Require Import Axioms Classical.
Require Import String Coqlib Decidableplus.
Require Import Errors Maps Integers Floats.
Require Import AST Values Memory Events Globalenvs Determinism.
Require Import Ctypes Cop Csyntax Csem.
Require Cstrategy.

Local Open Scope string_scope.
Local Open Scope list_scope.


Notation "'do' X <- A ; B" := (match A with Some X => B | None => None end)
  (at level 200, X ident, A at level 100, B at level 200)
  : option_monad_scope.

Notation "'do' X , Y <- A ; B" := (match A with Some (X, Y) => B | None => None end)
  (at level 200, X ident, Y ident, A at level 100, B at level 200)
  : option_monad_scope.

Notation "'do' X , Y , Z <- A ; B" := (match A with Some (X, Y, Z) => B | None => None end)
  (at level 200, X ident, Y ident, Z ident, A at level 100, B at level 200)
  : option_monad_scope.

Notation " 'check' A ; B" := (if A then B else None)
  (at level 200, A at level 100, B at level 200)
  : option_monad_scope.

Notation "'do' X <- A ; B" := (match A with Some X => B | None => nil end)
  (at level 200, X ident, A at level 100, B at level 200)
  : list_monad_scope.

Notation " 'check' A ; B" := (if A then B else nil)
  (at level 200, A at level 100, B at level 200)
  : list_monad_scope.

Definition is_val (a: expr) : option (val * type) :=
  match a with
  | Eval v ty => Some(v, ty)
  | _ => None
  end.

Lemma is_val_inv:
  forall a v ty, is_val a = Some(v, ty) -> a = Eval v ty.

Definition is_loc (a: expr) : option (block * ptrofs * type) :=
  match a with
  | Eloc b ofs ty => Some(b, ofs, ty)
  | _ => None
  end.

Lemma is_loc_inv:
  forall a b ofs ty, is_loc a = Some(b, ofs, ty) -> a = Eloc b ofs ty.

Local Open Scope option_monad_scope.

Fixpoint is_val_list (al: exprlist) : option (list (val * type)) :=
  match al with
  | Enil => Some nil
  | Econs a1 al => do vt1 <- is_val a1; do vtl <- is_val_list al; Some(vt1::vtl)
  end.

Definition is_skip (s: statement) : {s = Sskip} + {s <> Sskip}.

Section WITHEXTERNALCALLS.
Context `{external_calls_prf: ExternalCalls}.

Events, volatile memory accesses, and external functions.

Section EXEC.

Variable ge: genv.

Definition eventval_of_val (v: val) (t: typ) : option eventval :=
  match v with
  | Vint i => check (typ_eq t AST.Tint); Some (EVint i)
  | Vfloat f => check (typ_eq t AST.Tfloat); Some (EVfloat f)
  | Vsingle f => check (typ_eq t AST.Tsingle); Some (EVsingle f)
  | Vlong n => check (typ_eq t AST.Tlong); Some (EVlong n)
  | Vptr b ofs =>
      do id <- Genv.invert_symbol ge b;
      check (Genv.public_symbol ge id);
      check (typ_eq t AST.Tptr);
      Some (EVptr_global id ofs)
  | _ => None
  end.

Fixpoint list_eventval_of_val (vl: list val) (tl: list typ) : option (list eventval) :=
  match vl, tl with
  | nil, nil => Some nil
  | v1::vl, t1::tl =>
      do ev1 <- eventval_of_val v1 t1;
      do evl <- list_eventval_of_val vl tl;
      Some (ev1 :: evl)
  | _, _ => None
  end.

Definition val_of_eventval (ev: eventval) (t: typ) : option val :=
  match ev with
  | EVint i => check (typ_eq t AST.Tint); Some (Vint i)
  | EVfloat f => check (typ_eq t AST.Tfloat); Some (Vfloat f)
  | EVsingle f => check (typ_eq t AST.Tsingle); Some (Vsingle f)
  | EVlong n => check (typ_eq t AST.Tlong); Some (Vlong n)
  | EVptr_global id ofs =>
      check (Genv.public_symbol ge id);
      check (typ_eq t AST.Tptr);
      do b <- Genv.find_symbol ge id;
      Some (Vptr b ofs)
  end.

Ltac mydestr :=
  match goal with
  | [ |- None = Some _ -> _ ] => let X := fresh "X" in intro X; discriminate
  | [ |- Some _ = Some _ -> _ ] => let X := fresh "X" in intro X; inv X
  | [ |- match ?x with Some _ => _ | None => _ end = Some _ -> _ ] => destruct x eqn:?; mydestr
  | [ |- match ?x with true => _ | false => _ end = Some _ -> _ ] => destruct x eqn:?; mydestr
  | [ |- match ?x with left _ => _ | right _ => _ end = Some _ -> _ ] => destruct x; mydestr
  | _ => idtac
  end.

Lemma eventval_of_val_sound:
  forall v t ev, eventval_of_val v t = Some ev -> eventval_match ge ev t v.

Lemma eventval_of_val_complete:
  forall ev t v, eventval_match ge ev t v -> eventval_of_val v t = Some ev.

Lemma list_eventval_of_val_sound:
  forall vl tl evl, list_eventval_of_val vl tl = Some evl -> eventval_list_match ge evl tl vl.

Lemma list_eventval_of_val_complete:
  forall evl tl vl, eventval_list_match ge evl tl vl -> list_eventval_of_val vl tl = Some evl.

Lemma val_of_eventval_sound:
  forall ev t v, val_of_eventval ev t = Some v -> eventval_match ge ev t v.

Lemma val_of_eventval_complete:
  forall ev t v, eventval_match ge ev t v -> val_of_eventval ev t = Some v.


Definition do_volatile_load (w: world) (chunk: memory_chunk) (m: mem) (b: block) (ofs: ptrofs)
                             : option (world * trace * val) :=
  match Genv.block_is_volatile ge b with
  | Some true =>
    do id <- Genv.invert_symbol ge b;
      match nextworld_vload w chunk id ofs with
      | None => None
      | Some(res, w') =>
        do vres <- val_of_eventval res (type_of_chunk chunk);
          Some(w', Event_vload chunk id ofs res :: nil, Val.load_result chunk vres)
      end
  | Some false =>
    do v <- Mem.load chunk m b (Ptrofs.unsigned ofs);
      Some(w, E0, v)
  | None => None
  end.

Definition do_volatile_store (w: world) (chunk: memory_chunk) (m: mem) (b: block) (ofs: ptrofs) (v: val)
                             : option (world * trace * mem) :=
  match Genv.block_is_volatile ge b with
    Some true =>
    do id <- Genv.invert_symbol ge b;
      do ev <- eventval_of_val (Val.load_result chunk v) (type_of_chunk chunk);
      do w' <- nextworld_vstore w chunk id ofs ev;
      Some(w', Event_vstore chunk id ofs ev :: nil, m)
  | Some false =>
    do m' <- Mem.store chunk m b (Ptrofs.unsigned ofs) v;
      Some(w, E0, m')
  | None => None
  end.

Lemma do_volatile_load_sound:
  forall w chunk m b ofs w' t v,
  do_volatile_load w chunk m b ofs = Some(w', t, v) ->
  volatile_load ge chunk m b ofs t v /\ possible_trace w t w'.

Lemma do_volatile_load_complete:
  forall w chunk m b ofs w' t v,
  volatile_load ge chunk m b ofs t v -> possible_trace w t w' ->
  do_volatile_load w chunk m b ofs = Some(w', t, v).

Lemma do_volatile_store_sound:
  forall w chunk m b ofs v w' t m',
  do_volatile_store w chunk m b ofs v = Some(w', t, m') ->
  volatile_store ge chunk m b ofs v t m' /\ possible_trace w t w'.

