists ofs, eval_simple_lvalue e m a b ofs
  | RV => exists v, eval_simple_rvalue e m a v
  end.

Lemma simple_can_eval_rval:
  forall r C,
  simple r = true -> context RV RV C -> safe (ExprState f (C r) k e m) ->
  exists v, eval_simple_rvalue e m r v
        /\ safe (ExprState f (C (Eval v (typeof r))) k e m).

Lemma simple_can_eval_lval:
  forall l C,
  simple l = true -> context LV RV C -> safe (ExprState f (C l) k e m) ->
  exists b, exists ofs, eval_simple_lvalue e m l b ofs
         /\ safe (ExprState f (C (Eloc b ofs (typeof l))) k e m).

Fixpoint rval_list (vl: list val) (rl: exprlist) : exprlist :=
  match vl, rl with
  | v1 :: vl', Econs r1 rl' => Econs (Eval v1 (typeof r1)) (rval_list vl' rl')
  | _, _ => Enil
  end.

Inductive eval_simple_list': exprlist -> list val -> Prop :=
  | esrl'_nil:
      eval_simple_list' Enil nil
  | esrl'_cons: forall r rl v vl,
      eval_simple_rvalue e m r v ->
      eval_simple_list' rl vl ->
      eval_simple_list' (Econs r rl) (v :: vl).

Lemma eval_simple_list_implies:
  forall rl tyl vl,
  eval_simple_list e m rl tyl vl ->
  exists vl', cast_arguments m (rval_list vl' rl) tyl vl /\ eval_simple_list' rl vl'.

Lemma can_eval_simple_list:
  forall rl vl,
  eval_simple_list' rl vl ->
  forall tyl vl',
  cast_arguments m (rval_list vl rl) tyl vl' ->
  eval_simple_list e m rl tyl vl'.

Fixpoint exprlist_app (rl1 rl2: exprlist) : exprlist :=
  match rl1 with
  | Enil => rl2
  | Econs r1 rl1' => Econs r1 (exprlist_app rl1' rl2)
  end.

Lemma exprlist_app_assoc:
  forall rl2 rl3 rl1,
  exprlist_app (exprlist_app rl1 rl2) rl3 =
  exprlist_app rl1 (exprlist_app rl2 rl3).

Inductive contextlist' : (exprlist -> expr) -> Prop :=
  | contextlist'_call: forall r1 rl0 ty C,
      context RV RV C ->
      contextlist' (fun rl => C (Ecall r1 (exprlist_app rl0 rl) ty))
  | contextlist'_builtin: forall ef tyargs rl0 ty C,
      context RV RV C ->
      contextlist' (fun rl => C (Ebuiltin ef tyargs (exprlist_app rl0 rl) ty)).

Lemma exprlist_app_context:
  forall rl1 rl2,
  contextlist RV (fun x => exprlist_app rl1 (Econs x rl2)).

Lemma contextlist'_head:
  forall rl C,
  contextlist' C ->
  context RV RV (fun x => C (Econs x rl)).

Lemma contextlist'_tail:
  forall r1 C,
  contextlist' C ->
  contextlist' (fun x => C (Econs r1 x)).

Local Hint Resolve contextlist'_head contextlist'_tail : core.

Lemma eval_simple_list_steps:
  forall rl vl, eval_simple_list' rl vl ->
  forall C, contextlist' C ->
  star Csem.step ge (ExprState f (C rl) k e m)
                E0 (ExprState f (C (rval_list vl rl)) k e m).

Lemma simple_list_can_eval:
  forall rl C,
  simplelist rl = true ->
  contextlist' C ->
  safe (ExprState f (C rl) k e m) ->
  exists vl, eval_simple_list' rl vl.

Lemma rval_list_all_values:
  forall vl rl, exprlist_all_values (rval_list vl rl).

End SIMPLE_EVAL.


Section DECOMPOSITION.

Variable f: function.
Variable k: cont.
Variable e: env.
Variable m: mem.

Definition simple_side_effect (r: expr) : Prop :=
  match r with
  | Evalof l _ => simple l = true /\ type_is_volatile (typeof l) = true
  | Eseqand r1 r2 _ => simple r1 = true
  | Eseqor r1 r2 _ => simple r1 = true
  | Econdition r1 r2 r3 _ => simple r1 = true
  | Eassign l1 r2 _ => simple l1 = true /\ simple r2 = true
  | Eassignop _ l1 r2 _ _ => simple l1 = true /\ simple r2 = true
  | Epostincr _ l1 _ => simple l1 = true
  | Ecomma r1 r2 _ => simple r1 = true
  | Ecall r1 rl _ => simple r1 = true /\ simplelist rl = true
  | Ebuiltin ef tyargs rl _ => simplelist rl = true
  | Eparen r1 _ _ => simple r1 = true
  | _ => False
  end.

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

Local Hint Constructors leftcontext leftcontextlist : core.

Lemma decompose_expr:
  (forall a from C,
   context from RV C -> safe (ExprState f (C a) k e m) ->
       simple a = true
    \/ exists C', exists a', a = C' a' /\ simple_side_effect a' /\ leftcontext RV from C')
/\(forall rl C,
   contextlist' C -> safe (ExprState f (C rl) k e m) ->
       simplelist rl = true
    \/ exists C', exists a', rl = C' a' /\ simple_side_effect a' /\ leftcontextlist RV C').

Lemma decompose_topexpr:
  forall a,
  safe (ExprState f a k e m) ->
       simple a = true
    \/ exists C, exists a', a = C a' /\ simple_side_effect a' /\ leftcontext RV RV C.

End DECOMPOSITION.


Lemma estep_simulation:
  forall S t S',
  estep S t S' -> plus Csem.step ge S t S'.

