source: src/Clight/Csem.ma @ 950

(* *********************************************************************)
(*                                                                     *)
(*              The Compcert verified compiler                         *)
(*                                                                     *)
(*          Xavier Leroy, INRIA Paris-Rocquencourt                     *)
(*                                                                     *)
(*  Copyright Institut National de Recherche en Informatique et en     *)
(*  Automatique.  All rights reserved.  This file is distributed       *)
(*  under the terms of the GNU General Public License as published by  *)
(*  the Free Software Foundation, either version 2 of the License, or  *)
(*  (at your option) any later version.  This file is also distributed *)
(*  under the terms of the INRIA Non-Commercial License Agreement.     *)
(*                                                                     *)
(* *********************************************************************)

(* * Dynamic semantics for the Clight language *)

(*include "Coqlib.ma".*)
(*include "Errors.ma".*)
(*include "Integers.ma".*)
(*include "Floats.ma".*)
(*include "Values.ma".*)
(*include "AST.ma".*)
(*include "Mem.ma".*)
include "common/Globalenvs.ma".
include "Clight/Csyntax.ma".
include "common/Maps.ma".
(*include "Events.ma".*)
include "common/Smallstep.ma".

(* * * Semantics of type-dependent operations *)

(* * Interpretation of values as truth values.
  Non-zero integers, non-zero floats and non-null pointers are
  considered as true.  The integer zero (which also represents
  the null pointer) and the float 0.0 are false. *)

inductive is_false: val → type → Prop ≝
  | is_false_int: ∀sz,sg.
      is_false (Vint zero) (Tint sz sg)
  | is_false_pointer: ∀r,r',t.
      is_false (Vnull r) (Tpointer r' t)
 | is_false_float: ∀sz.
      is_false (Vfloat Fzero) (Tfloat sz).

inductive is_true: val → type → Prop ≝
  | is_true_int_int: ∀n,sz,sg.
      n ≠ zero →
      is_true (Vint n) (Tint sz sg)
  | is_true_pointer_pointer: ∀r,b,pc,ofs,s,t.
      is_true (Vptr r b pc ofs) (Tpointer s t)
  | is_true_float: ∀f,sz.
      f ≠ Fzero →
      is_true (Vfloat f) (Tfloat sz).

inductive bool_of_val : val → type → val → Prop ≝
  | bool_of_val_true: ∀v,ty. 
         is_true v ty → 
         bool_of_val v ty Vtrue
  | bool_of_val_false: ∀v,ty.
        is_false v ty →
        bool_of_val v ty Vfalse.

(* * The following [sem_] functions compute the result of an operator
  application.  Since operators are overloaded, the result depends
  both on the static types of the arguments and on their run-time values.
  Unlike in C, automatic conversions between integers and floats
  are not performed.  For instance, [e1 + e2] is undefined if [e1]
  is a float and [e2] an integer.  The Clight producer must have explicitly
  promoted [e2] to a float. *)

let rec sem_neg (v: val) (ty: type) : option val ≝
  match ty with
  [ Tint _ _ ⇒
      match v with
      [ Vint n ⇒ Some ? (Vint (two_complement_negation wordsize n))
      | _ ⇒ None ?
      ]
  | Tfloat _ ⇒
      match v with
      [ Vfloat f ⇒ Some ? (Vfloat (Fneg f))
      | _ ⇒ None ?
      ]
  | _ ⇒ None ?
  ].

let rec sem_notint (v: val) : option val ≝
  match v with
  [ Vint n ⇒ Some ? (Vint (xor n mone)) (* XXX *)
  | _ ⇒ None ?
  ].

let rec sem_notbool (v: val) (ty: type) : option val ≝
  match ty with
  [ Tint _ _ ⇒
      match v with
      [ Vint n ⇒ Some ? (of_bool (eq n zero))
      | _ ⇒ None ?
      ]
  | Tpointer _ _ ⇒
      match v with
      [ Vptr _ _ _ _ ⇒ Some ? Vfalse
      | Vnull _ ⇒ Some ? Vtrue
      | _ ⇒ None ?
      ]
  | Tfloat _ ⇒
      match v with
      [ Vfloat f ⇒ Some ? (of_bool (Fcmp Ceq f Fzero))
      | _ ⇒ None ?
      ]
  | _ ⇒ None ?
  ].

let rec sem_add (v1:val) (t1:type) (v2: val) (t2:type) : option val ≝
  match classify_add t1 t2 with 
  [ add_case_ii ⇒                       (**r integer addition *)
      match v1 with 
      [ Vint n1 ⇒ match v2 with
        [ Vint n2 ⇒ Some ? (Vint (addition_n wordsize n1 n2))
        | _ ⇒ None ? ]
      | _ ⇒ None ? ]
  | add_case_ff ⇒                       (**r float addition *)
      match v1 with
      [ Vfloat n1 ⇒ match v2 with
        [ Vfloat n2 ⇒ Some ? (Vfloat (Fadd n1 n2))
        | _ ⇒ None ? ]
      | _ ⇒ None ? ]
  | add_case_pi ty ⇒                    (**r pointer plus integer *)
      match v1 with
      [ Vptr r1 b1 p1 ofs1 ⇒ match v2 with
        [ Vint n2 ⇒ Some ? (Vptr r1 b1 p1 (shift_offset ofs1 (mul (repr (sizeof ty)) n2)))
        | _ ⇒ None ? ]
      | Vnull r ⇒ match v2 with
        [ Vint n2 ⇒ if eq n2 zero then Some ? (Vnull r) else None ?
        | _ ⇒ None ? ]
      | _ ⇒ None ? ]
  | add_case_ip ty ⇒                    (**r integer plus pointer *)
      match v1 with
      [ Vint n1 ⇒ match v2 with
        [ Vptr r2 b2 p2 ofs2 ⇒ Some ? (Vptr r2 b2 p2 (shift_offset ofs2 (mul (repr (sizeof ty)) n1)))
        | Vnull r ⇒ if eq n1 zero then Some ? (Vnull r) else None ?
        | _ ⇒ None ? ]
      | _ ⇒ None ? ]
  | add_default ⇒ None ?
].

let rec sem_sub (v1:val) (t1:type) (v2: val) (t2:type) : option val ≝
  match classify_sub t1 t2 with
  [ sub_case_ii ⇒                (**r integer subtraction *)
      match v1 with
      [ Vint n1 ⇒ match v2 with
        [ Vint n2 ⇒ Some ? (Vint (subtraction wordsize n1 n2))
        | _ ⇒ None ? ]
      | _ ⇒ None ? ]
  | sub_case_ff ⇒                (**r float subtraction *)
      match v1 with
      [ Vfloat f1 ⇒ match v2 with
        [ Vfloat f2 ⇒ Some ? (Vfloat (Fsub f1 f2))
        | _ ⇒ None ? ]
      | _ ⇒ None ? ]
  | sub_case_pi ty ⇒             (**r pointer minus integer *)
      match v1 with
      [ Vptr r1 b1 p1 ofs1 ⇒ match v2 with
        [ Vint n2 ⇒ Some ? (Vptr r1 b1 p1 (neg_shift_offset ofs1 (mul (repr (sizeof ty)) n2)))
        | _ ⇒ None ? ]
      | Vnull r ⇒ match v2 with
        [ Vint n2 ⇒ if eq n2 zero then Some ? (Vnull r) else None ?
        | _ ⇒ None ? ]
      | _ ⇒ None ? ]
  | sub_case_pp ty ⇒             (**r pointer minus pointer *)
      match v1 with
      [ Vptr r1 b1 p1 ofs1 ⇒ match v2 with
        [ Vptr r2 b2 p2 ofs2 ⇒
          if eq_block b1 b2 then
            if eq (repr (sizeof ty)) zero then None ?
            else match division_u ? (sub_offset ofs1 ofs2) (repr (sizeof ty)) with
                 [ None ⇒ None ?
                 | Some v ⇒ Some ? (Vint v)
                 ]
          else None ?
        | _ ⇒ None ? ]
      | Vnull r ⇒ match v2 with [ Vnull r' ⇒ Some ? (Vint zero) | _ ⇒ None ? ]
      | _ ⇒ None ? ]
  | sub_default ⇒ None ?
  ].

let rec sem_mul (v1:val) (t1:type) (v2: val) (t2:type) : option val ≝
 match classify_mul t1 t2 with
  [ mul_case_ii ⇒
      match v1 with
      [ Vint n1 ⇒ match v2 with
          [ Vint n2 ⇒ Some ? (Vint (mul n1 n2))
(*        [ Vint n2 ⇒ Some ? (Vint (\snd (split ? wordsize wordsize (multiplication ? n1 n2))))*)
        | _ ⇒ None ? ]
      | _ ⇒ None ? ]
  | mul_case_ff ⇒
      match v1 with
      [ Vfloat f1 ⇒ match v2 with
        [ Vfloat f2 ⇒ Some ? (Vfloat (Fmul f1 f2))
        | _ ⇒ None ? ]
      | _ ⇒ None ? ]
  | mul_default ⇒
      None ?
].

let rec sem_div (v1:val) (t1:type) (v2: val) (t2:type) : option val ≝
  match classify_div t1 t2 with
  [ div_case_I32unsi ⇒
      match v1 with
      [ Vint n1 ⇒ match v2 with
        [ Vint n2 ⇒ option_map … Vint (division_u ? n1 n2)
        | _ ⇒ None ? ]
      | _ ⇒ None ? ]
  | div_case_ii ⇒
      match v1 with
       [ Vint n1 ⇒ match v2 with
         [ Vint n2 ⇒ option_map … Vint (division_s ? n1 n2)
         | _ ⇒ None ? ]
      | _ ⇒ None ? ]
  | div_case_ff ⇒
      match v1 with
      [ Vfloat f1 ⇒ match v2 with
        [ Vfloat f2 ⇒ Some ? (Vfloat(Fdiv f1 f2))
        | _ ⇒ None ? ]
      | _ ⇒ None ? ]
  | div_default ⇒
      None ?
  ].

let rec sem_mod (v1:val) (t1:type) (v2: val) (t2:type) : option val ≝
  match classify_mod t1 t2 with
  [ mod_case_I32unsi ⇒
      match v1 with
      [ Vint n1 ⇒ match v2 with
        [ Vint n2 ⇒ option_map … Vint (modulus_u ? n1 n2)
        | _ ⇒ None ? ]
      | _ ⇒ None ? ]
  | mod_case_ii ⇒
      match v1 with
      [ Vint n1 ⇒ match v2 with
        [ Vint n2 ⇒ option_map … Vint (modulus_s ? n1 n2)
        | _ ⇒ None ? ]
      | _ ⇒ None ? ]
  | mod_default ⇒
      None ?
  ].

let rec sem_and (v1,v2: val) : option val ≝
  match v1 with
  [ Vint n1 ⇒ match v2 with
    [ Vint n2 ⇒ Some ? (Vint (conjunction_bv ? n1 n2))
    | _ ⇒ None ? ]
  | _ ⇒ None ?
  ].

let rec sem_or (v1,v2: val) : option val ≝
  match v1 with
  [ Vint n1 ⇒ match v2 with
    [ Vint n2 ⇒ Some ? (Vint (inclusive_disjunction_bv ? n1 n2))
    | _ ⇒ None ? ]
  | _ ⇒ None ?
  ].

let rec sem_xor (v1,v2: val) : option val ≝
  match v1 with
  [ Vint n1 ⇒ match v2 with
    [ Vint n2 ⇒ Some ? (Vint (exclusive_disjunction_bv ? n1 n2))
    | _ ⇒ None ? ]
  | _ ⇒ None ?
  ].

let rec sem_shl (v1,v2: val): option val ≝
  match v1 with
  [ Vint n1 ⇒ match v2 with
    [ Vint n2 ⇒
        if ltu n2 iwordsize then Some ? (Vint(shl n1 n2)) else None ?
    | _ ⇒ None ? ]
  | _ ⇒ None ? ].

let rec sem_shr (v1: val) (t1: type) (v2: val) (t2: type): option val ≝
  match classify_shr t1 t2 with 
  [ shr_case_I32unsi ⇒
      match v1 with 
      [ Vint n1 ⇒ match v2 with
        [ Vint n2 ⇒
            if ltu n2 iwordsize then Some ? (Vint (shru n1 n2)) else None ?
        | _ ⇒ None ? ]
      | _ ⇒ None ? ]
   | shr_case_ii => 
      match v1 with 
      [ Vint n1 ⇒ match v2 with
        [ Vint n2 ⇒
            if ltu n2 iwordsize then Some ? (Vint (shr n1 n2)) else None ?
        | _ ⇒ None ? ]
      | _ ⇒ None ? ]
   | shr_default ⇒
      None ?
   ].

let rec sem_cmp_mismatch (c: comparison): option val ≝
  match c with
  [ Ceq =>  Some ? Vfalse
  | Cne =>  Some ? Vtrue
  | _   => None ?
  ].

let rec sem_cmp_match (c: comparison): option val ≝
  match c with
  [ Ceq =>  Some ? Vtrue
  | Cne =>  Some ? Vfalse
  | _   => None ?
  ].
  