Lemma do_volatile_store_complete:
  forall w chunk m b ofs v w' t m',
  volatile_store ge chunk m b ofs v t m' -> possible_trace w t w' ->
  do_volatile_store w chunk m b ofs v = Some(w', t, m').


Definition do_deref_loc (w: world) (ty: type) (m: mem) (b: block) (ofs: ptrofs) : option (world * trace * val) :=
  match access_mode ty with
  | By_value chunk =>
      match type_is_volatile ty with
      | false => do v <- Mem.loadv chunk m (Vptr b ofs); Some(w, E0, v)
      | true => do_volatile_load w chunk m b ofs
      end
  | By_reference => Some(w, E0, Vptr b ofs)
  | By_copy => Some(w, E0, Vptr b ofs)
  | _ => None
  end.

Definition assign_copy_ok (ty: type) (b: block) (ofs: ptrofs) (b': block) (ofs': ptrofs) : Prop :=
  (alignof_blockcopy ge ty | Ptrofs.unsigned ofs') /\ (alignof_blockcopy ge ty | Ptrofs.unsigned ofs) /\
  (b' <> b \/ Ptrofs.unsigned ofs' = Ptrofs.unsigned ofs
           \/ Ptrofs.unsigned ofs' + sizeof ge ty <= Ptrofs.unsigned ofs
           \/ Ptrofs.unsigned ofs + sizeof ge ty <= Ptrofs.unsigned ofs').

Remark check_assign_copy:
  forall (ty: type) (b: block) (ofs: ptrofs) (b': block) (ofs': ptrofs),
  { assign_copy_ok ty b ofs b' ofs' } + {~ assign_copy_ok ty b ofs b' ofs' }.

Definition do_assign_loc (w: world) (ty: type) (m: mem) (b: block) (ofs: ptrofs) (v: val): option (world * trace * mem) :=
  match access_mode ty with
  | By_value chunk =>
      match type_is_volatile ty with
      | false => do m' <- Mem.storev chunk m (Vptr b ofs) v; Some(w, E0, m')
      | true => do_volatile_store w chunk m b ofs v
      end
  | By_copy =>
      match v with
      | Vptr b' ofs' =>
          if check_assign_copy ty b ofs b' ofs' then
            do bytes <- Mem.loadbytes m b' (Ptrofs.unsigned ofs') (sizeof ge ty);
            do m' <- Mem.storebytes m b (Ptrofs.unsigned ofs) bytes;
            Some(w, E0, m')
          else None
      | _ => None
      end
  | _ => None
  end.

Lemma do_deref_loc_sound:
  forall w ty m b ofs w' t v,
  do_deref_loc w ty m b ofs = Some(w', t, v) ->
  deref_loc ge ty m b ofs t v /\ possible_trace w t w'.

Lemma do_deref_loc_complete:
  forall w ty m b ofs w' t v,
  deref_loc ge ty m b ofs t v -> possible_trace w t w' ->
  do_deref_loc w ty m b ofs = Some(w', t, v).

Lemma do_assign_loc_sound:
  forall w ty m b ofs v w' t m',
  do_assign_loc w ty m b ofs v = Some(w', t, m') ->
  assign_loc ge ty m b ofs v t m' /\ possible_trace w t w'.

Lemma do_assign_loc_complete:
  forall w ty m b ofs v w' t m',
  assign_loc ge ty m b ofs v t m' -> possible_trace w t w' ->
  do_assign_loc w ty m b ofs v = Some(w', t, m').


Variable do_external_function:
  string -> signature -> Senv.t -> world -> list val -> mem -> option (world * trace * val * mem).

Hypothesis do_external_function_sound:
  forall id sg ge vargs m t vres m' w w',
  do_external_function id sg ge w vargs m = Some(w', t, vres, m') ->
  external_functions_sem id sg ge vargs m t vres m' /\ possible_trace w t w'.

Hypothesis do_external_function_complete:
  forall id sg ge vargs m t vres m' w w',
  external_functions_sem id sg ge vargs m t vres m' ->
  possible_trace w t w' ->
  do_external_function id sg ge w vargs m = Some(w', t, vres, m').

Variable do_builtin_function:
  string -> signature -> Senv.t -> world -> list val -> mem -> option (world * trace * val * mem).

Hypothesis do_builtin_function_sound:
  forall id sg ge vargs m t vres m' w w',
  do_builtin_function id sg ge w vargs m = Some(w', t, vres, m') ->
  builtin_functions_sem id sg ge vargs m t vres m' /\ possible_trace w t w'.

Hypothesis do_builtin_function_complete:
  forall id sg ge vargs m t vres m' w w',
  builtin_functions_sem id sg ge vargs m t vres m' ->
  possible_trace w t w' ->
  do_builtin_function id sg ge w vargs m = Some(w', t, vres, m').

Variable do_runtime_function:
  string -> signature -> Senv.t -> world -> list val -> mem -> option (world * trace * val * mem).

Hypothesis do_runtime_function_sound:
  forall id sg ge vargs m t vres m' w w',
  do_runtime_function id sg ge w vargs m = Some(w', t, vres, m') ->
  runtime_functions_sem id sg ge vargs m t vres m' /\ possible_trace w t w'.

Hypothesis do_runtime_function_complete:
  forall id sg ge vargs m t vres m' w w',
  runtime_functions_sem id sg ge vargs m t vres m' ->
  possible_trace w t w' ->
  do_runtime_function id sg ge w vargs m = Some(w', t, vres, m').

Variable do_inline_assembly:
  string -> signature -> Senv.t -> world -> list val -> mem -> option (world * trace * val * mem).

Hypothesis do_inline_assembly_sound:
  forall txt sg ge vargs m t vres m' w w',
  do_inline_assembly txt sg ge w vargs m = Some(w', t, vres, m') ->
  inline_assembly_sem txt sg ge vargs m t vres m' /\ possible_trace w t w'.

Hypothesis do_inline_assembly_complete:
  forall txt sg ge vargs m t vres m' w w',
  inline_assembly_sem txt sg ge vargs m t vres m' ->
  possible_trace w t w' ->
  do_inline_assembly txt sg ge w vargs m = Some(w', t, vres, m').

Definition do_ef_volatile_load (chunk: memory_chunk)
       (w: world) (vargs: list val) (m: mem) : option (world * trace * val * mem) :=
  match vargs with
  | Vptr b ofs :: nil => do w',t,v <- do_volatile_load w chunk m b ofs; Some(w',t,v,m)
  | _ => None
  end.

Definition do_ef_volatile_store (chunk: memory_chunk)
       (w: world) (vargs: list val) (m: mem) : option (world * trace * val * mem) :=
  match vargs with
  | Vptr b ofs :: v :: nil => do w',t,m' <- do_volatile_store w chunk m b ofs v; Some(w',t,Vundef,m')
  | _ => None
  end.

Definition do_ef_volatile_load_global (chunk: memory_chunk) (id: ident) (ofs: ptrofs)
       (w: world) (vargs: list val) (m: mem) : option (world * trace * val * mem) :=
  do b <- Genv.find_symbol ge id; do_ef_volatile_load chunk w (Vptr b ofs :: vargs) m.

Definition do_ef_volatile_store_global (chunk: memory_chunk) (id: ident) (ofs: ptrofs)
       (w: world) (vargs: list val) (m: mem) : option (world * trace * val * mem) :=
  do b <- Genv.find_symbol ge id; do_ef_volatile_store chunk w (Vptr b ofs :: vargs) m.

Definition do_alloc_size (v: val) : option ptrofs :=
  match v with
  | Vint n => if Archi.ptr64 then None else Some (Ptrofs.of_int n)
  | Vlong n => if Archi.ptr64 then Some (Ptrofs.of_int64 n) else None
  | _ => None
  end.