Lemma can_estep:
  forall f a k e m,
  safe (ExprState f a k e m) ->
  match a with Eval _ _ => False | _ => True end ->
  exists t, exists S, estep (ExprState f a k e m) t S.


Theorem step_simulation:
  forall S1 t S2,
  step S1 t S2 -> plus Csem.step ge S1 t S2.

Theorem progress:
  forall S,
  safe S -> (exists r, final_state S r) \/ (exists t, exists S', step S t S').

End STRATEGY.


Definition semantics (p: program) :=
  let ge := globalenv p in
  Semantics_gen step (initial_state p) final_state ge ge.


Remark deref_loc_trace:
  forall ge ty m b ofs t v,
  deref_loc ge ty m b ofs t v ->
  match t with nil => True | ev :: nil => True | _ => False end.

Remark deref_loc_receptive:
  forall ge ty m b ofs ev1 t1 v ev2,
  deref_loc ge ty m b ofs (ev1 :: t1) v ->
  match_traces ge (ev1 :: nil) (ev2 :: nil) ->
  t1 = nil /\ exists v', deref_loc ge ty m b ofs (ev2 :: nil) v'.

Remark assign_loc_trace:
  forall ge ty m b ofs t v m',
  assign_loc ge ty m b ofs v t m' ->
  match t with nil => True | ev :: nil => output_event ev | _ => False end.

Remark assign_loc_receptive:
  forall ge ty m b ofs ev1 t1 v m' ev2,
  assign_loc ge ty m b ofs v (ev1 :: t1) m' ->
  match_traces ge (ev1 :: nil) (ev2 :: nil) ->
  ev1 :: t1 = ev2 :: nil.

Lemma semantics_strongly_receptive:
  forall p, strongly_receptive (semantics p).


Theorem strategy_simulation:
  forall p, backward_simulation (Csem.semantics p) (semantics p).

A big-step semantics for CompCert C implementing the reduction strategy.

Section BIGSTEP.

Variable ge: genv.


Inductive outcome: Type :=
   | Out_break: outcome (* terminated by break *)
   | Out_continue: outcome (* terminated by continue *)
   | Out_normal: outcome (* terminated normally *)
   | Out_return: option (val * type) -> outcome. (* terminated by return *)

Inductive out_normal_or_continue : outcome -> Prop :=
  | Out_normal_or_continue_N: out_normal_or_continue Out_normal
  | Out_normal_or_continue_C: out_normal_or_continue Out_continue.

Inductive out_break_or_return : outcome -> outcome -> Prop :=
  | Out_break_or_return_B: out_break_or_return Out_break Out_normal
  | Out_break_or_return_R: forall ov,
      out_break_or_return (Out_return ov) (Out_return ov).

Definition outcome_switch (out: outcome) : outcome :=
  match out with
  | Out_break => Out_normal
  | o => o
  end.

Definition outcome_result_value (out: outcome) (t: type) (v: val) (m: mem) : Prop :=
  match out, t with
  | Out_normal, Tvoid => v = Vundef
  | Out_return None, Tvoid => v = Vundef
  | Out_return (Some (v', ty')), ty => ty <> Tvoid /\ sem_cast v' ty' ty m = Some v
  | _, _ => False
  end.


Inductive eval_expression: env -> mem -> expr -> trace -> mem -> val -> Prop :=
  | eval_expression_intro: forall e m a t m' a' v,
      eval_expr e m RV a t m' a' -> eval_simple_rvalue ge e m' a' v ->
      eval_expression e m a t m' v