let rec sem_cmp (c:comparison)
                  (v1: val) (t1: type) (v2: val) (t2: type)
                  (m: mem): option val ≝
  match classify_cmp t1 t2 with
  [ cmp_case_I32unsi ⇒
      match v1 with
      [ Vint n1 ⇒ match v2 with
        [ Vint n2 ⇒ Some ? (of_bool (cmpu c n1 n2))
        | _ ⇒ None ? ]
      | _ ⇒ None ? ]
  | cmp_case_ii ⇒
      match v1 with
      [ Vint n1 ⇒ match v2 with
         [ Vint n2 ⇒ Some ? (of_bool (cmp c n1 n2))
         | _ ⇒ None ?
         ]
      | _ ⇒ None ?      
      ]
  | cmp_case_pp ⇒
      match v1 with
      [ Vptr r1 b1 p1 ofs1 ⇒
        match v2 with
        [ Vptr r2 b2 p2 ofs2 ⇒
          if valid_pointer m r1 b1 ofs1
          ∧ valid_pointer m r2 b2 ofs2 then
            if eq_block b1 b2
            then Some ? (of_bool (cmp_offset c ofs1 ofs2))
            else sem_cmp_mismatch c
          else None ?
        | Vnull r2 ⇒ sem_cmp_mismatch c
        | _ ⇒ None ? ]
      | Vnull r1 ⇒
        match v2 with
        [ Vptr r2 b2 p2 ofs2 ⇒ sem_cmp_mismatch c
        | Vnull r2 ⇒ sem_cmp_match c
        | _ ⇒ None ?
        ]
      | _ ⇒ None ? ]
  | cmp_case_ff ⇒
      match v1 with
      [ Vfloat f1 ⇒
        match v2 with
        [ Vfloat f2 ⇒ Some ? (of_bool (Fcmp c f1 f2))
        | _ ⇒ None ? ]
      | _ ⇒ None ? ]
  | cmp_default ⇒ None ?
  ].

definition sem_unary_operation
            : unary_operation → val → type → option val ≝
  λop,v,ty.
  match op with
  [ Onotbool => sem_notbool v ty
  | Onotint => sem_notint v
  | Oneg => sem_neg v ty
  ].

let rec sem_binary_operation
    (op: binary_operation)
    (v1: val) (t1: type) (v2: val) (t2:type)
    (m: mem): option val ≝
  match op with
  [ Oadd ⇒ sem_add v1 t1 v2 t2
  | Osub ⇒ sem_sub v1 t1 v2 t2 
  | Omul ⇒ sem_mul v1 t1 v2 t2
  | Omod ⇒ sem_mod v1 t1 v2 t2
  | Odiv ⇒ sem_div v1 t1 v2 t2
  | Oand ⇒ sem_and v1 v2  
  | Oor  ⇒ sem_or v1 v2 
  | Oxor ⇒ sem_xor v1 v2 
  | Oshl ⇒ sem_shl v1 v2
  | Oshr ⇒ sem_shr v1 t1 v2 t2
  | Oeq ⇒ sem_cmp Ceq v1 t1 v2 t2 m
  | One ⇒ sem_cmp Cne v1 t1 v2 t2 m
  | Olt ⇒ sem_cmp Clt v1 t1 v2 t2 m
  | Ogt ⇒ sem_cmp Cgt v1 t1 v2 t2 m
  | Ole ⇒ sem_cmp Cle v1 t1 v2 t2 m
  | Oge ⇒ sem_cmp Cge v1 t1 v2 t2 m
  ].

(* * Semantic of casts.  [cast v1 t1 t2 v2] holds if value [v1],
  viewed with static type [t1], can be cast to type [t2],
  resulting in value [v2].  *)

let rec cast_int_int (sz: intsize) (sg: signedness) (i: int) : int ≝
  match sz with
  [ I8 ⇒ match sg with [ Signed ⇒ sign_ext 8 i | Unsigned ⇒ zero_ext 8 i ]
  | I16 ⇒ match sg with [ Signed => sign_ext 16 i | Unsigned ⇒ zero_ext 16 i ]
  | I32 ⇒ i
  ].

let rec cast_int_float (si : signedness) (i: int) : float ≝
  match si with
  [ Signed ⇒ floatofint i
  | Unsigned ⇒ floatofintu i
  ].

let rec cast_float_int (si : signedness) (f: float) : int ≝
  match si with
  [ Signed ⇒ intoffloat f
  | Unsigned ⇒ intuoffloat f
  ].

let rec cast_float_float (sz: floatsize) (f: float) : float ≝
  match sz with
  [ F32 ⇒ singleoffloat f
  | F64 ⇒ f
  ].

inductive type_region : type → region → Prop ≝
| type_rgn_pointer : ∀s,t. type_region (Tpointer s t) s
| type_rgn_array : ∀s,t,n. type_region (Tarray s t n) s
(* XXX Is the following necessary? *)
| type_rgn_code : ∀tys,ty. type_region (Tfunction tys ty) Code.

inductive cast : mem → val → type → type → val → Prop ≝
  | cast_ii:   ∀m,i,sz2,sz1,si1,si2.            (**r int to int  *)
      cast m (Vint i) (Tint sz1 si1) (Tint sz2 si2)
           (Vint (cast_int_int sz2 si2 i))
  | cast_fi:   ∀m,f,sz1,sz2,si2.                (**r float to int *)
      cast m (Vfloat f) (Tfloat sz1) (Tint sz2 si2)
           (Vint (cast_int_int sz2 si2 (cast_float_int si2 f)))
  | cast_if:   ∀m,i,sz1,sz2,si1.                (**r int to float  *)
      cast m (Vint i) (Tint sz1 si1) (Tfloat sz2)
          (Vfloat (cast_float_float sz2 (cast_int_float si1 i)))
  | cast_ff:   ∀m,f,sz1,sz2.                    (**r float to float *)
      cast m (Vfloat f) (Tfloat sz1) (Tfloat sz2)
           (Vfloat (cast_float_float sz2 f))
  | cast_pp: ∀m,r,r',ty,ty',b,pc,ofs.
      type_region ty r →
      type_region ty' r' →
      ∀pc':pointer_compat b r'.
      cast m (Vptr r b pc ofs) ty ty' (Vptr r' b pc' ofs)
  | cast_ip_z: ∀m,sz,sg,ty',r.
      type_region ty' r →
      cast m (Vint zero) (Tint sz sg) ty' (Vnull r)
  | cast_pp_z: ∀m,ty,ty',r,r'.
      type_region ty r →
      type_region ty' r' →
      cast m (Vnull r) ty ty' (Vnull r').

(* * * Operational semantics *)

(* * The semantics uses two environments.  The global environment
  maps names of functions and global variables to memory block references,
  and function pointers to their definitions.  (See module [Globalenvs].) *)

definition genv ≝ (genv_t Genv) clight_fundef.

(* * The local environment maps local variables to block references.
  The current value of the variable is stored in the associated memory
  block. *)

definition env ≝ (tree_t ? PTree) block. (* map variable -> location *)

definition empty_env: env ≝ (empty …).

(* * [load_value_of_type ty m b ofs] computes the value of a datum
  of type [ty] residing in memory [m] at block [b], offset [ofs].
  If the type [ty] indicates an access by value, the corresponding
  memory load is performed.  If the type [ty] indicates an access by
  reference, the pointer [Vptr b ofs] is returned. *)

let rec load_value_of_type (ty: type) (m: mem) (b: block) (ofs: offset) : option val ≝
  match access_mode ty with
  [ By_value chunk ⇒ loadv chunk m (Vptr Any b ? ofs)
  | By_reference r ⇒
    match pointer_compat_dec b r with
    [ inl p ⇒ Some ? (Vptr r b p ofs)
    | inr _ ⇒ None ?
    ]
  | By_nothing ⇒ None ?
  ].
cases b //
qed.

(* * Symmetrically, [store_value_of_type ty m b ofs v] returns the
  memory state after storing the value [v] in the datum
  of type [ty] residing in memory [m] at block [b], offset [ofs].
  This is allowed only if [ty] indicates an access by value. *)

let rec store_value_of_type (ty_dest: type) (m: mem) (loc: block) (ofs: offset) (v: val) : option mem ≝
  match access_mode ty_dest with
  [ By_value chunk ⇒ storev chunk m (Vptr Any loc ? ofs) v
  | By_reference _ ⇒ None ?
  | By_nothing ⇒ None ?
  ].
cases loc //
qed.

(* * Allocation of function-local variables.
  [alloc_variables e1 m1 vars e2 m2] allocates one memory block
  for each variable declared in [vars], and associates the variable
  name with this block.  [e1] and [m1] are the initial local environment
  and memory state.  [e2] and [m2] are the final local environment
  and memory state. *)

inductive alloc_variables: env → mem →
                            list (ident × type) →
                            env → mem → Prop ≝
  | alloc_variables_nil:
      ∀e,m.
      alloc_variables e m (nil ?) e m
  | alloc_variables_cons:
      ∀e,m,id,ty,vars,m1,b1,m2,e2.
      alloc m 0 (sizeof ty) Any = 〈m1, b1〉 →
      alloc_variables (set … id b1 e) m1 vars e2 m2 →
      alloc_variables e m (〈id, ty〉 :: vars) e2 m2.

(* * Initialization of local variables that are parameters to a function.
  [bind_parameters e m1 params args m2] stores the values [args]
  in the memory blocks corresponding to the variables [params].
  [m1] is the initial memory state and [m2] the final memory state. *)

inductive bind_parameters: env →
                           mem → list (ident × type) → list val →
                           mem → Prop ≝
  | bind_parameters_nil:
      ∀e,m.
      bind_parameters e m (nil ?) (nil ?) m
  | bind_parameters_cons:
      ∀e,m,id,ty,params,v1,vl,b,m1,m2.
      get ??? id e = Some ? b →
      store_value_of_type ty m b zero_offset v1 = Some ? m1 →
      bind_parameters e m1 params vl m2 →
      bind_parameters e m (〈id, ty〉 :: params) (v1 :: vl) m2.

(* * Return the list of blocks in the codomain of [e]. *)

definition blocks_of_env : env → list block ≝ λe.
  map ?? (λx. snd ?? x) (elements ??? e).

(* * Selection of the appropriate case of a [switch], given the value [n]
  of the selector expression. *)

let rec select_switch (n: int) (sl: labeled_statements)
                       on sl : labeled_statements ≝
  match sl with
  [ LSdefault _ ⇒ sl
  | LScase c s sl' ⇒ if eq c n then sl else select_switch n sl'
  ].

(* * Turn a labeled statement into a sequence *)

let rec seq_of_labeled_statement (sl: labeled_statements) : statement ≝
  match sl with
  [ LSdefault s ⇒ s
  | LScase c s sl' ⇒ Ssequence s (seq_of_labeled_statement sl')
  ].

(*
Section SEMANTICS.

Variable ge: genv.

(** ** Evaluation of expressions *)

Section EXPR.

Variable e: env.
Variable m: mem.
*)
(* * [eval_expr ge e m a v] defines the evaluation of expression [a]
  in r-value position.  [v] is the value of the expression.
  [e] is the current environment and [m] is the current memory state. *)

inductive eval_expr (ge:genv) (e:env) (m:mem) : expr → val → trace → Prop ≝
  | eval_Econst_int:   ∀i,ty.
      eval_expr ge e m (Expr (Econst_int i) ty) (Vint i) E0
  | eval_Econst_float:   ∀f,ty.
      eval_expr ge e m (Expr (Econst_float f) ty) (Vfloat f) E0
  | eval_Elvalue: ∀a,ty,loc,ofs,v,tr.
      eval_lvalue ge e m (Expr a ty) loc ofs tr →
      load_value_of_type ty m loc ofs = Some ? v →
      eval_expr ge e m (Expr a ty) v tr
  | eval_Eaddrof: ∀a,ty,r,loc,ofs,tr.
      eval_lvalue ge e m a loc ofs tr →
      ∀pc:pointer_compat loc r.
      eval_expr ge e m (Expr (Eaddrof a) (Tpointer r ty)) (Vptr r loc pc ofs) tr
  | eval_Esizeof: ∀ty',ty.
      eval_expr ge e m (Expr (Esizeof ty') ty) (Vint (repr (sizeof ty'))) E0
  | eval_Eunop:  ∀op,a,ty,v1,v,tr.
      eval_expr ge e m a v1 tr →
      sem_unary_operation op v1 (typeof a) = Some ? v →
      eval_expr ge e m (Expr (Eunop op a) ty) v tr
  | eval_Ebinop: ∀op,a1,a2,ty,v1,v2,v,tr1,tr2.
      eval_expr ge e m a1 v1 tr1 →
      eval_expr ge e m a2 v2 tr2 →
      sem_binary_operation op v1 (typeof a1) v2 (typeof a2) m = Some ? v →
      eval_expr ge e m (Expr (Ebinop op a1 a2) ty) v (tr1⧺tr2)
  | eval_Econdition_true: ∀a1,a2,a3,ty,v1,v2,tr1,tr2.
      eval_expr ge e m a1 v1 tr1 →
      is_true v1 (typeof a1) →
      eval_expr ge e m a2 v2 tr2 →
      eval_expr ge e m (Expr (Econdition a1 a2 a3) ty) v2 (tr1⧺tr2)
  | eval_Econdition_false: ∀a1,a2,a3,ty,v1,v3,tr1,tr2.
      eval_expr ge e m a1 v1 tr1 →
      is_false v1 (typeof a1) →
      eval_expr ge e m a3 v3 tr2 →
      eval_expr ge e m (Expr (Econdition a1 a2 a3) ty) v3 (tr1⧺tr2)
  | eval_Eorbool_1: ∀a1,a2,ty,v1,tr.
      eval_expr ge e m a1 v1 tr →
      is_true v1 (typeof a1) →
      eval_expr ge e m (Expr (Eorbool a1 a2) ty) Vtrue tr
  | eval_Eorbool_2: ∀a1,a2,ty,v1,v2,v,tr1,tr2.
      eval_expr ge e m a1 v1 tr1 →
      is_false v1 (typeof a1) →
      eval_expr ge e m a2 v2 tr2 →
      bool_of_val v2 (typeof a2) v →
      eval_expr ge e m (Expr (Eorbool a1 a2) ty) v (tr1⧺tr2)
  | eval_Eandbool_1: ∀a1,a2,ty,v1,tr.
      eval_expr ge e m a1 v1 tr →
      is_false v1 (typeof a1) →
      eval_expr ge e m (Expr (Eandbool a1 a2) ty) Vfalse tr
  | eval_Eandbool_2: ∀a1,a2,ty,v1,v2,v,tr1,tr2.
      eval_expr ge e m a1 v1 tr1 →
      is_true v1 (typeof a1) →
      eval_expr ge e m a2 v2 tr2 →
      bool_of_val v2 (typeof a2) v →
      eval_expr ge e m (Expr (Eandbool a1 a2) ty) v (tr1⧺tr2)
  | eval_Ecast:   ∀a,ty,ty',v1,v,tr.
      eval_expr ge e m a v1 tr →
      cast m v1 (typeof a) ty v →
      eval_expr ge e m (Expr (Ecast ty a) ty') v tr
  | eval_Ecost: ∀a,ty,v,l,tr.
      eval_expr ge e m a v tr →
      eval_expr ge e m (Expr (Ecost l a) ty) v (tr⧺Echarge l)