Definition do_ef_malloc
       (w: world) (vargs: list val) (m: mem) : option (world * trace * val * mem) :=
  match vargs with
  | v :: nil =>
      do sz <- do_alloc_size v;
      let (m', b) := Mem.alloc m (- size_chunk Mptr) (Ptrofs.unsigned sz) in
      do m'' <- Mem.store Mptr m' b (- size_chunk Mptr) v;
      Some(w, E0, Vptr b Ptrofs.zero, m'')
  | _ => None
  end.

Definition dec_not {A B}: {A}+{B} -> {B}+{A}.

Definition do_ef_free
       (w: world) (vargs: list val) (m: mem) : option (world * trace * val * mem) :=
  match vargs with
  | Vptr b lo :: nil =>
      do vsz <- Mem.load Mptr m b (Ptrofs.unsigned lo - size_chunk Mptr);
      do sz <- do_alloc_size vsz;
      check (zlt 0 (Ptrofs.unsigned sz));
      check (dec_not (in_stack_dec (Mem.stack m) b));
      do m' <- Mem.free m b (Ptrofs.unsigned lo - size_chunk Mptr) (Ptrofs.unsigned lo + Ptrofs.unsigned sz);
      Some(w, E0, Vundef, m')
  | _ => None
  end.

Definition memcpy_args_ok
  (sz al: Z) (bdst: block) (odst: Z) (bsrc: block) (osrc: Z) : Prop :=
      (al = 1 \/ al = 2 \/ al = 4 \/ al = 8)
   /\ sz >= 0 /\ (al | sz)
   /\ (sz > 0 -> (al | osrc))
   /\ (sz > 0 -> (al | odst))
   /\ (bsrc <> bdst \/ osrc = odst \/ osrc + sz <= odst \/ odst + sz <= osrc).

Definition do_ef_memcpy (sz al: Z)
       (w: world) (vargs: list val) (m: mem) : option (world * trace * val * mem) :=
  match vargs with
  | Vptr bdst odst :: Vptr bsrc osrc :: nil =>
      if decide (memcpy_args_ok sz al bdst (Ptrofs.unsigned odst) bsrc (Ptrofs.unsigned osrc)) then
        do bytes <- Mem.loadbytes m bsrc (Ptrofs.unsigned osrc) sz;
          do m' <- Mem.storebytes m bdst (Ptrofs.unsigned odst) bytes;
          Some(w, E0, Vundef, m')
      else None
  | _ => None
  end.

Definition do_ef_annot (text: string) (targs: list typ)
       (w: world) (vargs: list val) (m: mem) : option (world * trace * val * mem) :=
  do args <- list_eventval_of_val vargs targs;
  Some(w, Event_annot text args :: E0, Vundef, m).

Definition do_ef_annot_val (text: string) (targ: typ)
       (w: world) (vargs: list val) (m: mem) : option (world * trace * val * mem) :=
  match vargs with
  | varg :: nil =>
      do arg <- eventval_of_val varg targ;
      Some(w, Event_annot text (arg :: nil) :: E0, varg, m)
  | _ => None
  end.

Definition do_ef_debug (kind: positive) (text: ident) (targs: list typ)
       (w: world) (vargs: list val) (m: mem) : option (world * trace * val * mem) :=
  Some(w, E0, Vundef, m).

Definition do_external (ef: external_function):
       world -> list val -> mem -> option (world * trace * val * mem) :=
  match ef with
  | EF_external name sg => do_external_function name sg ge
  | EF_builtin name sg => do_builtin_function name sg ge
  | EF_runtime name sg => do_runtime_function name sg ge
  | EF_vload chunk => do_ef_volatile_load chunk
  | EF_vstore chunk => do_ef_volatile_store chunk
  | EF_malloc => do_ef_malloc
  | EF_memcpy sz al => do_ef_memcpy sz al
  | EF_annot text targs => do_ef_annot text targs
  | EF_annot_val text targ => do_ef_annot_val text targ
  | EF_inline_asm text sg clob => do_inline_assembly text sg ge
  | EF_debug kind text targs => do_ef_debug kind text targs
  end.

Lemma do_ef_external_sound:
  forall ef w vargs m w' t vres m',
  do_external ef w vargs m = Some(w', t, vres, m') ->
  external_call ef ge vargs m t vres m' /\ possible_trace w t w'.

Lemma do_ef_external_complete:
  forall ef w vargs m w' t vres m',
  external_call ef ge vargs m t vres m' -> possible_trace w t w' ->
  do_external ef w vargs m = Some(w', t, vres, m').

Reduction of expressions

Inductive reduction: Type :=
  | Lred (rule: string) (l': expr) (m': mem)
  | Rred (rule: string) (r': expr) (m': mem) (t: trace)
  | Callred (rule: string) (fd: fundef) (args: list val) (tyres: type) (m': mem) (i: ident)
  | Stuckred.

Section EXPRS.

Variable e: env.
Variable w: world.

Fixpoint sem_cast_arguments (vtl: list (val * type)) (tl: typelist) (m: mem) : option (list val) :=
  match vtl, tl with
  | nil, Tnil => Some nil
  | (v1,t1)::vtl, Tcons t1' tl =>
      do v <- sem_cast v1 t1 t1' m; do vl <- sem_cast_arguments vtl tl m; Some(v::vl)
  | _, _ => None
  end.


Definition reducts (A: Type): Type := list ((expr -> A) * reduction).

Definition topred (r: reduction) : reducts expr :=
  ((fun (x: expr) => x), r) :: nil.

Definition stuck : reducts expr :=
  ((fun (x: expr) => x), Stuckred) :: nil.

Definition incontext {A B: Type} (ctx: A -> B) (ll: reducts A) : reducts B :=
  map (fun z => ((fun (x: expr) => ctx(fst z x)), snd z)) ll.

Definition incontext2 {A1 A2 B: Type}
                     (ctx1: A1 -> B) (ll1: reducts A1)
                     (ctx2: A2 -> B) (ll2: reducts A2) : reducts B :=
  incontext ctx1 ll1 ++ incontext ctx2 ll2.

Notation "'do' X <- A ; B" := (match A with Some X => B | None => stuck end)
  (at level 200, X ident, A at level 100, B at level 200)
  : reducts_monad_scope.

Notation "'do' X , Y <- A ; B" := (match A with Some (X, Y) => B | None => stuck end)
  (at level 200, X ident, Y ident, A at level 100, B at level 200)
  : reducts_monad_scope.

Notation "'do' X , Y , Z <- A ; B" := (match A with Some (X, Y, Z) => B | None => stuck end)
  (at level 200, X ident, Y ident, Z ident, A at level 100, B at level 200)
  : reducts_monad_scope.

Notation " 'check' A ; B" := (if A then B else stuck)
  (at level 200, A at level 100, B at level 200)
  : reducts_monad_scope.

Local Open Scope reducts_monad_scope.

Definition builtin_is_enabled (ef: external_function) :
  {builtin_enabled ef} + {~ builtin_enabled ef}.

Definition invert_pointer vf :=
  match vf with
    Vptr b _ => Genv.invert_symbol ge b
  | _ => None
  end.