with eval_expr: env -> mem -> kind -> expr -> trace -> mem -> expr -> Prop :=
  | eval_val: forall e m v ty,
      eval_expr e m RV (Eval v ty) E0 m (Eval v ty)
  | eval_var: forall e m x ty,
      eval_expr e m LV (Evar x ty) E0 m (Evar x ty)
  | eval_field: forall e m a t m' a' f ty,
      eval_expr e m RV a t m' a' ->
      eval_expr e m LV (Efield a f ty) t m' (Efield a' f ty)
  | eval_valof: forall e m a t m' a' ty,
      type_is_volatile (typeof a) = false ->
      eval_expr e m LV a t m' a' ->
      eval_expr e m RV (Evalof a ty) t m' (Evalof a' ty)
  | eval_valof_volatile: forall e m a t1 m' a' ty b ofs t2 v,
      type_is_volatile (typeof a) = true ->
      eval_expr e m LV a t1 m' a' ->
      eval_simple_lvalue ge e m' a' b ofs ->
      deref_loc ge (typeof a) m' b ofs t2 v ->
      ty = typeof a ->
      eval_expr e m RV (Evalof a ty) (t1 ** t2) m' (Eval v ty)
  | eval_deref: forall e m a t m' a' ty,
      eval_expr e m RV a t m' a' ->
      eval_expr e m LV (Ederef a ty) t m' (Ederef a' ty)
  | eval_addrof: forall e m a t m' a' ty,
      eval_expr e m LV a t m' a' ->
      eval_expr e m RV (Eaddrof a ty) t m' (Eaddrof a' ty)
  | eval_unop: forall e m a t m' a' op ty,
      eval_expr e m RV a t m' a' ->
      eval_expr e m RV (Eunop op a ty) t m' (Eunop op a' ty)
  | eval_binop: forall e m a1 t1 m' a1' a2 t2 m'' a2' op ty,
      eval_expr e m RV a1 t1 m' a1' -> eval_expr e m' RV a2 t2 m'' a2' ->
      eval_expr e m RV (Ebinop op a1 a2 ty) (t1 ** t2) m'' (Ebinop op a1' a2' ty)
  | eval_cast: forall e m a t m' a' ty,
      eval_expr e m RV a t m' a' ->
      eval_expr e m RV (Ecast a ty) t m' (Ecast a' ty)
  | eval_seqand_true: forall e m a1 a2 ty t1 m' a1' v1 t2 m'' a2' v2 v,
      eval_expr e m RV a1 t1 m' a1' -> eval_simple_rvalue ge e m' a1' v1 ->
      bool_val v1 (typeof a1) m' = Some true ->
      eval_expr e m' RV a2 t2 m'' a2' -> eval_simple_rvalue ge e m'' a2' v2 ->
      sem_cast v2 (typeof a2) type_bool m'' = Some v ->
      eval_expr e m RV (Eseqand a1 a2 ty) (t1**t2) m'' (Eval v ty)
  | eval_seqand_false: forall e m a1 a2 ty t1 m' a1' v1,
      eval_expr e m RV a1 t1 m' a1' -> eval_simple_rvalue ge e m' a1' v1 ->
      bool_val v1 (typeof a1) m' = Some false ->
      eval_expr e m RV (Eseqand a1 a2 ty) t1 m' (Eval (Vint Int.zero) ty)
  | eval_seqor_false: forall e m a1 a2 ty t1 m' a1' v1 t2 m'' a2' v2 v,
      eval_expr e m RV a1 t1 m' a1' -> eval_simple_rvalue ge e m' a1' v1 ->
      bool_val v1 (typeof a1) m' = Some false ->
      eval_expr e m' RV a2 t2 m'' a2' -> eval_simple_rvalue ge e m'' a2' v2 ->
      sem_cast v2 (typeof a2) type_bool m'' = Some v ->
      eval_expr e m RV (Eseqor a1 a2 ty) (t1**t2) m'' (Eval v ty)
  | eval_seqor_true: forall e m a1 a2 ty t1 m' a1' v1,
      eval_expr e m RV a1 t1 m' a1' -> eval_simple_rvalue ge e m' a1' v1 ->
      bool_val v1 (typeof a1) m' = Some true ->
      eval_expr e m RV (Eseqor a1 a2 ty) t1 m' (Eval (Vint Int.one) ty)
  | eval_condition: forall e m a1 a2 a3 ty t1 m' a1' v1 t2 m'' a' v' b v,
      eval_expr e m RV a1 t1 m' a1' -> eval_simple_rvalue ge e m' a1' v1 ->
      bool_val v1 (typeof a1) m' = Some b ->
      eval_expr e m' RV (if b then a2 else a3) t2 m'' a' -> eval_simple_rvalue ge e m'' a' v' ->
      sem_cast v' (typeof (if b then a2 else a3)) ty m'' = Some v ->
      eval_expr e m RV (Econdition a1 a2 a3 ty) (t1**t2) m'' (Eval v ty)
  | eval_sizeof: forall e m ty' ty,
      eval_expr e m RV (Esizeof ty' ty) E0 m (Esizeof ty' ty)
  | eval_alignof: forall e m ty' ty,
      eval_expr e m RV (Ealignof ty' ty) E0 m (Ealignof ty' ty)
  | eval_assign: forall e m l r ty t1 m1 l' t2 m2 r' b ofs v v' t3 m3,
      eval_expr e m LV l t1 m1 l' -> eval_expr e m1 RV r t2 m2 r' ->
      eval_simple_lvalue ge e m2 l' b ofs ->
      eval_simple_rvalue ge e m2 r' v ->
      sem_cast v (typeof r) (typeof l) m2 = Some v' ->
      assign_loc ge (typeof l) m2 b ofs v' t3 m3 ->
      ty = typeof l ->
      eval_expr e m RV (Eassign l r ty) (t1**t2**t3) m3 (Eval v' ty)
  | eval_assignop: forall e m op l r tyres ty t1 m1 l' t2 m2 r' b ofs
                          v1 v2 v3 v4 t3 t4 m3,
      eval_expr e m LV l t1 m1 l' -> eval_expr e m1 RV r t2 m2 r' ->
      eval_simple_lvalue ge e m2 l' b ofs ->
      deref_loc ge (typeof l) m2 b ofs t3 v1 ->
      eval_simple_rvalue ge e m2 r' v2 ->
      sem_binary_operation ge op v1 (typeof l) v2 (typeof r) m2 = Some v3 ->
      sem_cast v3 tyres (typeof l) m2 = Some v4 ->
      assign_loc ge (typeof l) m2 b ofs v4 t4 m3 ->
      ty = typeof l ->
      eval_expr e m RV (Eassignop op l r tyres ty) (t1**t2**t3**t4) m3 (Eval v4 ty)
  | eval_postincr: forall e m id l ty t1 m1 l' b ofs v1 v2 v3 m2 t2 t3,
      eval_expr e m LV l t1 m1 l' ->
      eval_simple_lvalue ge e m1 l' b ofs ->
      deref_loc ge ty m1 b ofs t2 v1 ->
      sem_incrdecr ge id v1 ty m1 = Some v2 ->
      sem_cast v2 (incrdecr_type ty) ty m1 = Some v3 ->
      assign_loc ge ty m1 b ofs v3 t3 m2 ->
      ty = typeof l ->
      eval_expr e m RV (Epostincr id l ty) (t1**t2**t3) m2 (Eval v1 ty)
  | eval_comma: forall e m r1 r2 ty t1 m1 r1' v1 t2 m2 r2',
      eval_expr e m RV r1 t1 m1 r1' ->
      eval_simple_rvalue ge e m1 r1' v1 ->
      eval_expr e m1 RV r2 t2 m2 r2' ->
      ty = typeof r2 ->
      eval_expr e m RV (Ecomma r1 r2 ty) (t1**t2) m2 r2'
  | eval_call: forall e m rf rargs ty t1 m1 rf' t2 m2 rargs' vf vargs
                      targs tres cconv fd t3 m3 vres,
      eval_expr e m RV rf t1 m1 rf' -> eval_exprlist e m1 rargs t2 m2 rargs' ->
      eval_simple_rvalue ge e m2 rf' vf ->
      eval_simple_list ge e m2 rargs' targs vargs ->
      classify_fun (typeof rf) = fun_case_f targs tres cconv ->
      Genv.find_funct ge vf = Some fd ->
      type_of_fundef fd = Tfunction targs tres cconv ->
      eval_funcall m2 fd vargs t3 m3 vres ->
      eval_expr e m RV (Ecall rf rargs ty) (t1**t2**t3) m3 (Eval vres ty)