(* * [eval_lvalue ge e m a r b ofs] defines the evaluation of expression [a]
  in l-value position.  The result is the memory location [b, ofs]
  that contains the value of the expression [a].  The memory location should
  be representable in a pointer of region r. *)

with eval_lvalue (*(ge:genv) (e:env) (m:mem)*) : expr → block → offset → trace → Prop ≝
  | eval_Evar_local:   ∀id,l,ty.
      (* XXX notation? e!id*) get ??? id e = Some ? l →
      eval_lvalue ge e m (Expr (Evar id) ty) l zero_offset E0
  | eval_Evar_global: ∀id,l,ty.
      (* XXX e!id *) get ??? id e = None ? →
      find_symbol ?? ge id = Some ? l →
      eval_lvalue ge e m (Expr (Evar id) ty) l zero_offset E0
  | eval_Ederef: ∀a,ty,r,l,p,ofs,tr.
      eval_expr ge e m a (Vptr r l p ofs) tr →
      eval_lvalue ge e m (Expr (Ederef a) ty) l ofs tr
    (* Aside: note that each block of memory is entirely contained within one
       memory region; hence adding a field offset will not produce a location
       outside of the original location's region. *)
 | eval_Efield_struct:   ∀a,i,ty,l,ofs,id,fList,delta,tr.
      eval_lvalue ge e m a l ofs tr →
      typeof a = Tstruct id fList →
      field_offset i fList = OK ? delta →
      eval_lvalue ge e m (Expr (Efield a i) ty) l (shift_offset ofs (repr delta)) tr
 | eval_Efield_union:   ∀a,i,ty,l,ofs,id,fList,tr.
      eval_lvalue ge e m a l ofs tr →
      typeof a = Tunion id fList →
      eval_lvalue ge e m (Expr (Efield a i) ty) l ofs tr.

let rec eval_expr_ind (ge:genv) (e:env) (m:mem)
  (P:∀a,v,tr. eval_expr ge e m a v tr → Prop)
  (eci:∀i,ty. P ??? (eval_Econst_int ge e m i ty))
  (ecF:∀f,ty. P ??? (eval_Econst_float ge e m f ty))
  (elv:∀a,ty,loc,ofs,v,tr,H1,H2. P ??? (eval_Elvalue ge e m a ty loc ofs v tr H1 H2))
  (ead:∀a,ty,r,loc,pc,ofs,tr,H. P ??? (eval_Eaddrof ge e m a ty r loc pc ofs tr H))
  (esz:∀ty',ty. P ??? (eval_Esizeof ge e m ty' ty))
  (eun:∀op,a,ty,v1,v,tr,H1,H2. P a v1 tr H1 → P ??? (eval_Eunop ge e m op a ty v1 v tr H1 H2))
  (ebi:∀op,a1,a2,ty,v1,v2,v,tr1,tr2,H1,H2,H3. P a1 v1 tr1 H1 → P a2 v2 tr2 H2 → P ??? (eval_Ebinop ge e m op a1 a2 ty v1 v2 v tr1 tr2 H1 H2 H3))
  (ect:∀a1,a2,a3,ty,v1,v2,tr1,tr2,H1,H2,H3. P a1 v1 tr1 H1 → P a2 v2 tr2 H3 → P ??? (eval_Econdition_true ge e m a1 a2 a3 ty v1 v2 tr1 tr2 H1 H2 H3))
  (ecf:∀a1,a2,a3,ty,v1,v3,tr1,tr2,H1,H2,H3. P a1 v1 tr1 H1 → P a3 v3 tr2 H3 → P ??? (eval_Econdition_false ge e m a1 a2 a3 ty v1 v3 tr1 tr2 H1 H2 H3))
  (eo1:∀a1,a2,ty,v1,tr,H1,H2. P a1 v1 tr H1 → P ??? (eval_Eorbool_1 ge e m a1 a2 ty v1 tr H1 H2))
  (eo2:∀a1,a2,ty,v1,v2,v,tr1,tr2,H1,H2,H3,H4. P a1 v1 tr1 H1 → P a2 v2 tr2 H3 → P ??? (eval_Eorbool_2 ge e m a1 a2 ty v1 v2 v tr1 tr2 H1 H2 H3 H4))
  (ea1:∀a1,a2,ty,v1,tr,H1,H2. P a1 v1 tr H1 → P ??? (eval_Eandbool_1 ge e m a1 a2 ty v1 tr H1 H2))
  (ea2:∀a1,a2,ty,v1,v2,v,tr1,tr2,H1,H2,H3,H4. P a1 v1 tr1 H1 → P a2 v2 tr2 H3 → P ??? (eval_Eandbool_2 ge e m a1 a2 ty v1 v2 v tr1 tr2 H1 H2 H3 H4))
  (ecs:∀a,ty,ty',v1,v,tr,H1,H2. P a v1 tr H1 → P ??? (eval_Ecast ge e m a ty ty' v1 v tr H1 H2))
  (eco:∀a,ty,v,l,tr,H. P a v tr H → P ??? (eval_Ecost ge e m a ty v l tr H))
  (a:expr) (v:val) (tr:trace) (ev:eval_expr ge e m a v tr) on ev : P a v tr ev ≝
  match ev with
  [ eval_Econst_int i ty ⇒ eci i ty
  | eval_Econst_float f ty ⇒ ecF f ty
  | eval_Elvalue a ty loc ofs v tr H1 H2 ⇒ elv a ty loc ofs v tr H1 H2
  | eval_Eaddrof a ty r loc pc ofs tr H ⇒ ead a ty r loc pc ofs tr H
  | eval_Esizeof ty' ty ⇒ esz ty' ty
  | eval_Eunop op a ty v1 v tr H1 H2 ⇒ eun op a ty v1 v tr H1 H2 (eval_expr_ind ge e m P eci ecF elv ead esz eun ebi ect ecf eo1 eo2 ea1 ea2 ecs eco a v1 tr H1)
  | eval_Ebinop op a1 a2 ty v1 v2 v tr1 tr2 H1 H2 H3 ⇒ ebi op a1 a2 ty v1 v2 v tr1 tr2 H1 H2 H3 (eval_expr_ind ge e m P eci ecF elv ead esz eun ebi ect ecf eo1 eo2 ea1 ea2 ecs eco a1 v1 tr1 H1) (eval_expr_ind ge e m P eci ecF elv ead esz eun ebi ect ecf eo1 eo2 ea1 ea2 ecs eco a2 v2 tr2 H2)
  | eval_Econdition_true a1 a2 a3 ty v1 v2 tr1 tr2 H1 H2 H3 ⇒ ect a1 a2 a3 ty v1 v2 tr1 tr2 H1 H2 H3 (eval_expr_ind ge e m P eci ecF elv ead esz eun ebi ect ecf eo1 eo2 ea1 ea2 ecs eco a1 v1 tr1 H1) (eval_expr_ind ge e m P eci ecF elv ead esz eun ebi ect ecf eo1 eo2 ea1 ea2 ecs eco a2 v2 tr2 H3)
  | eval_Econdition_false a1 a2 a3 ty v1 v3 tr1 tr2 H1 H2 H3 ⇒ ecf a1 a2 a3 ty v1 v3 tr1 tr2 H1 H2 H3 (eval_expr_ind ge e m P eci ecF elv ead esz eun ebi ect ecf eo1 eo2 ea1 ea2 ecs eco a1 v1 tr1 H1) (eval_expr_ind ge e m P eci ecF elv ead esz eun ebi ect ecf eo1 eo2 ea1 ea2 ecs eco a3 v3 tr2 H3)
  | eval_Eorbool_1 a1 a2 ty v1 tr H1 H2 ⇒ eo1 a1 a2 ty v1 tr H1 H2 (eval_expr_ind ge e m P eci ecF elv ead esz eun ebi ect ecf eo1 eo2 ea1 ea2 ecs eco a1 v1 tr H1)
  | eval_Eorbool_2 a1 a2 ty v1 v2 v tr1 tr2 H1 H2 H3 H4 ⇒ eo2 a1 a2 ty v1 v2 v tr1 tr2 H1 H2 H3 H4 (eval_expr_ind ge e m P eci ecF elv ead esz eun ebi ect ecf eo1 eo2 ea1 ea2 ecs eco a1 v1 tr1 H1) (eval_expr_ind ge e m P eci ecF elv ead esz eun ebi ect ecf eo1 eo2 ea1 ea2 ecs eco a2 v2 tr2 H3)
  | eval_Eandbool_1 a1 a2 ty v1 tr H1 H2 ⇒ ea1 a1 a2 ty v1 tr H1 H2 (eval_expr_ind ge e m P eci ecF elv ead esz eun ebi ect ecf eo1 eo2 ea1 ea2 ecs eco a1 v1 tr H1)
  | eval_Eandbool_2 a1 a2 ty v1 v2 v tr1 tr2 H1 H2 H3 H4 ⇒ ea2 a1 a2 ty v1 v2 v tr1 tr2 H1 H2 H3 H4 (eval_expr_ind ge e m P eci ecF elv ead esz eun ebi ect ecf eo1 eo2 ea1 ea2 ecs eco a1 v1 tr1 H1) (eval_expr_ind ge e m P eci ecF elv ead esz eun ebi ect ecf eo1 eo2 ea1 ea2 ecs eco a2 v2 tr2 H3)
  | eval_Ecast a ty ty' v1 v tr H1 H2 ⇒ ecs a ty ty' v1 v tr H1 H2 (eval_expr_ind ge e m P eci ecF elv ead esz eun ebi ect ecf eo1 eo2 ea1 ea2 ecs eco a v1 tr H1)
  | eval_Ecost a ty v l tr H ⇒ eco a ty v l tr H (eval_expr_ind ge e m P eci ecF elv ead esz eun ebi ect ecf eo1 eo2 ea1 ea2 ecs eco a v tr H)
  ].

inverter eval_expr_inv_ind for eval_expr : Prop.

let rec eval_lvalue_ind (ge:genv) (e:env) (m:mem)
  (P:∀a,loc,ofs,tr. eval_lvalue ge e m a loc ofs tr → Prop)
  (lvl:∀id,l,ty,H. P ???? (eval_Evar_local ge e m id l ty H))
  (lvg:∀id,l,ty,H1,H2. P ???? (eval_Evar_global ge e m id l ty H1 H2))
  (lde:∀a,ty,r,l,pc,ofs,tr,H. P ???? (eval_Ederef ge e m a ty r l pc ofs tr H))
  (lfs:∀a,i,ty,l,ofs,id,fList,delta,tr,H1,H2,H3. P a l ofs tr H1 → P ???? (eval_Efield_struct ge e m a i ty l ofs id fList delta tr H1 H2 H3))
  (lfu:∀a,i,ty,l,ofs,id,fList,tr,H1,H2. P a l ofs tr H1 → P ???? (eval_Efield_union ge e m a i ty l ofs id fList tr H1 H2))
  (a:expr) (loc:block) (ofs:offset) (tr:trace) (ev:eval_lvalue ge e m a loc ofs tr) on ev : P a loc ofs tr ev ≝
  match ev with
  [ eval_Evar_local id l ty H ⇒ lvl id l ty H
  | eval_Evar_global id l ty H1 H2 ⇒ lvg id l ty H1 H2
  | eval_Ederef a ty r l pc ofs tr H ⇒ lde a ty r l pc ofs tr H 
  | eval_Efield_struct a i ty l ofs id fList delta tr H1 H2 H3 ⇒ lfs a i ty l ofs id fList delta tr H1 H2 H3 (eval_lvalue_ind ge e m P lvl lvg lde lfs lfu a l ofs tr H1)
  | eval_Efield_union a i ty l ofs id fList tr H1 H2 ⇒ lfu a i ty l ofs id fList tr H1 H2 (eval_lvalue_ind ge e m P lvl lvg lde lfs lfu a l ofs tr H1)
  ].