Fixpoint step_expr (k: kind) (a: expr) (m: mem): reducts expr :=
  match k, a with
  | LV, Eloc b ofs ty =>
      nil
  | LV, Evar x ty =>
      match e!x with
      | Some(b, ty') =>
          check type_eq ty ty';
          topred (Lred "red_var_local" (Eloc b Ptrofs.zero ty) m)
      | None =>
          do b <- Genv.find_symbol ge x;
          topred (Lred "red_var_global" (Eloc b Ptrofs.zero ty) m)
      end
  | LV, Ederef r ty =>
      match is_val r with
      | Some(Vptr b ofs, ty') =>
          topred (Lred "red_deref" (Eloc b ofs ty) m)
      | Some _ =>
          stuck
      | None =>
          incontext (fun x => Ederef x ty) (step_expr RV r m)
      end
  | LV, Efield r f ty =>
      match is_val r with
      | Some(Vptr b ofs, ty') =>
          match ty' with
          | Tstruct id _ =>
              do co <- ge.(genv_cenv)!id;
              match field_offset ge f (co_members co) with
              | Error _ => stuck
              | OK delta => topred (Lred "red_field_struct" (Eloc b (Ptrofs.add ofs (Ptrofs.repr delta)) ty) m)
              end
          | Tunion id _ =>
              do co <- ge.(genv_cenv)!id;
              topred (Lred "red_field_union" (Eloc b ofs ty) m)
          | _ => stuck
          end
      | Some _ =>
          stuck
      | None =>
          incontext (fun x => Efield x f ty) (step_expr RV r m)
      end
  | RV, Eval v ty =>
      nil
  | RV, Evalof l ty =>
      match is_loc l with
      | Some(b, ofs, ty') =>
          check type_eq ty ty';
          do w',t,v <- do_deref_loc w ty m b ofs;
          topred (Rred "red_rvalof" (Eval v ty) m t)
      | None =>
          incontext (fun x => Evalof x ty) (step_expr LV l m)
      end
  | RV, Eaddrof l ty =>
      match is_loc l with
      | Some(b, ofs, ty') => topred (Rred "red_addrof" (Eval (Vptr b ofs) ty) m E0)
      | None => incontext (fun x => Eaddrof x ty) (step_expr LV l m)
      end
  | RV, Eunop op r1 ty =>
      match is_val r1 with
      | Some(v1, ty1) =>
          do v <- sem_unary_operation op v1 ty1 m;
          topred (Rred "red_unop" (Eval v ty) m E0)
      | None =>
          incontext (fun x => Eunop op x ty) (step_expr RV r1 m)
      end
  | RV, Ebinop op r1 r2 ty =>
      match is_val r1, is_val r2 with
      | Some(v1, ty1), Some(v2, ty2) =>
          do v <- sem_binary_operation ge op v1 ty1 v2 ty2 m;
          topred (Rred "red_binop" (Eval v ty) m E0)
      | _, _ =>
         incontext2 (fun x => Ebinop op x r2 ty) (step_expr RV r1 m)
                    (fun x => Ebinop op r1 x ty) (step_expr RV r2 m)
      end
  | RV, Ecast r1 ty =>
      match is_val r1 with
      | Some(v1, ty1) =>
          do v <- sem_cast v1 ty1 ty m;
          topred (Rred "red_cast" (Eval v ty) m E0)
      | None =>
          incontext (fun x => Ecast x ty) (step_expr RV r1 m)
      end
  | RV, Eseqand r1 r2 ty =>
      match is_val r1 with
      | Some(v1, ty1) =>
          do b <- bool_val v1 ty1 m;
          if b then topred (Rred "red_seqand_true" (Eparen r2 type_bool ty) m E0)
               else topred (Rred "red_seqand_false" (Eval (Vint Int.zero) ty) m E0)
      | None =>
          incontext (fun x => Eseqand x r2 ty) (step_expr RV r1 m)
      end
  | RV, Eseqor r1 r2 ty =>
      match is_val r1 with
      | Some(v1, ty1) =>
          do b <- bool_val v1 ty1 m;
          if b then topred (Rred "red_seqor_true" (Eval (Vint Int.one) ty) m E0)
               else topred (Rred "red_seqor_false" (Eparen r2 type_bool ty) m E0)
      | None =>
          incontext (fun x => Eseqor x r2 ty) (step_expr RV r1 m)
      end
  | RV, Econdition r1 r2 r3 ty =>
      match is_val r1 with
      | Some(v1, ty1) =>
          do b <- bool_val v1 ty1 m;
          topred (Rred "red_condition" (Eparen (if b then r2 else r3) ty ty) m E0)
      | None =>
          incontext (fun x => Econdition x r2 r3 ty) (step_expr RV r1 m)
      end
  | RV, Esizeof ty' ty =>
      topred (Rred "red_sizeof" (Eval (Vptrofs (Ptrofs.repr (sizeof ge ty'))) ty) m E0)
  | RV, Ealignof ty' ty =>
      topred (Rred "red_alignof" (Eval (Vptrofs (Ptrofs.repr (alignof ge ty'))) ty) m E0)
  | RV, Eassign l1 r2 ty =>
      match is_loc l1, is_val r2 with
      | Some(b, ofs, ty1), Some(v2, ty2) =>
          check type_eq ty1 ty;
          do v <- sem_cast v2 ty2 ty1 m;
          do w',t,m' <- do_assign_loc w ty1 m b ofs v;
          topred (Rred "red_assign" (Eval v ty) m' t)
      | _, _ =>
         incontext2 (fun x => Eassign x r2 ty) (step_expr LV l1 m)
                    (fun x => Eassign l1 x ty) (step_expr RV r2 m)
      end
  | RV, Eassignop op l1 r2 tyres ty =>
      match is_loc l1, is_val r2 with
      | Some(b, ofs, ty1), Some(v2, ty2) =>
          check type_eq ty1 ty;
          do w',t,v1 <- do_deref_loc w ty1 m b ofs;
          let r' := Eassign (Eloc b ofs ty1)
                           (Ebinop op (Eval v1 ty1) (Eval v2 ty2) tyres) ty1 in
          topred (Rred "red_assignop" r' m t)
      | _, _ =>
         incontext2 (fun x => Eassignop op x r2 tyres ty) (step_expr LV l1 m)
                    (fun x => Eassignop op l1 x tyres ty) (step_expr RV r2 m)
      end
  | RV, Epostincr id l ty =>
      match is_loc l with
      | Some(b, ofs, ty1) =>
          check type_eq ty1 ty;
          do w',t, v1 <- do_deref_loc w ty m b ofs;
          let op := match id with Incr => Oadd | Decr => Osub end in
          let r' :=
            Ecomma (Eassign (Eloc b ofs ty)
                           (Ebinop op (Eval v1 ty) (Eval (Vint Int.one) type_int32s) (incrdecr_type ty))
                           ty)
                   (Eval v1 ty) ty in
          topred (Rred "red_postincr" r' m t)
      | None =>
          incontext (fun x => Epostincr id x ty) (step_expr LV l m)
      end
  | RV, Ecomma r1 r2 ty =>
      match is_val r1 with
      | Some _ =>
          check type_eq (typeof r2) ty;
          topred (Rred "red_comma" r2 m E0)
      | None =>
          incontext (fun x => Ecomma x r2 ty) (step_expr RV r1 m)
      end
  | RV, Eparen r1 tycast ty =>
      match is_val r1 with
      | Some (v1, ty1) =>
          do v <- sem_cast v1 ty1 tycast m;
          topred (Rred "red_paren" (Eval v ty) m E0)
      | None =>
          incontext (fun x => Eparen x tycast ty) (step_expr RV r1 m)
      end
  | RV, Ecall r1 rargs ty =>
      match is_val r1, is_val_list rargs with
      | Some(vf, tyf), Some vtl =>
          match classify_fun tyf with
          | fun_case_f tyargs tyres cconv =>
            do i <- invert_pointer vf;
              do fd <- Genv.find_funct ge vf;
              do vargs <- sem_cast_arguments vtl tyargs m;
              check type_eq (type_of_fundef fd) (Tfunction tyargs tyres cconv);
              topred (Callred "red_call" fd vargs ty (Mem.push_new_stage m) i)
          | _ => stuck
          end
      | _, _ =>
          incontext2 (fun x => Ecall x rargs ty) (step_expr RV r1 m)
                     (fun x => Ecall r1 x ty) (step_exprlist rargs m)
      end
  | RV, Ebuiltin ef tyargs rargs ty =>
      match is_val_list rargs with
      | Some vtl =>
        do vargs <- sem_cast_arguments vtl tyargs m;
          if builtin_is_enabled ef then
            match do_external ef w vargs (Mem.push_new_stage m) with
            | None => stuck
            | Some(w',t,v,m') =>
              do m'' <- Mem.unrecord_stack_block m' ;
                topred (Rred "red_builtin" (Eval v ty) m'' t)
            end
          else stuck
      | _ =>
          incontext (fun x => Ebuiltin ef tyargs x ty) (step_exprlist rargs m)
      end
  | _, _ => stuck
  end