with eval_exprlist: env -> mem -> exprlist -> trace -> mem -> exprlist -> Prop :=
  | eval_nil: forall e m,
      eval_exprlist e m Enil E0 m Enil
  | eval_cons: forall e m a1 al t1 m1 a1' t2 m2 al',
      eval_expr e m RV a1 t1 m1 a1' -> eval_exprlist e m1 al t2 m2 al' ->
      eval_exprlist e m (Econs a1 al) (t1**t2) m2 (Econs a1' al')


with exec_stmt: env -> mem -> statement -> trace -> mem -> outcome -> Prop :=
  | exec_Sskip: forall e m,
      exec_stmt e m Sskip
               E0 m Out_normal
  | exec_Sdo: forall e m a t m' v,
      eval_expression e m a t m' v ->
      exec_stmt e m (Sdo a)
                t m' Out_normal
  | exec_Sseq_1: forall e m s1 s2 t1 m1 t2 m2 out,
      exec_stmt e m s1 t1 m1 Out_normal ->
      exec_stmt e m1 s2 t2 m2 out ->
      exec_stmt e m (Ssequence s1 s2)
                (t1 ** t2) m2 out
  | exec_Sseq_2: forall e m s1 s2 t1 m1 out,
      exec_stmt e m s1 t1 m1 out ->
      out <> Out_normal ->
      exec_stmt e m (Ssequence s1 s2)
                t1 m1 out
  | exec_Sifthenelse: forall e m a s1 s2 t1 m1 v1 t2 m2 b out,
      eval_expression e m a t1 m1 v1 ->
      bool_val v1 (typeof a) m1 = Some b ->
      exec_stmt e m1 (if b then s1 else s2) t2 m2 out ->
      exec_stmt e m (Sifthenelse a s1 s2)
                (t1**t2) m2 out
  | exec_Sreturn_none: forall e m,
      exec_stmt e m (Sreturn None)
               E0 m (Out_return None)
  | exec_Sreturn_some: forall e m a t m' v,
      eval_expression e m a t m' v ->
      exec_stmt e m (Sreturn (Some a))
                t m' (Out_return (Some(v, typeof a)))
  | exec_Sbreak: forall e m,
      exec_stmt e m Sbreak
               E0 m Out_break
  | exec_Scontinue: forall e m,
      exec_stmt e m Scontinue
               E0 m Out_continue
  | exec_Swhile_false: forall e m a s t m' v,
      eval_expression e m a t m' v ->
      bool_val v (typeof a) m' = Some false ->
      exec_stmt e m (Swhile a s)
                t m' Out_normal
  | exec_Swhile_stop: forall e m a s t1 m1 v t2 m2 out' out,
      eval_expression e m a t1 m1 v ->
      bool_val v (typeof a) m1 = Some true ->
      exec_stmt e m1 s t2 m2 out' ->
      out_break_or_return out' out ->
      exec_stmt e m (Swhile a s)
                (t1**t2) m2 out
  | exec_Swhile_loop: forall e m a s t1 m1 v t2 m2 out1 t3 m3 out,
      eval_expression e m a t1 m1 v ->
      bool_val v (typeof a) m1 = Some true ->
      exec_stmt e m1 s t2 m2 out1 ->
      out_normal_or_continue out1 ->
      exec_stmt e m2 (Swhile a s) t3 m3 out ->
      exec_stmt e m (Swhile a s)
                (t1 ** t2 ** t3) m3 out
  | exec_Sdowhile_false: forall e m s a t1 m1 out1 t2 m2 v,
      exec_stmt e m s t1 m1 out1 ->
      out_normal_or_continue out1 ->
      eval_expression e m1 a t2 m2 v ->
      bool_val v (typeof a) m2 = Some false ->
      exec_stmt e m (Sdowhile a s)
                (t1 ** t2) m2 Out_normal
  | exec_Sdowhile_stop: forall e m s a t m1 out1 out,
      exec_stmt e m s t m1 out1 ->
      out_break_or_return out1 out ->
      exec_stmt e m (Sdowhile a s)
                t m1 out
  | exec_Sdowhile_loop: forall e m s a t1 m1 out1 t2 m2 v t3 m3 out,
      exec_stmt e m s t1 m1 out1 ->
      out_normal_or_continue out1 ->
      eval_expression e m1 a t2 m2 v ->
      bool_val v (typeof a) m2 = Some true ->
      exec_stmt e m2 (Sdowhile a s) t3 m3 out ->
      exec_stmt e m (Sdowhile a s)
                (t1 ** t2 ** t3) m3 out
  | exec_Sfor_start: forall e m s a1 a2 a3 out m1 m2 t1 t2,
      exec_stmt e m a1 t1 m1 Out_normal ->
      exec_stmt e m1 (Sfor Sskip a2 a3 s) t2 m2 out ->
      exec_stmt e m (Sfor a1 a2 a3 s)
                (t1 ** t2) m2 out
  | exec_Sfor_false: forall e m s a2 a3 t m' v,
      eval_expression e m a2 t m' v ->
      bool_val v (typeof a2) m' = Some false ->
      exec_stmt e m (Sfor Sskip a2 a3 s)
                t m' Out_normal
  | exec_Sfor_stop: forall e m s a2 a3 t1 m1 v t2 m2 out1 out,
      eval_expression e m a2 t1 m1 v ->
      bool_val v (typeof a2) m1 = Some true ->
      exec_stmt e m1 s t2 m2 out1 ->
      out_break_or_return out1 out ->
      exec_stmt e m (Sfor Sskip a2 a3 s)
                (t1 ** t2) m2 out
  | exec_Sfor_loop: forall e m s a2 a3 t1 m1 v t2 m2 out1 t3 m3 t4 m4 out,
      eval_expression e m a2 t1 m1 v ->
      bool_val v (typeof a2) m1 = Some true ->
      exec_stmt e m1 s t2 m2 out1 ->
      out_normal_or_continue out1 ->
      exec_stmt e m2 a3 t3 m3 Out_normal ->
      exec_stmt e m3 (Sfor Sskip a2 a3 s) t4 m4 out ->
      exec_stmt e m (Sfor Sskip a2 a3 s)
                (t1 ** t2 ** t3 ** t4) m4 out
  | exec_Sswitch: forall e m a sl t1 m1 v n t2 m2 out,
      eval_expression e m a t1 m1 v ->
      sem_switch_arg v (typeof a) = Some n ->
      exec_stmt e m1 (seq_of_labeled_statement (select_switch n sl)) t2 m2 out ->
      exec_stmt e m (Sswitch a sl)
                (t1 ** t2) m2 (outcome_switch out)