(*
ninverter eval_lvalue_inv_ind for eval_lvalue : Prop.
*)

definition eval_lvalue_inv_ind :
  ∀x1: genv.
   ∀x2: env.
    ∀x3: mem.
     ∀x4: expr.
       ∀x6: block.
        ∀x7: offset.
         ∀x8: trace.
          ∀P:
            ∀_z1430: expr.
              ∀_z1428: block. ∀_z1427: offset. ∀_z1426: trace. Prop.
           ∀_H1: ?.
            ∀_H2: ?.
             ∀_H3: ?.
              ∀_H4: ?.
               ∀_H5: ?.
                ∀_Hterm: eval_lvalue x1 x2 x3 x4 x6 x7 x8.
                 P x4 x6 x7 x8
:=
  (λx1:genv.
    (λx2:env.
      (λx3:mem.
        (λx4:expr.
            (λx6:block.
              (λx7:offset.
                (λx8:trace.
                  (λP:∀_z1430: expr.
                         ∀_z1428: block.
                          ∀_z1427: offset. ∀_z1426: trace. Prop.
                    (λH1:?.
                      (λH2:?.
                        (λH3:?.
                          (λH4:?.
                            (λH5:?.
                              (λHterm:eval_lvalue x1 x2 x3 x4 x6 x7 x8.
                                ((λHcut:∀z1435: eq expr x4 x4.
                                           ∀z1433: eq block x6 x6.
                                            ∀z1432: eq offset x7 x7.
                                             ∀z1431: eq trace x8 x8.
                                              P x4 x6 x7 x8.
                                   (Hcut (refl expr x4)
                                     (refl block x6)
                                     (refl offset x7) (refl trace x8)))
                                  ?))))))))))))))).
[ @(eval_lvalue_ind x1 x2 x3 (λa,loc,ofs,tr,e. ∀e1:eq ? x4 a. ∀e3:eq ? x6 loc. ∀e4:eq ? x7 ofs. ∀e5:eq ? x8 tr. P a loc ofs tr) … Hterm)
  [ @H1 | @H2 | @H3 | @H4 | @H5 ]
| *: skip
] qed.

let rec eval_expr_ind2 (ge:genv) (e:env) (m:mem)
  (P:∀a,v,tr. eval_expr ge e m a v tr → Prop)
  (Q:∀a,loc,ofs,tr. eval_lvalue ge e m a loc ofs tr → Prop)
  (eci:∀i,ty. P ??? (eval_Econst_int ge e m i ty))
  (ecF:∀f,ty. P ??? (eval_Econst_float ge e m f ty))
  (elv:∀a,ty,loc,ofs,v,tr,H1,H2. Q (Expr a ty) loc ofs tr H1 → P ??? (eval_Elvalue ge e m a ty loc ofs v tr H1 H2))
  (ead:∀a,ty,r,loc,pc,ofs,tr,H. Q a loc ofs tr H → P ??? (eval_Eaddrof ge e m a ty r loc ofs tr H pc))
  (esz:∀ty',ty. P ??? (eval_Esizeof ge e m ty' ty))
  (eun:∀op,a,ty,v1,v,tr,H1,H2. P a v1 tr H1 → P ??? (eval_Eunop ge e m op a ty v1 v tr H1 H2))
  (ebi:∀op,a1,a2,ty,v1,v2,v,tr1,tr2,H1,H2,H3. P a1 v1 tr1 H1 → P a2 v2 tr2 H2 → P ??? (eval_Ebinop ge e m op a1 a2 ty v1 v2 v tr1 tr2 H1 H2 H3))
  (ect:∀a1,a2,a3,ty,v1,v2,tr1,tr2,H1,H2,H3. P a1 v1 tr1 H1 → P a2 v2 tr2 H3 → P ??? (eval_Econdition_true ge e m a1 a2 a3 ty v1 v2 tr1 tr2 H1 H2 H3))
  (ecf:∀a1,a2,a3,ty,v1,v3,tr1,tr2,H1,H2,H3. P a1 v1 tr1 H1 → P a3 v3 tr2 H3 → P ??? (eval_Econdition_false ge e m a1 a2 a3 ty v1 v3 tr1 tr2 H1 H2 H3))
  (eo1:∀a1,a2,ty,v1,tr,H1,H2. P a1 v1 tr H1 → P ??? (eval_Eorbool_1 ge e m a1 a2 ty v1 tr H1 H2))
  (eo2:∀a1,a2,ty,v1,v2,v,tr1,tr2,H1,H2,H3,H4. P a1 v1 tr1 H1 → P a2 v2 tr2 H3 → P ??? (eval_Eorbool_2 ge e m a1 a2 ty v1 v2 v tr1 tr2 H1 H2 H3 H4))
  (ea1:∀a1,a2,ty,v1,tr,H1,H2. P a1 v1 tr H1 → P ??? (eval_Eandbool_1 ge e m a1 a2 ty v1 tr H1 H2))
  (ea2:∀a1,a2,ty,v1,v2,v,tr1,tr2,H1,H2,H3,H4. P a1 v1 tr1 H1 → P a2 v2 tr2 H3 → P ??? (eval_Eandbool_2 ge e m a1 a2 ty v1 v2 v tr1 tr2 H1 H2 H3 H4))
  (ecs:∀a,ty,ty',v1,v,tr,H1,H2. P a v1 tr H1 → P ??? (eval_Ecast ge e m a ty ty' v1 v tr H1 H2))
  (eco:∀a,ty,v,l,tr,H. P a v tr H → P ??? (eval_Ecost ge e m a ty v l tr H))
  (lvl:∀id,l,ty,H. Q ???? (eval_Evar_local ge e m id l ty H))
  (lvg:∀id,l,ty,H1,H2. Q ???? (eval_Evar_global ge e m id l ty H1 H2))
  (lde:∀a,ty,r,l,pc,ofs,tr,H. P a (Vptr r l pc ofs) tr H → Q ???? (eval_Ederef ge e m a ty r l pc ofs tr H))
  (lfs:∀a,i,ty,l,ofs,id,fList,delta,tr,H1,H2,H3. Q a l ofs tr H1 → Q ???? (eval_Efield_struct ge e m a i ty l ofs id fList delta tr H1 H2 H3))
  (lfu:∀a,i,ty,l,ofs,id,fList,tr,H1,H2. Q a l ofs tr H1 → Q ???? (eval_Efield_union ge e m a i ty l ofs id fList tr H1 H2))
  
  (a:expr) (v:val) (tr:trace) (ev:eval_expr ge e m a v tr) on ev : P a v tr ev ≝
  match ev with
  [ eval_Econst_int i ty ⇒ eci i ty
  | eval_Econst_float f ty ⇒ ecF f ty
  | eval_Elvalue a ty loc ofs v tr H1 H2 ⇒ elv a ty loc ofs v tr H1 H2 (eval_lvalue_ind2 ge e m P Q eci ecF elv ead esz eun ebi ect ecf eo1 eo2 ea1 ea2 ecs eco lvl lvg lde lfs lfu (Expr a ty) loc ofs tr H1)
  | eval_Eaddrof a ty r loc ofs tr H pc ⇒ ead a ty r loc pc ofs tr H (eval_lvalue_ind2 ge e m P Q eci ecF elv ead esz eun ebi ect ecf eo1 eo2 ea1 ea2 ecs eco lvl lvg lde lfs lfu a loc ofs tr H)
  | eval_Esizeof ty' ty ⇒ esz ty' ty
  | eval_Eunop op a ty v1 v tr H1 H2 ⇒ eun op a ty v1 v tr H1 H2 (eval_expr_ind2 ge e m P Q eci ecF elv ead esz eun ebi ect ecf eo1 eo2 ea1 ea2 ecs eco lvl lvg lde lfs lfu a v1 tr H1)
  | eval_Ebinop op a1 a2 ty v1 v2 v tr1 tr2 H1 H2 H3 ⇒ ebi op a1 a2 ty v1 v2 v tr1 tr2 H1 H2 H3 (eval_expr_ind2 ge e m P Q eci ecF elv ead esz eun ebi ect ecf eo1 eo2 ea1 ea2 ecs eco lvl lvg lde lfs lfu a1 v1 tr1 H1) (eval_expr_ind2 ge e m P Q eci ecF elv ead esz eun ebi ect ecf eo1 eo2 ea1 ea2 ecs eco lvl lvg lde lfs lfu a2 v2 tr2 H2)
  | eval_Econdition_true a1 a2 a3 ty v1 v2 tr1 tr2 H1 H2 H3 ⇒ ect a1 a2 a3 ty v1 v2 tr1 tr2 H1 H2 H3 (eval_expr_ind2 ge e m P Q eci ecF elv ead esz eun ebi ect ecf eo1 eo2 ea1 ea2 ecs eco lvl lvg lde lfs lfu a1 v1 tr1 H1) (eval_expr_ind2 ge e m P Q eci ecF elv ead esz eun ebi ect ecf eo1 eo2 ea1 ea2 ecs eco lvl lvg lde lfs lfu a2 v2 tr2 H3)
  | eval_Econdition_false a1 a2 a3 ty v1 v3 tr1 tr2 H1 H2 H3 ⇒ ecf a1 a2 a3 ty v1 v3 tr1 tr2 H1 H2 H3 (eval_expr_ind2 ge e m P Q eci ecF elv ead esz eun ebi ect ecf eo1 eo2 ea1 ea2 ecs eco lvl lvg lde lfs lfu a1 v1 tr1 H1) (eval_expr_ind2 ge e m P Q eci ecF elv ead esz eun ebi ect ecf eo1 eo2 ea1 ea2 ecs eco lvl lvg lde lfs lfu a3 v3 tr2 H3)
  | eval_Eorbool_1 a1 a2 ty v1 tr H1 H2 ⇒ eo1 a1 a2 ty v1 tr H1 H2 (eval_expr_ind2 ge e m P Q eci ecF elv ead esz eun ebi ect ecf eo1 eo2 ea1 ea2 ecs eco lvl lvg lde lfs lfu a1 v1 tr H1)
  | eval_Eorbool_2 a1 a2 ty v1 v2 v tr1 tr2 H1 H2 H3 H4 ⇒ eo2 a1 a2 ty v1 v2 v tr1 tr2 H1 H2 H3 H4 (eval_expr_ind2 ge e m P Q eci ecF elv ead esz eun ebi ect ecf eo1 eo2 ea1 ea2 ecs eco lvl lvg lde lfs lfu a1 v1 tr1 H1) (eval_expr_ind2 ge e m P Q eci ecF elv ead esz eun ebi ect ecf eo1 eo2 ea1 ea2 ecs eco lvl lvg lde lfs lfu a2 v2 tr2 H3)
  | eval_Eandbool_1 a1 a2 ty v1 tr H1 H2 ⇒ ea1 a1 a2 ty v1 tr H1 H2 (eval_expr_ind2 ge e m P Q eci ecF elv ead esz eun ebi ect ecf eo1 eo2 ea1 ea2 ecs eco lvl lvg lde lfs lfu a1 v1 tr H1)
  | eval_Eandbool_2 a1 a2 ty v1 v2 v tr1 tr2 H1 H2 H3 H4 ⇒ ea2 a1 a2 ty v1 v2 v tr1 tr2 H1 H2 H3 H4 (eval_expr_ind2 ge e m P Q eci ecF elv ead esz eun ebi ect ecf eo1 eo2 ea1 ea2 ecs eco lvl lvg lde lfs lfu a1 v1 tr1 H1) (eval_expr_ind2 ge e m P Q eci ecF elv ead esz eun ebi ect ecf eo1 eo2 ea1 ea2 ecs eco lvl lvg lde lfs lfu a2 v2 tr2 H3)
  | eval_Ecast a ty ty' v1 v tr H1 H2 ⇒ ecs a ty ty' v1 v tr H1 H2 (eval_expr_ind2 ge e m P Q eci ecF elv ead esz eun ebi ect ecf eo1 eo2 ea1 ea2 ecs eco lvl lvg lde lfs lfu a v1 tr H1)
  | eval_Ecost a ty v l tr H ⇒ eco a ty v l tr H (eval_expr_ind2 ge e m P Q eci ecF elv ead esz eun ebi ect ecf eo1 eo2 ea1 ea2 ecs eco lvl lvg lde lfs lfu a v tr H)
  ]
and eval_lvalue_ind2 (ge:genv) (e:env) (m:mem)
  (P:∀a,v,tr. eval_expr ge e m a v tr → Prop)
  (Q:∀a,loc,ofs,tr. eval_lvalue ge e m a loc ofs tr → Prop)
  (eci:∀i,ty. P ??? (eval_Econst_int ge e m i ty))
  (ecF:∀f,ty. P ??? (eval_Econst_float ge e m f ty))
  (elv:∀a,ty,loc,ofs,v,tr,H1,H2. Q (Expr a ty) loc ofs tr H1 → P ??? (eval_Elvalue ge e m a ty loc ofs v tr H1 H2))
  (ead:∀a,ty,r,loc,pc,ofs,tr,H. Q a loc ofs tr H → P ??? (eval_Eaddrof ge e m a ty r loc ofs tr H pc))
  (esz:∀ty',ty. P ??? (eval_Esizeof ge e m ty' ty))
  (eun:∀op,a,ty,v1,v,tr,H1,H2. P a v1 tr H1 → P ??? (eval_Eunop ge e m op a ty v1 v tr H1 H2))
  (ebi:∀op,a1,a2,ty,v1,v2,v,tr1,tr2,H1,H2,H3. P a1 v1 tr1 H1 → P a2 v2 tr2 H2 → P ??? (eval_Ebinop ge e m op a1 a2 ty v1 v2 v tr1 tr2 H1 H2 H3))
  (ect:∀a1,a2,a3,ty,v1,v2,tr1,tr2,H1,H2,H3. P a1 v1 tr1 H1 → P a2 v2 tr2 H3 → P ??? (eval_Econdition_true ge e m a1 a2 a3 ty v1 v2 tr1 tr2 H1 H2 H3))
  (ecf:∀a1,a2,a3,ty,v1,v3,tr1,tr2,H1,H2,H3. P a1 v1 tr1 H1 → P a3 v3 tr2 H3 → P ??? (eval_Econdition_false ge e m a1 a2 a3 ty v1 v3 tr1 tr2 H1 H2 H3))
  (eo1:∀a1,a2,ty,v1,tr,H1,H2. P a1 v1 tr H1 → P ??? (eval_Eorbool_1 ge e m a1 a2 ty v1 tr H1 H2))
  (eo2:∀a1,a2,ty,v1,v2,v,tr1,tr2,H1,H2,H3,H4. P a1 v1 tr1 H1 → P a2 v2 tr2 H3 → P ??? (eval_Eorbool_2 ge e m a1 a2 ty v1 v2 v tr1 tr2 H1 H2 H3 H4))
  (ea1:∀a1,a2,ty,v1,tr,H1,H2. P a1 v1 tr H1 → P ??? (eval_Eandbool_1 ge e m a1 a2 ty v1 tr H1 H2))
  (ea2:∀a1,a2,ty,v1,v2,v,tr1,tr2,H1,H2,H3,H4. P a1 v1 tr1 H1 → P a2 v2 tr2 H3 → P ??? (eval_Eandbool_2 ge e m a1 a2 ty v1 v2 v tr1 tr2 H1 H2 H3 H4))
  (ecs:∀a,ty,ty',v1,v,tr,H1,H2. P a v1 tr H1 → P ??? (eval_Ecast ge e m a ty ty' v1 v tr H1 H2))
  (eco:∀a,ty,v,l,tr,H. P a v tr H → P ??? (eval_Ecost ge e m a ty v l tr H))
  (lvl:∀id,l,ty,H. Q ???? (eval_Evar_local ge e m id l ty H))
  (lvg:∀id,l,ty,H1,H2. Q ???? (eval_Evar_global ge e m id l ty H1 H2))
  (lde:∀a,ty,r,l,pc,ofs,tr,H. P a (Vptr r l pc ofs) tr H → Q ???? (eval_Ederef ge e m a ty r l pc ofs tr H))
  (lfs:∀a,i,ty,l,ofs,id,fList,delta,tr,H1,H2,H3. Q a l ofs tr H1 → Q ???? (eval_Efield_struct ge e m a i ty l ofs id fList delta tr H1 H2 H3))
  (lfu:∀a,i,ty,l,ofs,id,fList,tr,H1,H2. Q a l ofs tr H1 → Q ???? (eval_Efield_union ge e m a i ty l ofs id fList tr H1 H2))
  (a:expr) (loc:block) (ofs:offset) (tr:trace) (ev:eval_lvalue ge e m a loc ofs tr) on ev : Q a loc ofs tr ev ≝
  match ev with
  [ eval_Evar_local id l ty H ⇒ lvl id l ty H
  | eval_Evar_global id l ty H1 H2 ⇒ lvg id l ty H1 H2
  | eval_Ederef a ty r l pc ofs tr H ⇒ lde a ty r l pc ofs tr H (eval_expr_ind2 ge e m P Q eci ecF elv ead esz eun ebi ect ecf eo1 eo2 ea1 ea2 ecs eco lvl lvg lde lfs lfu a (Vptr r l pc ofs) tr H)
  | eval_Efield_struct a i ty l ofs id fList delta tr H1 H2 H3 ⇒ lfs a i ty l ofs id fList delta tr H1 H2 H3 (eval_lvalue_ind2 ge e m P Q eci ecF elv ead esz eun ebi ect ecf eo1 eo2 ea1 ea2 ecs eco lvl lvg lde lfs lfu a l ofs tr H1)
  | eval_Efield_union a i ty l ofs id fList tr H1 H2 ⇒ lfu a i ty l ofs id fList tr H1 H2 (eval_lvalue_ind2 ge e m P Q eci ecF elv ead esz eun ebi ect ecf eo1 eo2 ea1 ea2 ecs eco lvl lvg lde lfs lfu a l ofs tr H1)
  ].