with step_exprlist (rl: exprlist) (m: mem): reducts exprlist :=
  match rl with
  | Enil =>
      nil
  | Econs r1 rs =>
      incontext2 (fun x => Econs x rs) (step_expr RV r1 m)
                 (fun x => Econs r1 x) (step_exprlist rs m)
  end.


Inductive imm_safe_t: kind -> expr -> mem -> Prop :=
  | imm_safe_t_val: forall v ty m,
      imm_safe_t RV (Eval v ty) m
  | imm_safe_t_loc: forall b ofs ty m,
      imm_safe_t LV (Eloc b ofs ty) m
  | imm_safe_t_lred: forall to C l m l' m',
      lred ge e l m l' m' ->
      context LV to C ->
      imm_safe_t to (C l) m
  | imm_safe_t_rred: forall to C r m t r' m' w',
      rred ge r m t r' m' -> possible_trace w t w' ->
      context RV to C ->
      imm_safe_t to (C r) m
  | imm_safe_t_callred: forall to C r m fd args ty i,
      callred ge r m fd args ty i ->
      context RV to C ->
      imm_safe_t to (C r) m.

Remark imm_safe_t_imm_safe:
  forall k a m, imm_safe_t k a m -> imm_safe ge e k a m.

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) = 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, exists w', deref_loc ge ty m b ofs t v /\ possible_trace w t w'
  | 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, exists w',
      ty = ty1 /\ sem_cast v2 ty2 ty1 m = Some v /\ assign_loc ge ty1 m b ofs v t m' /\ possible_trace w t w'
  | Eassignop op (Eloc b ofs ty1) (Eval v2 ty2) tyres ty =>
      exists t, exists v1, exists w',
      ty = ty1 /\ deref_loc ge ty1 m b ofs t v1 /\ possible_trace w t w'
  | Epostincr id (Eloc b ofs ty1) ty =>
      exists t, exists v1, exists w',
      ty = ty1 /\ deref_loc ge ty m b ofs t v1 /\ possible_trace w t w'
  | Ecomma (Eval v ty1) r2 ty =>
      typeof r2 = ty
  | Eparen (Eval v1 ty1) tycast ty =>
      exists v, sem_cast v1 ty1 tycast m = Some v
  | Ecall (Eval vf tyf) rargs ty =>
      exprlist_all_values rargs ->
      exists tyargs tyres cconv fd vl id,
        is_function_ident ge vf id
        /\ 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 ->
    builtin_enabled ef /\
    exists vargs t vres m' m'' w',
      cast_arguments m rargs tyargs vargs
      /\ external_call ef ge vargs (Mem.push_new_stage m) t vres m'
      /\ Mem.unrecord_stack_block m' = Some m''
      /\ possible_trace w t w'
  | _ => 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 w' r m t r' m', rred ge r m t r' m' -> possible_trace w t w' -> invert_expr_prop r m.

Lemma callred_invert:
  forall r fd args ty i m,
  callred ge r m fd args ty i ->
  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_t_inv:
  forall k a m,
  imm_safe_t k a m ->
  match a with
  | Eloc _ _ _ => True
  | Eval _ _ => True
  | _ => invert_expr_prop a m
  end.


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)).

Hint Constructors context contextlist.
Hint Resolve context_compose contextlist_compose.

Definition reduction_ok (k: kind) (a: expr) (m: mem) (rd: reduction) : Prop :=
  match k, rd with
  | LV, Lred _ l' m' => lred ge e a m l' m'
  | RV, Rred _ r' m' t => rred ge a m t r' m' /\ exists w', possible_trace w t w'
  | RV, Callred _ fd args tyres m' i => callred ge a m fd args tyres i /\ m' = Mem.push_new_stage m
  | LV, Stuckred => ~imm_safe_t k a m
  | RV, Stuckred => ~imm_safe_t k a m
  | _, _ => False
  end.

Definition reducts_ok (k: kind) (a: expr) (m: mem) (ll: reducts expr) : Prop :=
  (forall C rd,
      In (C, rd) ll ->
      exists a', exists k', context k' k C /\ a = C a' /\ reduction_ok k' a' m rd)
  /\ (ll = nil -> match k with LV => is_loc a <> None | RV => is_val a <> None end).

Definition list_reducts_ok (al: exprlist) (m: mem) (ll: reducts exprlist) : Prop :=
  (forall C rd,
      In (C, rd) ll ->
      exists a', exists k', contextlist k' C /\ al = C a' /\ reduction_ok k' a' m rd)
  /\ (ll = nil -> is_val_list al <> None).

Ltac monadInv :=
  match goal with
  | [ H: match ?x with Some _ => _ | None => None end = Some ?y |- _ ] =>
      destruct x eqn:?; [monadInv|discriminate]
  | [ H: match ?x with left _ => _ | right _ => None end = Some ?y |- _ ] =>
      destruct x; [monadInv|discriminate]
  | _ => idtac
  end.

Lemma sem_cast_arguments_sound:
  forall m rargs vtl tyargs vargs,
  is_val_list rargs = Some vtl ->
  sem_cast_arguments vtl tyargs m = Some vargs ->
  cast_arguments m rargs tyargs vargs.

Lemma sem_cast_arguments_complete:
  forall m al tyl vl,
  cast_arguments m al tyl vl ->
  exists vtl, is_val_list al = Some vtl /\ sem_cast_arguments vtl tyl m = Some vl.

Lemma topred_ok:
  forall k a m rd,
  reduction_ok k a m rd ->
  reducts_ok k a m (topred rd).

Lemma stuck_ok:
  forall k a m,
  ~imm_safe_t k a m ->
  reducts_ok k a m stuck.

Lemma wrong_kind_ok:
  forall k a m,
  k <> Cstrategy.expr_kind a ->
  reducts_ok k a m stuck.

Lemma not_invert_ok:
  forall k a m,
  match a with
  | Eloc _ _ _ => False
  | Eval _ _ => False
  | _ => invert_expr_prop a m -> False
  end ->
  reducts_ok k a m stuck.

Lemma incontext_ok:
  forall k a m C res k' a',
  reducts_ok k' a' m res ->
  a = C a' ->
  context k' k C ->
  match k' with LV => is_loc a' = None | RV => is_val a' = None end ->
  reducts_ok k a m (incontext C res).

Lemma incontext2_ok:
  forall k a m k1 a1 res1 k2 a2 res2 C1 C2,
  reducts_ok k1 a1 m res1 ->
  reducts_ok k2 a2 m res2 ->
  a = C1 a1 -> a = C2 a2 ->
  context k1 k C1 -> context k2 k C2 ->
  match k1 with LV => is_loc a1 = None | RV => is_val a1 = None end
  \/ match k2 with LV => is_loc a2 = None | RV => is_val a2 = None end ->
  reducts_ok k a m (incontext2 C1 res1 C2 res2).