with eval_funcall: mem -> fundef -> list val -> trace -> mem -> val -> Prop :=
  | eval_funcall_internal: forall m f vargs t e m1 m2 m3 out vres m4,
      list_norepet (var_names f.(fn_params) ++ var_names f.(fn_vars)) ->
      alloc_variables ge empty_env m (f.(fn_params) ++ f.(fn_vars)) e m1 ->
      bind_parameters ge e m1 f.(fn_params) vargs m2 ->
      exec_stmt e m2 f.(fn_body) t m3 out ->
      outcome_result_value out f.(fn_return) vres m3 ->
      Mem.free_list m3 (blocks_of_env ge e) = Some m4 ->
      eval_funcall m (Internal f) vargs t m4 vres
  | eval_funcall_external: forall m ef targs tres cconv vargs t vres m',
      external_call ef ge vargs m t vres m' ->
      eval_funcall m (External ef targs tres cconv) vargs t m' vres.

Scheme eval_expression_ind5 := Minimality for eval_expression Sort Prop
  with eval_expr_ind5 := Minimality for eval_expr Sort Prop
  with eval_exprlist_ind5 := Minimality for eval_exprlist Sort Prop
  with exec_stmt_ind5 := Minimality for exec_stmt Sort Prop
  with eval_funcall_ind5 := Minimality for eval_funcall Sort Prop.

Combined Scheme bigstep_induction from
        eval_expression_ind5, eval_expr_ind5, eval_exprlist_ind5,
        exec_stmt_ind5, eval_funcall_ind5.