definition combined_expr_lvalue_ind ≝
λge,e,m,P,Q,eci,ecF,elv,ead,esz,eun,ebi,ect,ecf,eo1,eo2,ea1,ea2,ecs,eco,lvl,lvg,lde,lfs,lfu.  
conj ??
  (eval_expr_ind2 ge e m P Q eci ecF elv ead esz eun ebi ect ecf eo1 eo2 ea1 ea2 ecs eco lvl lvg lde lfs lfu)
  (eval_lvalue_ind2 ge e m P Q eci ecF elv ead esz eun ebi ect ecf eo1 eo2 ea1 ea2 ecs eco lvl lvg lde lfs lfu).

(* * [eval_lvalue ge e m a b ofs] defines the evaluation of expression [a]
  in l-value position.  The result is the memory location [b, ofs]
  that contains the value of the expression [a]. *)

(*
Scheme eval_expr_ind22 := Minimality for eval_expr Sort Prop
  with eval_lvalue_ind2 := Minimality for eval_lvalue Sort Prop.
*)

(* * [eval_exprlist ge e m al vl] evaluates a list of r-value
  expressions [al] to their values [vl]. *)

inductive eval_exprlist (ge:genv) (e:env) (m:mem) : list expr → list val → trace → Prop ≝
  | eval_Enil:
      eval_exprlist ge e m (nil ?) (nil ?) E0
  | eval_Econs:   ∀a,bl,v,vl,tr1,tr2.
      eval_expr ge e m a v tr1 →
      eval_exprlist ge e m bl vl tr2 →
      eval_exprlist ge e m (a :: bl) (v :: vl) (tr1⧺tr2).

(*End EXPR.*)

(* * ** Transition semantics for statements and functions *)

(* * Continuations *)

inductive cont: Type[0] :=
  | Kstop: cont
  | Kseq: statement -> cont -> cont
       (**r [Kseq s2 k] = after [s1] in [s1;s2] *)
  | Kwhile: expr -> statement -> cont -> cont
       (**r [Kwhile e s k] = after [s] in [while (e) s] *)
  | Kdowhile: expr -> statement -> cont -> cont
       (**r [Kdowhile e s k] = after [s] in [do s while (e)] *)
  | Kfor2: expr -> statement -> statement -> cont -> cont
       (**r [Kfor2 e2 e3 s k] = after [s] in [for(e1;e2;e3) s] *)
  | Kfor3: expr -> statement -> statement -> cont -> cont
       (**r [Kfor3 e2 e3 s k] = after [e3] in [for(e1;e2;e3) s] *)
  | Kswitch: cont -> cont
       (**r catches [break] statements arising out of [switch] *)
  | Kcall: option (block × offset × type) -> (**r where to store result *)
           function ->                       (**r calling function *)
           env ->                            (**r local env of calling function *)
           cont -> cont.

(* * Pop continuation until a call or stop *)

let rec call_cont (k: cont) : cont :=
  match k with
  [ Kseq s k => call_cont k
  | Kwhile e s k => call_cont k
  | Kdowhile e s k => call_cont k
  | Kfor2 e2 e3 s k => call_cont k
  | Kfor3 e2 e3 s k => call_cont k
  | Kswitch k => call_cont k
  | _ => k
  ].

definition is_call_cont : cont → Prop ≝ λk.
  match k with
  [ Kstop => True
  | Kcall _ _ _ _ => True
  | _ => False
  ].

(* * States *)

inductive state: Type[0] :=
  | State:
      ∀f: function.
      ∀s: statement.
      ∀k: cont.
      ∀e: env.
      ∀m: mem.  state
  | Callstate:
      ∀fd: clight_fundef.
      ∀args: list val.
      ∀k: cont.
      ∀m: mem. state
  | Returnstate:
      ∀res: val.
      ∀k: cont.
      ∀m: mem. state.
                 
(* * Find the statement and manufacture the continuation 
  corresponding to a label *)

let rec find_label (lbl: label) (s: statement) (k: cont) 
                    on s: option (statement × cont) :=
  match s with
  [ Ssequence s1 s2 =>
      match find_label lbl s1 (Kseq s2 k) with
      [ Some sk => Some ? sk
      | None => find_label lbl s2 k
      ]
  | Sifthenelse a s1 s2 =>
      match find_label lbl s1 k with
      [ Some sk => Some ? sk
      | None => find_label lbl s2 k
      ]
  | Swhile a s1 =>
      find_label lbl s1 (Kwhile a s1 k)
  | Sdowhile a s1 =>
      find_label lbl s1 (Kdowhile a s1 k)
  | Sfor a1 a2 a3 s1 =>
      match find_label lbl a1 (Kseq (Sfor Sskip a2 a3 s1) k) with
      [ Some sk => Some ? sk
      | None =>
          match find_label lbl s1 (Kfor2 a2 a3 s1 k) with
          [ Some sk => Some ? sk
          | None => find_label lbl a3 (Kfor3 a2 a3 s1 k)
          ]
      ]
  | Sswitch e sl =>
      find_label_ls lbl sl (Kswitch k)
  | Slabel lbl' s' =>
      match ident_eq lbl lbl' with
      [ inl _ ⇒ Some ? 〈s', k〉
      | inr _ ⇒ find_label lbl s' k
      ]
  | _ => None ?
  ]

and find_label_ls (lbl: label) (sl: labeled_statements) (k: cont) 
                    on sl: option (statement × cont) :=
  match sl with
  [ LSdefault s => find_label lbl s k
  | LScase _ s sl' =>
      match find_label lbl s (Kseq (seq_of_labeled_statement sl') k) with
      [ Some sk => Some ? sk
      | None => find_label_ls lbl sl' k
      ]
  ].

(* * Transition relation *)

(* Strip off outer pointer for use when comparing function types. *)
definition fun_typeof ≝
λe. match typeof e with
[ Tvoid ⇒ Tvoid
| Tint a b ⇒ Tint a b
| Tfloat a ⇒ Tfloat a
| Tpointer _ ty ⇒ ty
| Tarray a b c ⇒ Tarray a b c
| Tfunction a b ⇒ Tfunction a b
| Tstruct a b ⇒ Tstruct a b
| Tunion a b ⇒ Tunion a b
| Tcomp_ptr a b ⇒ Tcomp_ptr a b
].