Lemma incontext_list_ok:
  forall ef tyargs al ty m res,
  list_reducts_ok al m res ->
  is_val_list al = None ->
  reducts_ok RV (Ebuiltin ef tyargs al ty) m
                (incontext (fun x => Ebuiltin ef tyargs x ty) res).

Lemma incontext2_list_ok:
  forall a1 a2 ty m res1 res2,
  reducts_ok RV a1 m res1 ->
  list_reducts_ok a2 m res2 ->
  is_val a1 = None \/ is_val_list a2 = None ->
  reducts_ok RV (Ecall a1 a2 ty) m
               (incontext2 (fun x => Ecall x a2 ty) res1
                           (fun x => Ecall a1 x ty) res2).

Lemma incontext2_list_ok':
  forall a1 a2 m res1 res2,
  reducts_ok RV a1 m res1 ->
  list_reducts_ok a2 m res2 ->
  list_reducts_ok (Econs a1 a2) m
               (incontext2 (fun x => Econs x a2) res1
                           (fun x => Econs a1 x) res2).

Lemma is_val_list_all_values:
  forall al vtl, is_val_list al = Some vtl -> exprlist_all_values al.

Ltac myinv :=
  match goal with
  | [ H: False |- _ ] => destruct H
  | [ H: _ /\ _ |- _ ] => destruct H; myinv
  | [ H: exists _, _ |- _ ] => destruct H; myinv
  | _ => idtac
  end.

Theorem step_expr_sound:
  forall a k m, reducts_ok k a m (step_expr k a m)
with step_exprlist_sound:
  forall al m, list_reducts_ok al m (step_exprlist al m).

Lemma step_exprlist_val_list:
  forall m al, is_val_list al <> None -> step_exprlist al m = nil.


Lemma lred_topred:
  forall l1 m1 l2 m2,
  lred ge e l1 m1 l2 m2 ->
  exists rule, step_expr LV l1 m1 = topred (Lred rule l2 m2).

Lemma rred_topred:
  forall w' r1 m1 t r2 m2,
  rred ge r1 m1 t r2 m2 -> possible_trace w t w' ->
  exists rule, step_expr RV r1 m1 = topred (Rred rule r2 m2 t).

Lemma callred_topred:
  forall a fd args ty i m,
  callred ge a m fd args ty i ->
  exists rule, step_expr RV a m = topred (Callred rule fd args ty (Mem.push_new_stage m) i).

Definition reducts_incl {A B: Type} (C: A -> B) (res1: reducts A) (res2: reducts B) : Prop :=
  forall C1 rd, In (C1, rd) res1 -> In ((fun x => C(C1 x)), rd) res2.

Lemma reducts_incl_trans:
  forall (A1 A2: Type) (C: A1 -> A2) res1 res2, reducts_incl C res1 res2 ->
  forall (A3: Type) (C': A2 -> A3) res3,
  reducts_incl C' res2 res3 ->
  reducts_incl (fun x => C'(C x)) res1 res3.

Lemma reducts_incl_nil:
  forall (A B: Type) (C: A -> B) res,
  reducts_incl C nil res.

Lemma reducts_incl_val:
  forall (A: Type) a m v ty (C: expr -> A) res,
  is_val a = Some(v, ty) -> reducts_incl C (step_expr RV a m) res.

Lemma reducts_incl_loc:
  forall (A: Type) a m b ofs ty (C: expr -> A) res,
  is_loc a = Some(b, ofs, ty) -> reducts_incl C (step_expr LV a m) res.

Lemma reducts_incl_listval:
  forall (A: Type) a m vtl (C: exprlist -> A) res,
  is_val_list a = Some vtl -> reducts_incl C (step_exprlist a m) res.

Lemma reducts_incl_incontext:
  forall (A B: Type) (C: A -> B) res,
  reducts_incl C res (incontext C res).

Lemma reducts_incl_incontext2_left:
  forall (A1 A2 B: Type) (C1: A1 -> B) res1 (C2: A2 -> B) res2,
  reducts_incl C1 res1 (incontext2 C1 res1 C2 res2).

Lemma reducts_incl_incontext2_right:
  forall (A1 A2 B: Type) (C1: A1 -> B) res1 (C2: A2 -> B) res2,
  reducts_incl C2 res2 (incontext2 C1 res1 C2 res2).

Hint Resolve reducts_incl_val reducts_incl_loc reducts_incl_listval
             reducts_incl_incontext reducts_incl_incontext2_left reducts_incl_incontext2_right.

Lemma step_expr_context:
  forall from to C, context from to C ->
  forall a m, reducts_incl C (step_expr from a m) (step_expr to (C a) m)
with step_exprlist_context:
  forall from C, contextlist from C ->
  forall a m, reducts_incl C (step_expr from a m) (step_exprlist (C a) m).


Lemma not_stuckred_imm_safe:
  forall m a k,
  (forall C, ~In (C, Stuckred) (step_expr k a m)) -> imm_safe_t k a m.

Lemma not_imm_safe_stuck_red:
  forall m a k C,
  context k RV C ->
  ~imm_safe_t k a m ->
  exists C', In (C', Stuckred) (step_expr RV (C a) m).


Lemma imm_safe_imm_safe_t:
  forall k a m,
  imm_safe ge e k a m ->
  imm_safe_t k a m \/
  exists C, exists a1, exists t, exists a1', exists m',
    context RV k C /\ a = C a1 /\ rred ge a1 m t a1' m' /\ forall w', ~possible_trace w t w'.


Variable fn_stack_requirements: ident -> Z.

Definition can_crash_world (w: world) (S: state) : Prop :=
  exists t, exists S', Csem.step fn_stack_requirements ge S t S' /\ forall w', ~possible_trace w t w'.

Theorem not_imm_safe_t:
  forall K C a m f k,
  context K RV C ->
  ~imm_safe_t K a m ->
  Csem.step fn_stack_requirements ge (ExprState f (C a) k e m) E0 Stuckstate \/ can_crash_world w (ExprState f (C a) k e m).

End EXPRS.

Transitions over states.

Fixpoint do_alloc_variables (e: env) (m: mem) (l: list (ident * type)) {struct l} : env * mem :=
  match l with
  | nil => (e,m)
  | (id, ty) :: l' =>
      let (m1,b1) := Mem.alloc m 0 (sizeof ge ty) in
      do_alloc_variables (PTree.set id (b1, ty) e) m1 l'
end.

Lemma do_alloc_variables_sound:
  forall l e m, alloc_variables ge e m l (fst (do_alloc_variables e m l)) (snd (do_alloc_variables e m l)).

Lemma do_alloc_variables_complete:
  forall e1 m1 l e2 m2, alloc_variables ge e1 m1 l e2 m2 ->
  do_alloc_variables e1 m1 l = (e2, m2).

Lemma alloc_variables_perm_1:
  forall (ge: genv) e1 m1 vars e2 m2,
    alloc_variables ge e1 m1 vars e2 m2 ->
    forall b o k p,
      Mem.perm m1 b o k p ->
      Mem.perm m2 b o k p.


Lemma alloc_variables_perm_2:
  forall (ge: genv) e1 m1 vars e2 m2,
    alloc_variables ge e1 m1 vars e2 m2 ->
    forall b o k p,
      Mem.valid_block m1 b ->
      Mem.perm m2 b o k p ->
      Mem.perm m1 b o k p.

Lemma alloc_variables_not_in_vars:
  forall ge e1 m1 vars e2 m2,
    alloc_variables ge e1 m1 vars e2 m2 ->
    forall id,
      ~ In id (map fst vars) ->
      e1 ! id = e2 ! id.