CoInductive evalinf_expr: env -> mem -> kind -> expr -> traceinf -> Prop :=
  | evalinf_field: forall e m a t f ty,
      evalinf_expr e m RV a t ->
      evalinf_expr e m LV (Efield a f ty) t
  | evalinf_valof: forall e m a t ty,
      evalinf_expr e m LV a t ->
      evalinf_expr e m RV (Evalof a ty) t
  | evalinf_deref: forall e m a t ty,
      evalinf_expr e m RV a t ->
      evalinf_expr e m LV (Ederef a ty) t
  | evalinf_addrof: forall e m a t ty,
      evalinf_expr e m LV a t ->
      evalinf_expr e m RV (Eaddrof a ty) t
  | evalinf_unop: forall e m a t op ty,
      evalinf_expr e m RV a t ->
      evalinf_expr e m RV (Eunop op a ty) t
  | evalinf_binop_left: forall e m a1 t1 a2 op ty,
      evalinf_expr e m RV a1 t1 ->
      evalinf_expr e m RV (Ebinop op a1 a2 ty) t1
  | evalinf_binop_right: forall e m a1 t1 m' a1' a2 t2 op ty,
      eval_expr e m RV a1 t1 m' a1' -> evalinf_expr e m' RV a2 t2 ->
      evalinf_expr e m RV (Ebinop op a1 a2 ty) (t1 *** t2)
  | evalinf_cast: forall e m a t ty,
      evalinf_expr e m RV a t ->
      evalinf_expr e m RV (Ecast a ty) t
  | evalinf_seqand: forall e m a1 a2 ty t1,
      evalinf_expr e m RV a1 t1 ->
      evalinf_expr e m RV (Eseqand a1 a2 ty) t1
  | evalinf_seqand_2: forall e m a1 a2 ty t1 m' a1' v1 t2,
      eval_expr e m RV a1 t1 m' a1' -> eval_simple_rvalue ge e m' a1' v1 ->
      bool_val v1 (typeof a1) m' = Some true ->
      evalinf_expr e m' RV a2 t2 ->
      evalinf_expr e m RV (Eseqand a1 a2 ty) (t1***t2)
  | evalinf_seqor: forall e m a1 a2 ty t1,
      evalinf_expr e m RV a1 t1 ->
      evalinf_expr e m RV (Eseqor a1 a2 ty) t1
  | evalinf_seqor_2: forall e m a1 a2 ty t1 m' a1' v1 t2,
      eval_expr e m RV a1 t1 m' a1' -> eval_simple_rvalue ge e m' a1' v1 ->
      bool_val v1 (typeof a1) m' = Some false ->
      evalinf_expr e m' RV a2 t2 ->
      evalinf_expr e m RV (Eseqor a1 a2 ty) (t1***t2)
  | evalinf_condition: forall e m a1 a2 a3 ty t1,
      evalinf_expr e m RV a1 t1 ->
      evalinf_expr e m RV (Econdition a1 a2 a3 ty) t1
  | evalinf_condition_2: forall e m a1 a2 a3 ty t1 m' a1' v1 t2 b,
      eval_expr e m RV a1 t1 m' a1' -> eval_simple_rvalue ge e m' a1' v1 ->
      bool_val v1 (typeof a1) m' = Some b ->
      evalinf_expr e m' RV (if b then a2 else a3) t2 ->
      evalinf_expr e m RV (Econdition a1 a2 a3 ty) (t1***t2)
  | evalinf_assign_left: forall e m a1 t1 a2 ty,
      evalinf_expr e m LV a1 t1 ->
      evalinf_expr e m RV (Eassign a1 a2 ty) t1
  | evalinf_assign_right: forall e m a1 t1 m' a1' a2 t2 ty,
      eval_expr e m LV a1 t1 m' a1' -> evalinf_expr e m' RV a2 t2 ->
      evalinf_expr e m RV (Eassign a1 a2 ty) (t1 *** t2)
  | evalinf_assignop_left: forall e m a1 t1 a2 op tyres ty,
      evalinf_expr e m LV a1 t1 ->
      evalinf_expr e m RV (Eassignop op a1 a2 tyres ty) t1
  | evalinf_assignop_right: forall e m a1 t1 m' a1' a2 t2 op tyres ty,
      eval_expr e m LV a1 t1 m' a1' -> evalinf_expr e m' RV a2 t2 ->
      evalinf_expr e m RV (Eassignop op a1 a2 tyres ty) (t1 *** t2)
  | evalinf_postincr: forall e m a t id ty,
      evalinf_expr e m LV a t ->
      evalinf_expr e m RV (Epostincr id a ty) t
  | evalinf_comma_left: forall e m a1 t1 a2 ty,
      evalinf_expr e m RV a1 t1 ->
      evalinf_expr e m RV (Ecomma a1 a2 ty) t1
  | evalinf_comma_right: forall e m a1 t1 m1 a1' v1 a2 t2 ty,
      eval_expr e m RV a1 t1 m1 a1' -> eval_simple_rvalue ge e m1 a1' v1 ->
      ty = typeof a2 ->
      evalinf_expr e m1 RV a2 t2 ->
      evalinf_expr e m RV (Ecomma a1 a2 ty) (t1 *** t2)
  | evalinf_call_left: forall e m a1 t1 a2 ty,
      evalinf_expr e m RV a1 t1 ->
      evalinf_expr e m RV (Ecall a1 a2 ty) t1
  | evalinf_call_right: forall e m a1 t1 m1 a1' a2 t2 ty,
      eval_expr e m RV a1 t1 m1 a1' ->
      evalinf_exprlist e m1 a2 t2 ->
      evalinf_expr e m RV (Ecall a1 a2 ty) (t1 *** t2)
  | evalinf_call: forall e m rf rargs ty t1 m1 rf' t2 m2 rargs' vf vargs
                      targs tres cconv fd t3,
      eval_expr e m RV rf t1 m1 rf' -> eval_exprlist e m1 rargs t2 m2 rargs' ->
      eval_simple_rvalue ge e m2 rf' vf ->
      eval_simple_list ge e m2 rargs' targs vargs ->
      classify_fun (typeof rf) = fun_case_f targs tres cconv ->
      Genv.find_funct ge vf = Some fd ->
      type_of_fundef fd = Tfunction targs tres cconv ->
      evalinf_funcall m2 fd vargs t3 ->
      evalinf_expr e m RV (Ecall rf rargs ty) (t1***t2***t3)

with evalinf_exprlist: env -> mem -> exprlist -> traceinf -> Prop :=
  | evalinf_cons_left: forall e m a1 al t1,
      evalinf_expr e m RV a1 t1 ->
      evalinf_exprlist e m (Econs a1 al) t1
  | evalinf_cons_right: forall e m a1 al t1 m1 a1' t2,
      eval_expr e m RV a1 t1 m1 a1' -> evalinf_exprlist e m1 al t2 ->
      evalinf_exprlist e m (Econs a1 al) (t1***t2)