(* XXX: note that cost labels in exprs expose a particular eval order. *)

inductive step (ge:genv) : state → trace → state → Prop ≝

  | step_assign:   ∀f,a1,a2,k,e,m,loc,ofs,v2,m',tr1,tr2.
      eval_lvalue ge e m a1 loc ofs tr1 →
      eval_expr ge e m a2 v2 tr2 →
      store_value_of_type (typeof a1) m loc ofs v2 = Some ? m' →
      step ge (State f (Sassign a1 a2) k e m)
           (tr1⧺tr2) (State f Sskip k e m')

  | step_call_none:   ∀f,a,al,k,e,m,vf,vargs,fd,tr1,tr2.
      eval_expr ge e m a vf tr1 →
      eval_exprlist ge e m al vargs tr2 →
      find_funct ?? ge vf = Some ? fd →
      type_of_fundef fd = fun_typeof a →
      step ge (State f (Scall (None ?) a al) k e m)
           (tr1⧺tr2) (Callstate fd vargs (Kcall (None ?) f e k) m)

  | step_call_some:   ∀f,lhs,a,al,k,e,m,loc,ofs,vf,vargs,fd,tr1,tr2,tr3.
      eval_lvalue ge e m lhs loc ofs tr1 →
      eval_expr ge e m a vf tr2 →
      eval_exprlist ge e m al vargs tr3 →
      find_funct ?? ge vf = Some ? fd →
      type_of_fundef fd = fun_typeof a →
      step ge (State f (Scall (Some ? lhs) a al) k e m)
           (tr1⧺tr2⧺tr3) (Callstate fd vargs (Kcall (Some ? 〈〈loc, ofs〉, typeof lhs〉) f e k) m)

  | step_seq:  ∀f,s1,s2,k,e,m.
      step ge (State f (Ssequence s1 s2) k e m)
           E0 (State f s1 (Kseq s2 k) e m)
  | step_skip_seq: ∀f,s,k,e,m.
      step ge (State f Sskip (Kseq s k) e m)
           E0 (State f s k e m)
  | step_continue_seq: ∀f,s,k,e,m.
      step ge (State f Scontinue (Kseq s k) e m)
           E0 (State f Scontinue k e m)
  | step_break_seq: ∀f,s,k,e,m.
      step ge (State f Sbreak (Kseq s k) e m)
           E0 (State f Sbreak k e m)

  | step_ifthenelse_true:  ∀f,a,s1,s2,k,e,m,v1,tr.
      eval_expr ge e m a v1 tr →
      is_true v1 (typeof a) →
      step ge (State f (Sifthenelse a s1 s2) k e m)
           tr (State f s1 k e m)
  | step_ifthenelse_false: ∀f,a,s1,s2,k,e,m,v1,tr.
      eval_expr ge e m a v1 tr →
      is_false v1 (typeof a) →
      step ge (State f (Sifthenelse a s1 s2) k e m)
           tr (State f s2 k e m)

  | step_while_false: ∀f,a,s,k,e,m,v,tr.
      eval_expr ge e m a v tr →
      is_false v (typeof a) →
      step ge (State f (Swhile a s) k e m)
           tr (State f Sskip k e m)
  | step_while_true: ∀f,a,s,k,e,m,v,tr.
      eval_expr ge e m a v tr →
      is_true v (typeof a) →
      step ge (State f (Swhile a s) k e m)
           tr (State f s (Kwhile a s k) e m)
  | step_skip_or_continue_while: ∀f,x,a,s,k,e,m.
      x = Sskip ∨ x = Scontinue →
      step ge (State f x (Kwhile a s k) e m)
           E0 (State f (Swhile a s) k e m)
  | step_break_while: ∀f,a,s,k,e,m.
      step ge (State f Sbreak (Kwhile a s k) e m)
           E0 (State f Sskip k e m)

  | step_dowhile: ∀f,a,s,k,e,m.
      step ge (State f (Sdowhile a s) k e m)
        E0 (State f s (Kdowhile a s k) e m)
  | step_skip_or_continue_dowhile_false: ∀f,x,a,s,k,e,m,v,tr.
      x = Sskip ∨ x = Scontinue →
      eval_expr ge e m a v tr →
      is_false v (typeof a) →
      step ge (State f x (Kdowhile a s k) e m)
           tr (State f Sskip k e m)
  | step_skip_or_continue_dowhile_true: ∀f,x,a,s,k,e,m,v,tr.
      x = Sskip ∨ x = Scontinue →
      eval_expr ge e m a v tr →
      is_true v (typeof a) →
      step ge (State f x (Kdowhile a s k) e m)
           tr (State f (Sdowhile a s) k e m)
  | step_break_dowhile: ∀f,a,s,k,e,m.
      step ge (State f Sbreak (Kdowhile a s k) e m)
           E0 (State f Sskip k e m)

  | step_for_start: ∀f,a1,a2,a3,s,k,e,m.
      a1 ≠ Sskip →
      step ge (State f (Sfor a1 a2 a3 s) k e m)
           E0 (State f a1 (Kseq (Sfor Sskip a2 a3 s) k) e m)
  | step_for_false: ∀f,a2,a3,s,k,e,m,v,tr.
      eval_expr ge e m a2 v tr →
      is_false v (typeof a2) →
      step ge (State f (Sfor Sskip a2 a3 s) k e m)
           tr (State f Sskip k e m)
  | step_for_true: ∀f,a2,a3,s,k,e,m,v,tr.
      eval_expr ge e m a2 v tr →
      is_true v (typeof a2) →
      step ge (State f (Sfor Sskip a2 a3 s) k e m)
           tr (State f s (Kfor2 a2 a3 s k) e m)
  | step_skip_or_continue_for2: ∀f,x,a2,a3,s,k,e,m.
      x = Sskip ∨ x = Scontinue →
      step ge (State f x (Kfor2 a2 a3 s k) e m)
           E0 (State f a3 (Kfor3 a2 a3 s k) e m)
  | step_break_for2: ∀f,a2,a3,s,k,e,m.
      step ge (State f Sbreak (Kfor2 a2 a3 s k) e m)
           E0 (State f Sskip k e m)
  | step_skip_for3: ∀f,a2,a3,s,k,e,m.
      step ge (State f Sskip (Kfor3 a2 a3 s k) e m)
           E0 (State f (Sfor Sskip a2 a3 s) k e m)

  | step_return_0: ∀f,k,e,m.
      fn_return f = Tvoid →
      step ge (State f (Sreturn (None ?)) k e m)
           E0 (Returnstate Vundef (call_cont k) (free_list m (blocks_of_env e)))
  | step_return_1: ∀f,a,k,e,m,v,tr.
      fn_return f ≠ Tvoid →
      eval_expr ge e m a v tr →
      step ge (State f (Sreturn (Some ? a)) k e m)
           tr (Returnstate v (call_cont k) (free_list m (blocks_of_env e)))
  | step_skip_call: ∀f,k,e,m.
      is_call_cont k →
      fn_return f = Tvoid →
      step ge (State f Sskip k e m)
           E0 (Returnstate Vundef k (free_list m (blocks_of_env e)))

  | step_switch: ∀f,a,sl,k,e,m,n,tr.
      eval_expr ge e m a (Vint n) tr →
      step ge (State f (Sswitch a sl) k e m)
           tr (State f (seq_of_labeled_statement (select_switch n sl)) (Kswitch k) e m)
  | step_skip_break_switch: ∀f,x,k,e,m.
      x = Sskip ∨ x = Sbreak →
      step ge (State f x (Kswitch k) e m)
           E0 (State f Sskip k e m)
  | step_continue_switch: ∀f,k,e,m.
      step ge (State f Scontinue (Kswitch k) e m)
           E0 (State f Scontinue k e m)

  | step_label: ∀f,lbl,s,k,e,m.
      step ge (State f (Slabel lbl s) k e m)
           E0 (State f s k e m)

  | step_goto: ∀f,lbl,k,e,m,s',k'.
      find_label lbl (fn_body f) (call_cont k) = Some ? 〈s', k'〉 →
      step ge (State f (Sgoto lbl) k e m)
           E0 (State f s' k' e m)

  | step_internal_function: ∀f,vargs,k,m,e,m1,m2.
      alloc_variables empty_env m ((fn_params f) @ (fn_vars f)) e m1 →
      bind_parameters e m1 (fn_params f) vargs m2 →
      step ge (Callstate (CL_Internal f) vargs k m)
           E0 (State f (fn_body f) k e m2)

  | step_external_function: ∀id,targs,tres,vargs,k,m,vres,t.
      event_match (external_function id targs tres) vargs t vres →
      step ge (Callstate (CL_External id targs tres) vargs k m)
            t (Returnstate vres k m)

  | step_returnstate_0: ∀v,f,e,k,m.
      step ge (Returnstate v (Kcall (None ?) f e k) m)
           E0 (State f Sskip k e m)

  | step_returnstate_1: ∀v,f,e,k,m,m',loc,ofs,ty.
      store_value_of_type ty m loc ofs v = Some ? m' →
      step ge (Returnstate v (Kcall (Some ? 〈〈loc, ofs〉, ty〉) f e k) m)
           E0 (State f Sskip k e m')
           
  | step_cost: ∀f,lbl,s,k,e,m.
      step ge (State f (Scost lbl s) k e m)
           (Echarge lbl) (State f s k e m).
(*
(** * Alternate big-step semantics *)

(** ** Big-step semantics for terminating statements and functions *)

(** The execution of a statement produces an ``outcome'', indicating
  how the execution terminated: either normally or prematurely
  through the execution of a [break], [continue] or [return] statement. *)

inductive outcome: Type[0] :=
   | Out_break: outcome                 (**r terminated by [break] *)
   | Out_continue: outcome              (**r terminated by [continue] *)
   | Out_normal: outcome                (**r terminated normally *)
   | Out_return: option val -> outcome. (**r 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: ∀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) : Prop :=
  match out, t with
  | Out_normal, Tvoid => v = Vundef
  | Out_return None, Tvoid => v = Vundef
  | Out_return (Some v'), ty => ty <> Tvoid /\ v'=v
  | _, _ => False
  end. 

(** [exec_stmt ge e m1 s t m2 out] describes the execution of 
  the statement [s].  [out] is the outcome for this execution.
  [m1] is the initial memory state, [m2] the final memory state.
  [t] is the trace of input/output events performed during this
  evaluation. *)

inductive exec_stmt: env -> mem -> statement -> trace -> mem -> outcome -> Prop :=
  | exec_Sskip:   ∀e,m.
      exec_stmt e m Sskip
               E0 m Out_normal
  | exec_Sassign:   ∀e,m,a1,a2,loc,ofs,v2,m'.
      eval_lvalue e m a1 loc ofs ->
      eval_expr e m a2 v2 ->
      store_value_of_type (typeof a1) m loc ofs v2 = Some m' ->
      exec_stmt e m (Sassign a1 a2)
               E0 m' Out_normal
  | exec_Scall_none:   ∀e,m,a,al,vf,vargs,f,t,m',vres.
      eval_expr e m a vf ->
      eval_exprlist e m al vargs ->
      Genv.find_funct ge vf = Some f ->
      type_of_fundef f = typeof a ->
      eval_funcall m f vargs t m' vres ->
      exec_stmt e m (Scall None a al)
                t m' Out_normal
  | exec_Scall_some:   ∀e,m,lhs,a,al,loc,ofs,vf,vargs,f,t,m',vres,m''.
      eval_lvalue e m lhs loc ofs ->
      eval_expr e m a vf ->
      eval_exprlist e m al vargs ->
      Genv.find_funct ge vf = Some f ->
      type_of_fundef f = typeof a ->
      eval_funcall m f vargs t m' vres ->
      store_value_of_type (typeof lhs) m' loc ofs vres = Some m'' ->
      exec_stmt e m (Scall (Some lhs) a al)
                t m'' Out_normal
  | exec_Sseq_1:   ∀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:   ∀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_true: ∀e,m,a,s1,s2,v1,t,m',out.
      eval_expr e m a v1 ->
      is_true v1 (typeof a) ->
      exec_stmt e m s1 t m' out ->
      exec_stmt e m (Sifthenelse a s1 s2)
                t m' out
  | exec_Sifthenelse_false: ∀e,m,a,s1,s2,v1,t,m',out.
      eval_expr e m a v1 ->
      is_false v1 (typeof a) ->
      exec_stmt e m s2 t m' out ->
      exec_stmt e m (Sifthenelse a s1 s2)
                t m' out
  | exec_Sreturn_none:   ∀e,m.
      exec_stmt e m (Sreturn None)
               E0 m (Out_return None)
  | exec_Sreturn_some: ∀e,m,a,v.
      eval_expr e m a v ->
      exec_stmt e m (Sreturn (Some a))
               E0 m (Out_return (Some v))
  | exec_Sbreak:   ∀e,m.
      exec_stmt e m Sbreak
               E0 m Out_break
  | exec_Scontinue:   ∀e,m.
      exec_stmt e m Scontinue
               E0 m Out_continue
  | exec_Swhile_false: ∀e,m,a,s,v.
      eval_expr e m a v ->
      is_false v (typeof a) ->
      exec_stmt e m (Swhile a s)
               E0 m Out_normal
  | exec_Swhile_stop: ∀e,m,a,v,s,t,m',out',out.
      eval_expr e m a v ->
      is_true v (typeof a) ->
      exec_stmt e m s t m' out' ->
      out_break_or_return out' out ->
      exec_stmt e m (Swhile a s)
                t m' out
  | exec_Swhile_loop: ∀e,m,a,s,v,t1,m1,out1,t2,m2,out.
      eval_expr e m a v ->
      is_true v (typeof a) ->
      exec_stmt e m s t1 m1 out1 ->
      out_normal_or_continue out1 ->
      exec_stmt e m1 (Swhile a s) t2 m2 out ->
      exec_stmt e m (Swhile a s)
                (t1 ** t2) m2 out
  | exec_Sdowhile_false: ∀e,m,s,a,t,m1,out1,v.
      exec_stmt e m s t m1 out1 ->
      out_normal_or_continue out1 ->
      eval_expr e m1 a v ->
      is_false v (typeof a) ->
      exec_stmt e m (Sdowhile a s)
                t m1 Out_normal
  | exec_Sdowhile_stop: ∀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: ∀e,m,s,a,m1,m2,t1,t2,out,out1,v.
      exec_stmt e m s t1 m1 out1 ->
      out_normal_or_continue out1 ->
      eval_expr e m1 a v ->
      is_true v (typeof a) ->
      exec_stmt e m1 (Sdowhile a s) t2 m2 out ->
      exec_stmt e m (Sdowhile a s) 
                (t1 ** t2) m2 out
  | exec_Sfor_start: ∀e,m,s,a1,a2,a3,out,m1,m2,t1,t2.
      a1 <> Sskip ->
      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: ∀e,m,s,a2,a3,v.
      eval_expr e m a2 v ->
      is_false v (typeof a2) ->
      exec_stmt e m (Sfor Sskip a2 a3 s)
               E0 m Out_normal
  | exec_Sfor_stop: ∀e,m,s,a2,a3,v,m1,t,out1,out.
      eval_expr e m a2 v ->
      is_true v (typeof a2) ->
      exec_stmt e m s t m1 out1 ->
      out_break_or_return out1 out ->
      exec_stmt e m (Sfor Sskip a2 a3 s)
                t m1 out
  | exec_Sfor_loop: ∀e,m,s,a2,a3,v,m1,m2,m3,t1,t2,t3,out1,out.
      eval_expr e m a2 v ->
      is_true v (typeof a2) ->
      exec_stmt e m s t1 m1 out1 ->
      out_normal_or_continue out1 ->
      exec_stmt e m1 a3 t2 m2 Out_normal ->
      exec_stmt e m2 (Sfor Sskip a2 a3 s) t3 m3 out ->
      exec_stmt e m (Sfor Sskip a2 a3 s)
                (t1 ** t2 ** t3) m3 out
  | exec_Sswitch:   ∀e,m,a,t,n,sl,m1,out.
      eval_expr e m a (Vint n) ->
      exec_stmt e m (seq_of_labeled_statement (select_switch n sl)) t m1 out ->
      exec_stmt e m (Sswitch a sl)
                t m1 (outcome_switch out)

(** [eval_funcall m1 fd args t m2 res] describes the invocation of
  function [fd] with arguments [args].  [res] is the value returned
  by the call.  *)

with eval_funcall: mem -> fundef -> list val -> trace -> mem -> val -> Prop :=
  | eval_funcall_internal: ∀m,f,vargs,t,e,m1,m2,m3,out,vres.
      alloc_variables empty_env m (f.(fn_params) ++ f.(fn_vars)) e m1 ->
      bind_parameters 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 ->
      eval_funcall m (Internal f) vargs t (Mem.free_list m3 (blocks_of_env e)) vres
  | eval_funcall_external: ∀m,id,targs,tres,vargs,t,vres.
      event_match (external_function id targs tres) vargs t vres ->
      eval_funcall m (External id targs tres) vargs t m vres.