Lemma alloc_variables_perm:
  forall ge e1 m1 vars e2 m2,
    alloc_variables ge e1 m1 vars e2 m2 ->
    list_norepet (map fst vars) ->
    forall id b ty o k p,
      e1 ! id = None ->
      e2 ! id = Some (b,ty) ->
      Mem.perm m2 b o k p ->
      0 <= o < sizeof ge ty.

Lemma do_alloc_variables_perms:
  forall m1 l e2 m2,
    alloc_variables ge empty_env m1 l e2 m2 ->
    list_norepet (map fst l) ->
    Forall
      (fun b : block * frame_info =>
         forall (o : Z) (k : perm_kind) (p : permission),
           Mem.perm m2 (fst b) o k p -> 0 <= o < frame_size (snd b)) (blocks_with_info ge e2).

Lemma list_norepet_map:
  forall {A B} (f: A -> B) (l: list A) (lnr: list_norepet (map f l)),
    list_norepet l.

Lemma do_alloc_variables_norepet:
  forall
    v (lnr: list_norepet (map fst v)) e m e' m'
    (DAV: do_alloc_variables e m v = (e', m'))
    (REC: forall i1 i2 (diff: i1 <> i2) b1 b2 t1 t2
              (EQ1: e ! i1 = Some (b1,t1))
              (EQ2: e ! i2 = Some (b2,t2)),
          b1 <> b2)
      (VALID: forall i b t, e ! i = Some (b, t) -> Plt b (Mem.nextblock m))
      i1 i2 (diff: i1 <> i2) b1 b2 t1 t2
      (EQ1: e' ! i1 = Some (b1,t1))
      (EQ2: e' ! i2 = Some (b2,t2)),
      b1 <> b2.

Program Definition alloc_variables_record f m sz : option (env * mem) :=
  check (list_norepet_dec ident_eq (var_names (fn_params f) ++ var_names (fn_vars f)));
    let '(e,m1) := do_alloc_variables empty_env m (f.(fn_params) ++ f.(fn_vars)) in
    do m1 <- Mem.record_stack_blocks m1 (Build_frame_adt (blocks_with_info ge e) (Z.max 0 sz) _ _);
      Some (e,m1).

Lemma destr_dep_match:
  forall {A B: Type} (a: option A) (x: B)
    (T: forall x (pf: Some x = a), B)
    (MATCH: match a as ano return (ano = a -> option B) with
            | Some m1 => fun Heq: Some m1 = a => Some (T _ Heq)
            | None => fun Heq => None
            end eq_refl = Some x) ,
  forall P: B -> Prop,
    (forall m (pf: Some m = a) x, T m pf = x -> P x) ->
    P x.

Lemma constr_let:
  forall {A1 A2 B: Type} (a: A1 * A2)
    m b (EQ: a = (m,b))
    (T: forall m b (pf: (m,b) = a), option B)
    X
    (EQ: T _ _ (eq_sym EQ) = Some X),
    (let (m0,b0) as ano return (ano = a -> option B) := a in
     fun Heq : (m0,b0) = a => T m0 b0 Heq) eq_refl = Some X.

Lemma constr_match:
  forall {A B: Type} (a: option A)
    x (EQ: a = Some x)
    (T: forall x (pf: Some x = a), option B)
    X
    (EQ: T _ (eq_sym EQ) = Some X),
    (match a as ano return (ano = a -> option B) with
       Some m1 =>
       fun Heq : Some m1 = a => T m1 Heq
     | None => fun _ => None
     end) eq_refl = Some X.


Lemma destr_dep_let:
  forall {A1 A2 B: Type} (a: A1*A2) (bres: B) (T: forall m b (p: (m,b) = a), option B),
    (let (m,b) as ano return (ano = a -> option B) := a in
     fun Heq : (m,b) = a => T _ _ Heq) eq_refl = Some bres ->
    forall (P: B -> Prop),
      (forall m b (pf: (m, b) = a) x, T _ _ pf = Some x -> P x) ->
      P bres.

Lemma alloc_variables_record_spec:
  forall f m sz e m',
    alloc_variables_record f m sz = Some (e,m') ->
    list_norepet (var_names (fn_params f) ++ var_names (fn_vars f)) /\
    exists m1 fa,
      alloc_variables ge empty_env m (f.(fn_params) ++ f.(fn_vars)) e m1 /\
      frame_adt_blocks fa = blocks_with_info ge e /\
      frame_adt_size fa = Z.max 0 sz /\
      Mem.record_stack_blocks m1 fa = Some m'.

Function sem_bind_parameters (w: world) (e: env) (m: mem) (l: list (ident * type)) (lv: list val)
                          {struct l} : option mem :=
  match l, lv with
  | nil, nil => Some m
  | (id, ty) :: params, v1::lv =>
      match PTree.get id e with
         | Some (b, ty') =>
             check (type_eq ty ty');
             do w', t, m1 <- do_assign_loc w ty m b Ptrofs.zero v1;
             match t with nil => sem_bind_parameters w e m1 params lv | _ => None end
        | None => None
      end
   | _, _ => None
end.

Lemma sem_bind_parameters_sound : forall w e m l lv m',
  sem_bind_parameters w e m l lv = Some m' ->
  bind_parameters ge e m l lv m'.

Lemma sem_bind_parameters_complete : forall w e m l lv m',
  bind_parameters ge e m l lv m' ->
  sem_bind_parameters w e m l lv = Some m'.

Inductive transition : Type := TR (rule: string) (t: trace) (S': state).

Variable fn_stack_requirements: ident -> Z.

Definition expr_final_state (f: function) (k: cont) (e: env) (C_rd: (expr -> expr) * reduction) :=
  match snd C_rd with
  | Lred rule a m => TR rule E0 (ExprState f (fst C_rd a) k e m)
  | Rred rule a m t => TR rule t (ExprState f (fst C_rd a) k e m)
  | Callred rule fd vargs ty m i => TR rule E0 (Callstate fd vargs (Kcall f e (fst C_rd) ty k) m (fn_stack_requirements i))
  | Stuckred => TR "step_stuck" E0 Stuckstate
  end.

Local Open Scope list_monad_scope.

Definition ret (rule: string) (S: state) : list transition :=
  TR rule E0 S :: nil.

Definition do_step (w: world) (s: state) : list transition :=
  match s with

  | ExprState f a k e m =>
      match is_val a with
      | Some(v, ty) =>
        match k with
        | Kdo k => ret "step_do_2" (State f Sskip k e m )
        | Kifthenelse s1 s2 k =>
            do b <- bool_val v ty m;
            ret "step_ifthenelse_2" (State f (if b then s1 else s2) k e m)
        | Kwhile1 x s k =>
            do b <- bool_val v ty m;
            if b
            then ret "step_while_true" (State f s (Kwhile2 x s k) e m)
            else ret "step_while_false" (State f Sskip k e m)
        | Kdowhile2 x s k =>
            do b <- bool_val v ty m;
            if b
            then ret "step_dowhile_true" (State f (Sdowhile x s) k e m)
            else ret "step_dowhile_false" (State f Sskip k e m)
        | Kfor2 a2 a3 s k =>
            do b <- bool_val v ty m;
            if b
            then ret "step_for_true" (State f s (Kfor3 a2 a3 s k) e m)
            else ret "step_for_false" (State f Sskip k e m)
        | Kreturn k =>
            do v' <- sem_cast v ty f.(fn_return) m;
              do m' <- Mem.free_list m (blocks_of_env ge e);
            ret "step_return_2" (Returnstate v' (call_cont k) m')
        | Kswitch1 sl k =>
            do n <- sem_switch_arg v ty;
            ret "step_expr_switch" (State f (seq_of_labeled_statement (select_switch n sl)) (Kswitch2 k) e m)
        | _ => nil
        end