with execinf_stmt: env -> mem -> statement -> traceinf -> Prop :=
  | execinf_Sdo: forall e m a t,
      evalinf_expr e m RV a t ->
      execinf_stmt e m (Sdo a) t
  | execinf_Sseq_1: forall e m s1 s2 t1,
      execinf_stmt e m s1 t1 ->
      execinf_stmt e m (Ssequence s1 s2) t1
  | execinf_Sseq_2: forall e m s1 s2 t1 m1 t2,
      exec_stmt e m s1 t1 m1 Out_normal ->
      execinf_stmt e m1 s2 t2 ->
      execinf_stmt e m (Ssequence s1 s2) (t1***t2)
  | execinf_Sifthenelse_test: forall e m a s1 s2 t1,
      evalinf_expr e m RV a t1 ->
      execinf_stmt e m (Sifthenelse a s1 s2) t1
  | execinf_Sifthenelse: forall e m a s1 s2 t1 m1 v1 t2 b,
      eval_expression e m a t1 m1 v1 ->
      bool_val v1 (typeof a) m1 = Some b ->
      execinf_stmt e m1 (if b then s1 else s2) t2 ->
      execinf_stmt e m (Sifthenelse a s1 s2) (t1***t2)
  | execinf_Sreturn_some: forall e m a t,
      evalinf_expr e m RV a t ->
      execinf_stmt e m (Sreturn (Some a)) t
  | execinf_Swhile_test: forall e m a s t1,
      evalinf_expr e m RV a t1 ->
      execinf_stmt e m (Swhile a s) t1
  | execinf_Swhile_body: forall e m a s t1 m1 v t2,
      eval_expression e m a t1 m1 v ->
      bool_val v (typeof a) m1 = Some true ->
      execinf_stmt e m1 s t2 ->
      execinf_stmt e m (Swhile a s) (t1***t2)
  | execinf_Swhile_loop: forall e m a s t1 m1 v t2 m2 out1 t3,
      eval_expression e m a t1 m1 v ->
      bool_val v (typeof a) m1 = Some true ->
      exec_stmt e m1 s t2 m2 out1 ->
      out_normal_or_continue out1 ->
      execinf_stmt e m2 (Swhile a s) t3 ->
      execinf_stmt e m (Swhile a s) (t1***t2***t3)
  | execinf_Sdowhile_body: forall e m s a t1,
      execinf_stmt e m s t1 ->
      execinf_stmt e m (Sdowhile a s) t1
  | execinf_Sdowhile_test: forall e m s a t1 m1 out1 t2,
      exec_stmt e m s t1 m1 out1 ->
      out_normal_or_continue out1 ->
      evalinf_expr e m1 RV a t2 ->
      execinf_stmt e m (Sdowhile a s) (t1***t2)
  | execinf_Sdowhile_loop: forall e m s a t1 m1 out1 t2 m2 v t3,
      exec_stmt e m s t1 m1 out1 ->
      out_normal_or_continue out1 ->
      eval_expression e m1 a t2 m2 v ->
      bool_val v (typeof a) m2 = Some true ->
      execinf_stmt e m2 (Sdowhile a s) t3 ->
      execinf_stmt e m (Sdowhile a s) (t1***t2***t3)
  | execinf_Sfor_start_1: forall e m s a1 a2 a3 t1,
      execinf_stmt e m a1 t1 ->
      execinf_stmt e m (Sfor a1 a2 a3 s) t1
  | execinf_Sfor_start_2: forall e m s a1 a2 a3 m1 t1 t2,
      exec_stmt e m a1 t1 m1 Out_normal -> a1 <> Sskip ->
      execinf_stmt e m1 (Sfor Sskip a2 a3 s) t2 ->
      execinf_stmt e m (Sfor a1 a2 a3 s) (t1***t2)
  | execinf_Sfor_test: forall e m s a2 a3 t,
      evalinf_expr e m RV a2 t ->
      execinf_stmt e m (Sfor Sskip a2 a3 s) t
  | execinf_Sfor_body: forall e m s a2 a3 t1 m1 v t2,
      eval_expression e m a2 t1 m1 v ->
      bool_val v (typeof a2) m1 = Some true ->
      execinf_stmt e m1 s t2 ->
      execinf_stmt e m (Sfor Sskip a2 a3 s) (t1***t2)
  | execinf_Sfor_next: forall e m s a2 a3 t1 m1 v t2 m2 out1 t3,
      eval_expression e m a2 t1 m1 v ->
      bool_val v (typeof a2) m1 = Some true ->
      exec_stmt e m1 s t2 m2 out1 ->
      out_normal_or_continue out1 ->
      execinf_stmt e m2 a3 t3 ->
      execinf_stmt e m (Sfor Sskip a2 a3 s) (t1***t2***t3)
  | execinf_Sfor_loop: forall e m s a2 a3 t1 m1 v t2 m2 out1 t3 m3 t4,
      eval_expression e m a2 t1 m1 v ->
      bool_val v (typeof a2) m1 = Some true ->
      exec_stmt e m1 s t2 m2 out1 ->
      out_normal_or_continue out1 ->
      exec_stmt e m2 a3 t3 m3 Out_normal ->
      execinf_stmt e m3 (Sfor Sskip a2 a3 s) t4 ->
      execinf_stmt e m (Sfor Sskip a2 a3 s) (t1***t2***t3***t4)
  | execinf_Sswitch_expr: forall e m a sl t1,
      evalinf_expr e m RV a t1 ->
      execinf_stmt e m (Sswitch a sl) t1
  | execinf_Sswitch_body: forall e m a sl t1 m1 v n t2,
      eval_expression e m a t1 m1 v ->
      sem_switch_arg v (typeof a) = Some n ->
      execinf_stmt e m1 (seq_of_labeled_statement (select_switch n sl)) t2 ->
      execinf_stmt e m (Sswitch a sl) (t1***t2)


with evalinf_funcall: mem -> fundef -> list val -> traceinf -> Prop :=
  | evalinf_funcall_internal: forall m f vargs t e m1 m2,
      list_norepet (var_names f.(fn_params) ++ var_names f.(fn_vars)) ->
      alloc_variables ge empty_env m (f.(fn_params) ++ f.(fn_vars)) e m1 ->
      bind_parameters ge e m1 f.(fn_params) vargs m2 ->
      execinf_stmt e m2 f.(fn_body) t ->
      evalinf_funcall m (Internal f) vargs t.

Implication from big-step semantics to transition semantics

Inductive outcome_state_match
       (e: env) (m: mem) (f: function) (k: cont): outcome -> state -> Prop :=
  | osm_normal:
      outcome_state_match e m f k Out_normal (State f Sskip k e m)
  | osm_break:
      outcome_state_match e m f k Out_break (State f Sbreak k e m)
  | osm_continue:
      outcome_state_match e m f k Out_continue (State f Scontinue k e m)
  | osm_return_none: forall k',
      call_cont k' = call_cont k ->
      outcome_state_match e m f k
        (Out_return None) (State f (Sreturn None) k' e m)
  | osm_return_some: forall v ty k',
      call_cont k' = call_cont k ->
      outcome_state_match e m f k
        (Out_return (Some (v, ty))) (ExprState f (Eval v ty) (Kreturn k') e m).

Lemma is_call_cont_call_cont:
  forall k, is_call_cont k -> call_cont k = k.