Scheme exec_stmt_ind2 := Minimality for exec_stmt Sort Prop
  with eval_funcall_ind2 := Minimality for eval_funcall Sort Prop.

(** ** Big-step semantics for diverging statements and functions *)

(** Coinductive semantics for divergence.
  [execinf_stmt ge e m s t] holds if the execution of statement [s]
  diverges, i.e. loops infinitely.  [t] is the possibly infinite
  trace of observable events performed during the execution. *)

Coinductive execinf_stmt: env -> mem -> statement -> traceinf -> Prop :=
  | execinf_Scall_none:   ∀e,m,a,al,vf,vargs,f,t.
      eval_expr e m a vf ->
      eval_exprlist e m al vargs ->
      Genv.find_funct ge vf = Some f ->
      type_of_fundef f = typeof a ->
      evalinf_funcall m f vargs t ->
      execinf_stmt e m (Scall None a al) t
  | execinf_Scall_some:   ∀e,m,lhs,a,al,loc,ofs,vf,vargs,f,t.
      eval_lvalue e m lhs loc ofs ->
      eval_expr e m a vf ->
      eval_exprlist e m al vargs ->
      Genv.find_funct ge vf = Some f ->
      type_of_fundef f = typeof a ->
      evalinf_funcall m f vargs t ->
      execinf_stmt e m (Scall (Some lhs) a al) t
  | execinf_Sseq_1:   ∀e,m,s1,s2,t.
      execinf_stmt e m s1 t ->
      execinf_stmt e m (Ssequence s1 s2) t
  | execinf_Sseq_2:   ∀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_true: ∀e,m,a,s1,s2,v1,t.
      eval_expr e m a v1 ->
      is_true v1 (typeof a) ->
      execinf_stmt e m s1 t ->
      execinf_stmt e m (Sifthenelse a s1 s2) t
  | execinf_Sifthenelse_false: ∀e,m,a,s1,s2,v1,t.
      eval_expr e m a v1 ->
      is_false v1 (typeof a) ->
      execinf_stmt e m s2 t ->
      execinf_stmt e m (Sifthenelse a s1 s2) t
  | execinf_Swhile_body: ∀e,m,a,v,s,t.
      eval_expr e m a v ->
      is_true v (typeof a) ->
      execinf_stmt e m s t ->
      execinf_stmt e m (Swhile a s) t
  | execinf_Swhile_loop: ∀e,m,a,s,v,t1,m1,out1,t2.
      eval_expr e m a v ->
      is_true v (typeof a) ->
      exec_stmt e m s t1 m1 out1 ->
      out_normal_or_continue out1 ->
      execinf_stmt e m1 (Swhile a s) t2 ->
      execinf_stmt e m (Swhile a s) (t1 *** t2)
  | execinf_Sdowhile_body: ∀e,m,s,a,t.
      execinf_stmt e m s t ->
      execinf_stmt e m (Sdowhile a s) t
  | execinf_Sdowhile_loop: ∀e,m,s,a,m1,t1,t2,out1,v.
      exec_stmt e m s t1 m1 out1 ->
      out_normal_or_continue out1 ->
      eval_expr e m1 a v ->
      is_true v (typeof a) ->
      execinf_stmt e m1 (Sdowhile a s) t2 ->
      execinf_stmt e m (Sdowhile a s) (t1 *** t2)
  | execinf_Sfor_start_1: ∀e,m,s,a1,a2,a3,t.
      execinf_stmt e m a1 t ->
      execinf_stmt e m (Sfor a1 a2 a3 s) t
  | execinf_Sfor_start_2: ∀e,m,s,a1,a2,a3,m1,t1,t2.
      a1 <> Sskip ->
      exec_stmt e m a1 t1 m1 Out_normal ->
      execinf_stmt e m1 (Sfor Sskip a2 a3 s) t2 ->
      execinf_stmt e m (Sfor a1 a2 a3 s) (t1 *** t2)
  | execinf_Sfor_body: ∀e,m,s,a2,a3,v,t.
      eval_expr e m a2 v ->
      is_true v (typeof a2) ->
      execinf_stmt e m s t ->
      execinf_stmt e m (Sfor Sskip a2 a3 s) t
  | execinf_Sfor_next: ∀e,m,s,a2,a3,v,m1,t1,t2,out1.
      eval_expr e m a2 v ->
      is_true v (typeof a2) ->
      exec_stmt e m s t1 m1 out1 ->
      out_normal_or_continue out1 ->
      execinf_stmt e m1 a3 t2 ->
      execinf_stmt e m (Sfor Sskip a2 a3 s) (t1 *** t2)
  | execinf_Sfor_loop: ∀e,m,s,a2,a3,v,m1,m2,t1,t2,t3,out1.
      eval_expr e m a2 v ->
      is_true v (typeof a2) ->
      exec_stmt e m s t1 m1 out1 ->
      out_normal_or_continue out1 ->
      exec_stmt e m1 a3 t2 m2 Out_normal ->
      execinf_stmt e m2 (Sfor Sskip a2 a3 s) t3 ->
      execinf_stmt e m (Sfor Sskip a2 a3 s) (t1 *** t2 *** t3)
  | execinf_Sswitch:   ∀e,m,a,t,n,sl.
      eval_expr e m a (Vint n) ->
      execinf_stmt e m (seq_of_labeled_statement (select_switch n sl)) t ->
      execinf_stmt e m (Sswitch a sl) t

(** [evalinf_funcall ge m fd args t] holds if the invocation of function
    [fd] on arguments [args] diverges, with observable trace [t]. *)

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

End SEMANTICS.
*)
(* * * Whole-program semantics *)

(* * Execution of whole programs are described as sequences of transitions
  from an initial state to a final state.  An initial state is a [Callstate]
  corresponding to the invocation of the ``main'' function of the program
  without arguments and with an empty continuation. *)

inductive initial_state (p: clight_program): state -> Prop :=
  | initial_state_intro: ∀b,f,ge,m0.
      globalenv Genv ?? p = OK ? ge →
      init_mem Genv ?? p = OK ? m0 →
      find_symbol ?? ge (prog_main ?? p) = Some ? b →
      find_funct_ptr ?? ge b = Some ? f →
      initial_state p (Callstate f (nil ?) Kstop m0).

(* * A final state is a [Returnstate] with an empty continuation. *)

inductive final_state: state -> int -> Prop :=
  | final_state_intro: ∀r,m.
      final_state (Returnstate (Vint r) Kstop m) r.

(* * Execution of a whole program: [exec_program p beh]
  holds if the application of [p]'s main function to no arguments
  in the initial memory state for [p] has [beh] as observable
  behavior. *)

definition exec_program : clight_program → program_behavior → Prop ≝ λp,beh.
  ∀ge. globalenv ??? p = OK ? ge →
  program_behaves (mk_transrel ?? step) (initial_state p) final_state ge beh.
(*
(** Big-step execution of a whole program.  *)

inductive bigstep_program_terminates (p: program): trace -> int -> Prop :=
  | bigstep_program_terminates_intro: ∀b,f,m1,t,r.
      let ge := Genv.globalenv p in 
      let m0 := Genv.init_mem p in
      Genv.find_symbol ge p.(prog_main) = Some b ->
      Genv.find_funct_ptr ge b = Some f ->
      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: ∀b,f,t.
      let ge := Genv.globalenv p in 
      let m0 := Genv.init_mem p in
      Genv.find_symbol ge p.(prog_main) = Some b ->
      Genv.find_funct_ptr ge b = Some f ->
      evalinf_funcall ge m0 f nil t ->
      bigstep_program_diverges p t.

(** * Implication from big-step semantics to transition semantics *)

Section BIGSTEP_TO_TRANSITIONS.

Variable prog: program.
Let ge : genv := Genv.globalenv prog.

Definition exec_stmt_eval_funcall_ind
  (PS: env -> mem -> statement -> trace -> mem -> outcome -> Prop)
  (PF: mem -> fundef -> list val -> trace -> mem -> val -> Prop) :=
  fun a b c d e f g h i j k l m n o p q r s t u v w x y =>
  conj (exec_stmt_ind2 ge PS PF a b c d e f g h i j k l m n o p q r s t u v w x y)
       (eval_funcall_ind2 ge PS PF a b c d e f g h i j k l m n o p q r s t u v w x y).

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: ∀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: ∀a,v,k'.
      call_cont k' = call_cont k ->
      eval_expr ge e m a v ->
      outcome_state_match e m f k
        (Out_return (Some v)) (State f (Sreturn (Some a)) k' e m).

Lemma is_call_cont_call_cont:
  ∀k. is_call_cont k -> call_cont k = k.
Proof.
  destruct k; simpl; intros; contradiction || auto.
Qed.

Lemma exec_stmt_eval_funcall_steps:
  (∀e,m,s,t,m',out.
   exec_stmt ge e m s t m' out ->
   ∀f,k. exists S,
   star step ge (State f s k e m) t S
   /\ outcome_state_match e m' f k out S)
/\
  (∀m,fd,args,t,m',res.
   eval_funcall ge m fd args t m' res ->
   ∀k.
   is_call_cont k ->
   star step ge (Callstate fd args k m) t (Returnstate res k m')).
Proof.
  apply exec_stmt_eval_funcall_ind; intros.

(* skip *)
  econstructor; split. apply star_refl. constructor.

(* assign *)
  econstructor; split. apply star_one. econstructor; eauto. constructor.

(* call none *)
  econstructor; split.
  eapply star_left. econstructor; eauto. 
  eapply star_right. apply H4. simpl; auto. econstructor. reflexivity. traceEq.
  constructor.

(* call some *)
  econstructor; split.
  eapply star_left. econstructor; eauto. 
  eapply star_right. apply H5. simpl; auto. econstructor; eauto. reflexivity. traceEq.
  constructor.

(* sequence 2 *)
  destruct (H0 f (Kseq s2 k)) as [S1 [A1 B1]]. inv B1.
  destruct (H2 f k) as [S2 [A2 B2]]. 
  econstructor; split.
  eapply star_left. econstructor.
  eapply star_trans. eexact A1. 
  eapply star_left. constructor. eexact A2.
  reflexivity. reflexivity. traceEq.
  auto.

(* sequence 1 *)
  destruct (H0 f (Kseq s2 k)) as [S1 [A1 B1]].
  set (S2 :=
    match out with
    | Out_break => State f Sbreak k e m1
    | Out_continue => State f Scontinue k e m1
    | _ => S1
    end).
  exists S2; split.
  eapply star_left. econstructor.
  eapply star_trans. eexact A1.
  unfold S2; inv B1.
    congruence.
    apply star_one. apply step_break_seq.
    apply star_one. apply step_continue_seq.
    apply star_refl.
    apply star_refl.
  reflexivity. traceEq.
  unfold S2; inv B1; congruence || econstructor; eauto.

(* ifthenelse true *)
  destruct (H2 f k) as [S1 [A1 B1]].
  exists S1; split.
  eapply star_left. eapply step_ifthenelse_true; eauto. eexact A1. traceEq.
  auto.

(* ifthenelse false *)
  destruct (H2 f k) as [S1 [A1 B1]].
  exists S1; split.
  eapply star_left. eapply step_ifthenelse_false; eauto. eexact A1. traceEq.
  auto.

(* return none *)
  econstructor; split. apply star_refl. constructor. auto.