      | None =>
          map (expr_final_state f k e) (step_expr e w RV a m)
      end

  | State f (Sdo x) k e m =>
      ret "step_do_1" (ExprState f x (Kdo k) e m)
  | State f (Ssequence s1 s2) k e m =>
      ret "step_seq" (State f s1 (Kseq s2 k) e m)
  | State f Sskip (Kseq s k) e m =>
      ret "step_skip_seq" (State f s k e m)
  | State f Scontinue (Kseq s k) e m =>
      ret "step_continue_seq" (State f Scontinue k e m)
  | State f Sbreak (Kseq s k) e m =>
      ret "step_break_seq" (State f Sbreak k e m)

  | State f (Sifthenelse a s1 s2) k e m =>
      ret "step_ifthenelse_1" (ExprState f a (Kifthenelse s1 s2 k) e m)

  | State f (Swhile x s) k e m =>
      ret "step_while" (ExprState f x (Kwhile1 x s k) e m)
  | State f (Sskip|Scontinue) (Kwhile2 x s k) e m =>
      ret "step_skip_or_continue_while" (State f (Swhile x s) k e m)
  | State f Sbreak (Kwhile2 x s k) e m =>
      ret "step_break_while" (State f Sskip k e m)

  | State f (Sdowhile a s) k e m =>
      ret "step_dowhile" (State f s (Kdowhile1 a s k) e m)
  | State f (Sskip|Scontinue) (Kdowhile1 x s k) e m =>
      ret "step_skip_or_continue_dowhile" (ExprState f x (Kdowhile2 x s k) e m)
  | State f Sbreak (Kdowhile1 x s k) e m =>
      ret "step_break_dowhile" (State f Sskip k e m)

  | State f (Sfor a1 a2 a3 s) k e m =>
      if is_skip a1
      then ret "step_for" (ExprState f a2 (Kfor2 a2 a3 s k) e m)
      else ret "step_for_start" (State f a1 (Kseq (Sfor Sskip a2 a3 s) k) e m)
  | State f (Sskip|Scontinue) (Kfor3 a2 a3 s k) e m =>
      ret "step_skip_or_continue_for3" (State f a3 (Kfor4 a2 a3 s k) e m)
  | State f Sbreak (Kfor3 a2 a3 s k) e m =>
      ret "step_break_for3" (State f Sskip k e m)
  | State f Sskip (Kfor4 a2 a3 s k) e m =>
      ret "step_skip_for4" (State f (Sfor Sskip a2 a3 s) k e m)

  | State f (Sreturn None) k e m =>
    do m' <- Mem.free_list m (blocks_of_env ge e);
      ret "step_return_0" (Returnstate Vundef (call_cont k) m')
  | State f (Sreturn (Some x)) k e m =>
      ret "step_return_1" (ExprState f x (Kreturn k) e m)
  | State f Sskip ((Kstop | Kcall _ _ _ _ _) as k) e m =>
    do m' <- Mem.free_list m (blocks_of_env ge e);
      ret "step_skip_call" (Returnstate Vundef k m')

  | State f (Sswitch x sl) k e m =>
      ret "step_switch" (ExprState f x (Kswitch1 sl k) e m)
  | State f (Sskip|Sbreak) (Kswitch2 k) e m =>
      ret "step_skip_break_switch" (State f Sskip k e m)
  | State f Scontinue (Kswitch2 k) e m =>
      ret "step_continue_switch" (State f Scontinue k e m)

  | State f (Slabel lbl s) k e m =>
      ret "step_label" (State f s k e m)
  | State f (Sgoto lbl) k e m =>
      match find_label lbl f.(fn_body) (call_cont k) with
      | Some(s', k') => ret "step_goto" (State f s' k' e m)
      | None => nil
      end

  | Callstate (Internal f) vargs k m sz =>
    match alloc_variables_record f m (Z.max 0 sz) with
      Some (e,m1) =>
      do m2 <- sem_bind_parameters w e m1 f.(fn_params) vargs;
        ret "step_internal_function" (State f f.(fn_body) k e m2)
    | None => nil
    end
  | Callstate (External ef _ _ _) vargs k m _ =>
      match do_external ef w vargs m with
      | None => nil
      | Some(w',t,v,m') => TR "step_external_function" t (Returnstate v k m') :: nil
      end

  | Returnstate v (Kcall f e C ty k) m =>
    do m <- Mem.unrecord_stack_block m ;
      ret "step_returnstate" (ExprState f (C (Eval v ty)) k e m)

  | _ => nil
  end.

Ltac myinv :=
  match goal with
  | [ |- In _ nil -> _ ] => let X := fresh "X" in intro X; elim X
  | [ |- In _ (ret _ _) -> _ ] =>
        let X := fresh "X" in
        intro X; elim X; clear X;
        [let EQ := fresh "EQ" in intro EQ; unfold ret in EQ; inv EQ; myinv | myinv]
  | [ |- In _ (_ :: nil) -> _ ] =>
        let X := fresh "X" in
        intro X; elim X; clear X; [let EQ := fresh "EQ" in intro EQ; inv EQ; myinv | myinv]
  | [ |- In _ (match ?x with Some _ => _ | None => _ end) -> _ ] => destruct x eqn:?; myinv
  | [ |- In _ (match ?x with false => _ | true => _ end) -> _ ] => destruct x eqn:?; myinv
  | [ |- In _ (match ?x with left _ => _ | right _ => _ end) -> _ ] => destruct x; myinv
  | _ => idtac
  end.

Hint Extern 3 => exact I.

Theorem do_step_sound:
  forall w S rule t S',
  In (TR rule t S') (do_step w S) ->
  Csem.step fn_stack_requirements ge S t S' \/ (t = E0 /\ S' = Stuckstate /\ can_crash_world fn_stack_requirements w S).

Remark estep_not_val:
  forall f a k e m t S, estep fn_stack_requirements ge (ExprState f a k e m) t S -> is_val a = None.

Lemma frame_adt_eq:
  forall fa1 fa2,
    frame_adt_blocks fa1 = frame_adt_blocks fa2 ->
    frame_adt_size fa1 = frame_adt_size fa2 ->
    fa1 = fa2.

Lemma alloc_variables_record_complete:
  forall f m e m1 fa m1',
    list_norepet (var_names (fn_params f) ++ var_names (fn_vars f)) ->
    alloc_variables ge empty_env m (fn_params f ++ fn_vars f) e m1 ->
    frame_adt_blocks fa = blocks_with_info ge e ->
    Mem.record_stack_blocks m1 fa = Some m1' ->
    alloc_variables_record f m (frame_adt_size fa) = Some (e,m1').

Theorem do_step_complete:
  forall w S t S' w',
    possible_trace w t w' -> Csem.step fn_stack_requirements ge S t S' ->
    exists rule, In (TR rule t S') (do_step w S).

End EXEC.

Local Open Scope option_monad_scope.

Variable fn_stack_requirements: ident -> Z.

Definition do_initial_state (p: program): option (genv * state) :=
  let ge := globalenv p in
  do m0 <- Genv.init_mem p;
  do b <- Genv.find_symbol ge p.(prog_main);
  do f <- Genv.find_funct_ptr ge b;
  check (type_eq (type_of_fundef f) (Tfunction Tnil type_int32s cc_default));
  Some (ge, Callstate f nil Kstop (Mem.push_new_stage m0) (fn_stack_requirements (prog_main p))).

Definition at_final_state (S: state): option int :=
  match S with
  | Returnstate (Vint r) Kstop m => Some r
  | _ => None
  end.

End WITHEXTERNALCALLS.