Lemma leftcontext_compose:
  forall k2 k3 C2, leftcontext k2 k3 C2 ->
  forall k1 C1, leftcontext k1 k2 C1 ->
  leftcontext k1 k3 (fun x => C2(C1 x))
with leftcontextlist_compose:
  forall k2 C2, leftcontextlist k2 C2 ->
  forall k1 C1, leftcontext k1 k2 C1 ->
  leftcontextlist k1 (fun x => C2(C1 x)).

Lemma exprlist_app_leftcontext:
  forall rl1 rl2,
  simplelist rl1 = true -> leftcontextlist RV (fun x => exprlist_app rl1 (Econs x rl2)).

Lemma exprlist_app_simple:
  forall rl1 rl2,
  simplelist (exprlist_app rl1 rl2) = simplelist rl1 && simplelist rl2.

Lemma bigstep_to_steps:
  (forall e m a t m' v,
   eval_expression e m a t m' v ->
   forall f k,
   star step ge (ExprState f a k e m) t (ExprState f (Eval v (typeof a)) k e m'))
/\(forall e m K a t m' a',
   eval_expr e m K a t m' a' ->
   forall C f k, leftcontext K RV C ->
   simple a' = true /\ typeof a' = typeof a /\
   star step ge (ExprState f (C a) k e m) t (ExprState f (C a') k e m'))
/\(forall e m al t m' al',
   eval_exprlist e m al t m' al' ->
   forall a1 al2 ty C f k, leftcontext RV RV C -> simple a1 = true -> simplelist al2 = true ->
   simplelist al' = true /\
   star step ge (ExprState f (C (Ecall a1 (exprlist_app al2 al) ty)) k e m)
              t (ExprState f (C (Ecall a1 (exprlist_app al2 al') ty)) k e m'))
/\(forall e m s t m' out,
   exec_stmt e m s t m' out ->
   forall f k,
   exists S,
   star step ge (State f s k e m) t S /\ outcome_state_match e m' f k out S)
/\(forall m fd args t m' res,
   eval_funcall m fd args t m' res ->
   forall k,
   is_call_cont k ->
   star step ge (Callstate fd args k m) t (Returnstate res k m')).

Lemma eval_expression_to_steps:
   forall e m a t m' v,
   eval_expression e m a t m' v ->
   forall f k,
   star step ge (ExprState f a k e m) t (ExprState f (Eval v (typeof a)) k e m').
Proof (proj1 bigstep_to_steps).

Lemma eval_expr_to_steps:
   forall e m K a t m' a',
   eval_expr e m K a t m' a' ->
   forall C f k, leftcontext K RV C ->
   simple a' = true /\ typeof a' = typeof a /\
   star step ge (ExprState f (C a) k e m) t (ExprState f (C a') k e m').
Proof (proj1 (proj2 bigstep_to_steps)).

Lemma eval_exprlist_to_steps:
   forall e m al t m' al',
   eval_exprlist e m al t m' al' ->
   forall a1 al2 ty C f k, leftcontext RV RV C -> simple a1 = true -> simplelist al2 = true ->
   simplelist al' = true /\
   star step ge (ExprState f (C (Ecall a1 (exprlist_app al2 al) ty)) k e m)
              t (ExprState f (C (Ecall a1 (exprlist_app al2 al') ty)) k e m').
Proof (proj1 (proj2 (proj2 bigstep_to_steps))).

Lemma exec_stmt_to_steps:
   forall e m s t m' out,
   exec_stmt e m s t m' out ->
   forall f k,
   exists S,
   star step ge (State f s k e m) t S /\ outcome_state_match e m' f k out S.
Proof (proj1 (proj2 (proj2 (proj2 bigstep_to_steps)))).

Lemma eval_funcall_to_steps:
  forall m fd args t m' res,
  eval_funcall m fd args t m' res ->
  forall k,
  is_call_cont k ->
  star step ge (Callstate fd args k m) t (Returnstate res k m').
Proof (proj2 (proj2 (proj2 (proj2 bigstep_to_steps)))).

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

with esizelist (el: exprlist) : nat :=
  match el with
  | Enil => O
  | Econs r1 rl2 => S(esize r1 + esizelist rl2)%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 evalinf_funcall_steps:
  forall m fd args t k,
  evalinf_funcall m fd args t ->
  forever_N step lt ge O (Callstate fd args k m) t.

End BIGSTEP.

Whole-program behaviors, big-step style.

Inductive bigstep_program_terminates (p: program): trace -> int -> Prop :=
  | bigstep_program_terminates_intro: forall b f m0 m1 t r,
      let ge := globalenv p in
      Genv.init_mem p = Some m0 ->
      Genv.find_symbol ge p.(prog_main) = Some b ->
      Genv.find_funct_ptr ge b = Some f ->
      type_of_fundef f = Tfunction Tnil type_int32s cc_default ->
      eval_funcall ge m0 f nil t m1 (Vint r) ->
      bigstep_program_terminates p t r.

Inductive bigstep_program_diverges (p: program): traceinf -> Prop :=
  | bigstep_program_diverges_intro: forall b f m0 t,
      let ge := globalenv p in
      Genv.init_mem p = Some m0 ->
      Genv.find_symbol ge p.(prog_main) = Some b ->
      Genv.find_funct_ptr ge b = Some f ->
      type_of_fundef f = Tfunction Tnil type_int32s cc_default ->
      evalinf_funcall ge m0 f nil t ->
      bigstep_program_diverges p t.

Definition bigstep_semantics (p: program) :=
  Bigstep_semantics (bigstep_program_terminates p) (bigstep_program_diverges p).

Theorem bigstep_semantics_sound:
  forall p, bigstep_sound (bigstep_semantics p) (semantics p).