(* return some *)
  econstructor; split. apply star_refl. econstructor; eauto.

(* break *)
  econstructor; split. apply star_refl. constructor.

(* continue *)
  econstructor; split. apply star_refl. constructor.

(* while false *)
  econstructor; split.
  apply star_one. eapply step_while_false; eauto. 
  constructor.

(* while stop *)
  destruct (H2 f (Kwhile a s k)) as [S1 [A1 B1]].
  set (S2 :=
    match out' with
    | Out_break => State f Sskip k e m'
    | _ => S1
    end).
  exists S2; split.
  eapply star_left. eapply step_while_true; eauto. 
  eapply star_trans. eexact A1.
  unfold S2. inversion H3; subst.
  inv B1. apply star_one. constructor.    
  apply star_refl.
  reflexivity. traceEq.
  unfold S2. inversion H3; subst. constructor. inv B1; econstructor; eauto.

(* while loop *)
  destruct (H2 f (Kwhile a s k)) as [S1 [A1 B1]].
  destruct (H5 f k) as [S2 [A2 B2]].
  exists S2; split.
  eapply star_left. eapply step_while_true; eauto.
  eapply star_trans. eexact A1.
  eapply star_left.
  inv H3; inv B1; apply step_skip_or_continue_while; auto.
  eexact A2.
  reflexivity. reflexivity. traceEq.
  auto.

(* dowhile false *)
  destruct (H0 f (Kdowhile a s k)) as [S1 [A1 B1]].
  exists (State f Sskip k e m1); split.
  eapply star_left. constructor. 
  eapply star_right. eexact A1.
  inv H1; inv B1; eapply step_skip_or_continue_dowhile_false; eauto.
  reflexivity. traceEq. 
  constructor.

(* dowhile stop *)
  destruct (H0 f (Kdowhile a s k)) as [S1 [A1 B1]].
  set (S2 :=
    match out1 with
    | Out_break => State f Sskip k e m1
    | _ => S1
    end).
  exists S2; split.
  eapply star_left. apply step_dowhile. 
  eapply star_trans. eexact A1.
  unfold S2. inversion H1; subst.
  inv B1. apply star_one. constructor.
  apply star_refl.
  reflexivity. traceEq.
  unfold S2. inversion H1; subst. constructor. inv B1; econstructor; eauto.

(* dowhile loop *)
  destruct (H0 f (Kdowhile a s k)) as [S1 [A1 B1]].
  destruct (H5 f k) as [S2 [A2 B2]].
  exists S2; split.
  eapply star_left. apply step_dowhile. 
  eapply star_trans. eexact A1.
  eapply star_left.
  inv H1; inv B1; eapply step_skip_or_continue_dowhile_true; eauto.
  eexact A2.
  reflexivity. reflexivity. traceEq.
  auto.

(* for start *)
  destruct (H1 f (Kseq (Sfor Sskip a2 a3 s) k)) as [S1 [A1 B1]]. inv B1.
  destruct (H3 f k) as [S2 [A2 B2]].
  exists S2; split.
  eapply star_left. apply step_for_start; auto.   
  eapply star_trans. eexact A1.
  eapply star_left. constructor. eexact A2.
  reflexivity. reflexivity. traceEq.
  auto.

(* for false *)
  econstructor; split.
  eapply star_one. eapply step_for_false; eauto. 
  constructor.

(* for stop *)
  destruct (H2 f (Kfor2 a2 a3 s k)) as [S1 [A1 B1]].
  set (S2 :=
    match out1 with
    | Out_break => State f Sskip k e m1
    | _ => S1
    end).
  exists S2; split.
  eapply star_left. eapply step_for_true; eauto. 
  eapply star_trans. eexact A1.
  unfold S2. inversion H3; subst.
  inv B1. apply star_one. constructor. 
  apply star_refl.
  reflexivity. traceEq.
  unfold S2. inversion H3; subst. constructor. inv B1; econstructor; eauto.

(* for loop *)
  destruct (H2 f (Kfor2 a2 a3 s k)) as [S1 [A1 B1]].
  destruct (H5 f (Kfor3 a2 a3 s k)) as [S2 [A2 B2]]. inv B2.
  destruct (H7 f k) as [S3 [A3 B3]].
  exists S3; split.
  eapply star_left. eapply step_for_true; eauto. 
  eapply star_trans. eexact A1.
  eapply star_trans with (s2 := State f a3 (Kfor3 a2 a3 s k) e m1).
  inv H3; inv B1.
  apply star_one. constructor. auto. 
  apply star_one. constructor. auto. 
  eapply star_trans. eexact A2. 
  eapply star_left. constructor.
  eexact A3.
  reflexivity. reflexivity. reflexivity. reflexivity. traceEq.
  auto.

(* switch *)
  destruct (H1 f (Kswitch k)) as [S1 [A1 B1]].
  set (S2 :=
    match out with
    | Out_normal => State f Sskip k e m1
    | Out_break => State f Sskip k e m1
    | Out_continue => State f Scontinue k e m1
    | _ => S1
    end).
  exists S2; split.
  eapply star_left. eapply step_switch; eauto. 
  eapply star_trans. eexact A1. 
  unfold S2; inv B1.
    apply star_one. constructor. auto. 
    apply star_one. constructor. auto. 
    apply star_one. constructor. 
    apply star_refl.
    apply star_refl.
  reflexivity. traceEq.
  unfold S2. inv B1; simpl; econstructor; eauto.

(* call internal *)
  destruct (H2 f k) as [S1 [A1 B1]].
  eapply star_left. eapply step_internal_function; eauto.
  eapply star_right. eexact A1. 
  inv B1; simpl in H3; try contradiction.
  (* Out_normal *)
  assert (fn_return f = Tvoid /\ vres = Vundef).
    destruct (fn_return f); auto || contradiction.
  destruct H5. subst vres. apply step_skip_call; auto.
  (* Out_return None *)
  assert (fn_return f = Tvoid /\ vres = Vundef).
    destruct (fn_return f); auto || contradiction.
  destruct H6. subst vres.
  rewrite <- (is_call_cont_call_cont k H4). rewrite <- H5.
  apply step_return_0; auto.
  (* Out_return Some *)
  destruct H3. subst vres.
  rewrite <- (is_call_cont_call_cont k H4). rewrite <- H5.
  eapply step_return_1; eauto.
  reflexivity. traceEq.

(* call external *)
  apply star_one. apply step_external_function; auto. 
Qed.

Lemma exec_stmt_steps:
   ∀e,m,s,t,m',out.
   exec_stmt ge e m s t m' out ->
   ∀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 exec_stmt_eval_funcall_steps).

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

Definition order (x y: unit) := False.

Lemma evalinf_funcall_forever:
  ∀m,fd,args,T,k.
  evalinf_funcall ge m fd args T ->
  forever_N step order ge tt (Callstate fd args k m) T.
Proof.
  cofix CIH_FUN.
  assert (∀e,m,s,T,f,k.
          execinf_stmt ge e m s T ->
          forever_N step order ge tt (State f s k e m) T).
  cofix CIH_STMT.
  intros. inv H.

(* call none *)
  eapply forever_N_plus.
  apply plus_one. eapply step_call_none; eauto. 
  apply CIH_FUN. eauto. traceEq.
(* call some *)
  eapply forever_N_plus.
  apply plus_one. eapply step_call_some; eauto. 
  apply CIH_FUN. eauto. traceEq.

(* seq 1 *)
  eapply forever_N_plus.
  apply plus_one. econstructor.
  apply CIH_STMT; eauto. traceEq.
(* seq 2 *)
  destruct (exec_stmt_steps _ _ _ _ _ _ H0 f (Kseq s2 k)) as [S1 [A1 B1]].
  inv B1.
  eapply forever_N_plus.
  eapply plus_left. constructor. eapply star_trans. eexact A1. 
  apply star_one. constructor. reflexivity. reflexivity.
  apply CIH_STMT; eauto. traceEq.

(* ifthenelse true *)
  eapply forever_N_plus.
  apply plus_one. eapply step_ifthenelse_true; eauto. 
  apply CIH_STMT; eauto. traceEq.
(* ifthenelse false *)
  eapply forever_N_plus.
  apply plus_one. eapply step_ifthenelse_false; eauto. 
  apply CIH_STMT; eauto. traceEq.

(* while body *)
  eapply forever_N_plus.
  eapply plus_one. eapply step_while_true; eauto.
  apply CIH_STMT; eauto. traceEq.
(* while loop *)
  destruct (exec_stmt_steps _ _ _ _ _ _ H2 f (Kwhile a s0 k)) as [S1 [A1 B1]].
  eapply forever_N_plus with (s2 := State f (Swhile a s0) k e m1).
  eapply plus_left. eapply step_while_true; eauto.
  eapply star_right. eexact A1.
  inv H3; inv B1; apply step_skip_or_continue_while; auto. 
  reflexivity. reflexivity.
  apply CIH_STMT; eauto. traceEq.

(* dowhile body *)
  eapply forever_N_plus.
  eapply plus_one. eapply step_dowhile.
  apply CIH_STMT; eauto.
  traceEq.

(* dowhile loop *)
  destruct (exec_stmt_steps _ _ _ _ _ _ H0 f (Kdowhile a s0 k)) as [S1 [A1 B1]].
  eapply forever_N_plus with (s2 := State f (Sdowhile a s0) k e m1).
  eapply plus_left. eapply step_dowhile. 
  eapply star_right. eexact A1.
  inv H1; inv B1; eapply step_skip_or_continue_dowhile_true; eauto. 
  reflexivity. reflexivity.
  apply CIH_STMT. eauto. 
  traceEq.

(* for start 1 *)
  assert (a1 <> Sskip). red; intros; subst. inv H0.
  eapply forever_N_plus.
  eapply plus_one. apply step_for_start; auto. 
  apply CIH_STMT; eauto.
  traceEq.

(* for start 2 *)
  destruct (exec_stmt_steps _ _ _ _ _ _ H1 f (Kseq (Sfor Sskip a2 a3 s0) k)) as [S1 [A1 B1]].
  inv B1.
  eapply forever_N_plus.
  eapply plus_left. eapply step_for_start; eauto. 
  eapply star_right. eexact A1.
  apply step_skip_seq. 
  reflexivity. reflexivity.
  apply CIH_STMT; eauto.
  traceEq.

(* for body *)
  eapply forever_N_plus.
  apply plus_one. eapply step_for_true; eauto. 
  apply CIH_STMT; eauto.
  traceEq.

(* for next *)
  destruct (exec_stmt_steps _ _ _ _ _ _ H2 f (Kfor2 a2 a3 s0 k)) as [S1 [A1 B1]].
  eapply forever_N_plus.
  eapply plus_left. eapply step_for_true; eauto.
  eapply star_trans. eexact A1.
  apply star_one.
  inv H3; inv B1; apply step_skip_or_continue_for2; auto.
  reflexivity. reflexivity. 
  apply CIH_STMT; eauto.
  traceEq.

(* for body *)
  destruct (exec_stmt_steps _ _ _ _ _ _ H2 f (Kfor2 a2 a3 s0 k)) as [S1 [A1 B1]].
  destruct (exec_stmt_steps _ _ _ _ _ _ H4 f (Kfor3 a2 a3 s0 k)) as [S2 [A2 B2]].
  inv B2.
  eapply forever_N_plus.
  eapply plus_left. eapply step_for_true; eauto. 
  eapply star_trans. eexact A1.
  eapply star_left. inv H3; inv B1; apply step_skip_or_continue_for2; auto.
  eapply star_right. eexact A2. 
  constructor. 
  reflexivity. reflexivity. reflexivity. reflexivity.  
  apply CIH_STMT; eauto.
  traceEq.

(* switch *)
  eapply forever_N_plus.
  eapply plus_one. eapply step_switch; eauto.
  apply CIH_STMT; eauto.
  traceEq.

(* call internal *)
  intros. inv H0.
  eapply forever_N_plus.
  eapply plus_one. econstructor; eauto. 
  apply H; eauto.
  traceEq.
Qed.

Theorem bigstep_program_terminates_exec:
  ∀t,r. bigstep_program_terminates prog t r -> exec_program prog (Terminates t r).
Proof.
  intros. inv H. unfold ge0, m0 in *. 
  econstructor.
  econstructor. eauto. eauto.
  apply eval_funcall_steps. eauto. red; auto. 
  econstructor.
Qed.

Theorem bigstep_program_diverges_exec:
  ∀T. bigstep_program_diverges prog T ->
  exec_program prog (Reacts T) \/
  exists t, exec_program prog (Diverges t) /\ traceinf_prefix t T.
Proof.
  intros. inv H.
  set (st := Callstate f nil Kstop m0).
  assert (forever step ge0 st T). 
    eapply forever_N_forever with (order := order).
    red; intros. constructor; intros. red in H. elim H.
    eapply evalinf_funcall_forever; eauto. 
  destruct (forever_silent_or_reactive _ _ _ _ _ _ H)
  as [A | [t [s' [T' [B [C D]]]]]]. 
  left. econstructor. econstructor. eauto. eauto. auto.
  right. exists t. split.
  econstructor. econstructor; eauto. eauto. auto. 
  subst T. rewrite <- (E0_right t) at 1. apply traceinf_prefix_app. constructor.
Qed.

End BIGSTEP_TO_TRANSITIONS.



*)