Definition find_function_ptr
        (ge: genv) (ros: mreg + ident) (rs: regset) : option block :=
  match ros with
  | inl r =>
      match rs r with
      | Vptr b ofs => if Ptrofs.eq ofs Ptrofs.zero then Some b else None
      | _ => None
      end
  | inr symb =>
      Genv.find_symbol ge symb
  end.(** Extract the values of the arguments to an external call. *)

Inductive extcall_arg (rs: regset) (m: mem) (sp: val): loc -> val -> Prop :=
  | extcall_arg_reg: forall r,
      extcall_arg rs m sp (R r) (rs r)
  | extcall_arg_stack: forall ofs ty v,
      load_stack m sp ty (Ptrofs.repr (Stacklayout.fe_ofs_arg + 4 * ofs)) = Some v ->
      extcall_arg rs m sp (S Outgoing ofs ty) v.(** Extract the values of the arguments to an external call. *)

Inductive extcall_arg (rs: regset) (m: mem) (sp: val): loc -> val -> Prop :=
  | extcall_arg_reg: forall r,
      extcall_arg rs m sp (R r) (rs r)
  | extcall_arg_stack: forall ofs ty v,
      load_stack m sp ty (Ptrofs.repr (Stacklayout.fe_ofs_arg + 4 * ofs)) = Some v ->
      extcall_arg rs m sp (S Outgoing ofs ty) v.(** Extract the values of the arguments to an external call. *)

Inductive extcall_arg (rs: regset) (m: mem) (sp: val): loc -> val -> Prop :=
  | extcall_arg_reg: forall r,
      extcall_arg rs m sp (R r) (rs r)
  | extcall_arg_stack: forall ofs ty v,
      load_stack m sp ty (Ptrofs.repr (Stacklayout.fe_ofs_arg + 4 * ofs)) = Some v ->
      extcall_arg rs m sp (S Outgoing ofs ty) v.Inductive extcall_arg_pair (rs: regset) (m: mem) (sp: val): rpair loc -> val -> Prop :=
  | extcall_arg_one: forall l v,
      extcall_arg rs m sp l v ->
      extcall_arg_pair rs m sp (One l) v
  | extcall_arg_twolong: forall hi lo vhi vlo,
      extcall_arg rs m sp hi vhi ->
      extcall_arg rs m sp lo vlo ->
      extcall_arg_pair rs m sp (Twolong hi lo) (Val.longofwords vhi vlo).Inductive extcall_arg_pair (rs: regset) (m: mem) (sp: val): rpair loc -> val -> Prop :=
  | extcall_arg_one: forall l v,
      extcall_arg rs m sp l v ->
      extcall_arg_pair rs m sp (One l) v
  | extcall_arg_twolong: forall hi lo vhi vlo,
      extcall_arg rs m sp hi vhi ->
      extcall_arg rs m sp lo vlo ->
      extcall_arg_pair rs m sp (Twolong hi lo) (Val.longofwords vhi vlo).Inductive extcall_arg_pair (rs: regset) (m: mem) (sp: val): rpair loc -> val -> Prop :=
  | extcall_arg_one: forall l v,
      extcall_arg rs m sp l v ->
      extcall_arg_pair rs m sp (One l) v
  | extcall_arg_twolong: forall hi lo vhi vlo,
      extcall_arg rs m sp hi vhi ->
      extcall_arg rs m sp lo vlo ->
      extcall_arg_pair rs m sp (Twolong hi lo) (Val.longofwords vhi vlo).Definition extcall_arguments
    (rs: regset) (m: mem) (sp: val) (sg: signature) (args: list val) : Prop :=
  list_forall2 (extcall_arg_pair rs m sp) (loc_arguments sg) args.(** Mach execution states. *)

(** Mach execution states. *)

Inductive stackframe: Type :=
  | Stackframe:
      forall (f: block)       (**r pointer to calling function *)
             (sp: val)        (**r stack pointer in calling function *)
             (retaddr: val)   (**r Asm return address in calling function *)
             (c: code),       (**r program point in calling function *)
      stackframe.(** Mach execution states. *)

(** Mach execution states. *)

Inductive stackframe: Type :=
  | Stackframe:
      forall (f: block)       (**r pointer to calling function *)
             (sp: val)        (**r stack pointer in calling function *)
             (retaddr: val)   (**r Asm return address in calling function *)
             (c: code),       (**r program point in calling function *)
      stackframe.Inductive state: Type :=
  | State:
      forall (stack: list stackframe)  (**r call stack *)
             (f: block)                (**r pointer to current function *)
             (sp: val)                 (**r stack pointer *)
             (c: code)                 (**r current program point *)
             (rs: regset)              (**r register state *)
             (m: mem),                 (**r memory state *)
      state
  | Callstate:
      forall (stack: list stackframe)  (**r call stack *)
             (f: block)                (**r pointer to function to call *)
             (rs: regset)              (**r register state *)
             (m: mem),                 (**r memory state *)
      state
  | Returnstate:
      forall (stack: list stackframe)  (**r call stack *)
             (rs: regset)              (**r register state *)
             (m: mem),                 (**r memory state *)
      state.Inductive state: Type :=
  | State:
      forall (stack: list stackframe)  (**r call stack *)
             (f: block)                (**r pointer to current function *)
             (sp: val)                 (**r stack pointer *)
             (c: code)                 (**r current program point *)
             (rs: regset)              (**r register state *)
             (m: mem),                 (**r memory state *)
      state
  | Callstate:
      forall (stack: list stackframe)  (**r call stack *)
             (f: block)                (**r pointer to function to call *)
             (rs: regset)              (**r register state *)
             (m: mem),                 (**r memory state *)
      state
  | Returnstate:
      forall (stack: list stackframe)  (**r call stack *)
             (rs: regset)              (**r register state *)
             (m: mem),                 (**r memory state *)
      state.Inductive state: Type :=
  | State:
      forall (stack: list stackframe)  (**r call stack *)
             (f: block)                (**r pointer to current function *)
             (sp: val)                 (**r stack pointer *)
             (c: code)                 (**r current program point *)
             (rs: regset)              (**r register state *)
             (m: mem),                 (**r memory state *)
      state
  | Callstate:
      forall (stack: list stackframe)  (**r call stack *)
             (f: block)                (**r pointer to function to call *)
             (rs: regset)              (**r register state *)
             (m: mem),                 (**r memory state *)
      state
  | Returnstate:
      forall (stack: list stackframe)  (**r call stack *)
             (rs: regset)              (**r register state *)
             (m: mem),                 (**r memory state *)
      state.Inductive state: Type :=
  | State:
      forall (stack: list stackframe)  (**r call stack *)
             (f: block)                (**r pointer to current function *)
             (sp: val)                 (**r stack pointer *)
             (c: code)                 (**r current program point *)
             (rs: regset)              (**r register state *)
             (m: mem),                 (**r memory state *)
      state
  | Callstate:
      forall (stack: list stackframe)  (**r call stack *)
             (f: block)                (**r pointer to function to call *)
             (rs: regset)              (**r register state *)
             (m: mem),                 (**r memory state *)
      state
  | Returnstate:
      forall (stack: list stackframe)  (**r call stack *)
             (rs: regset)              (**r register state *)
             (m: mem),                 (**r memory state *)
      state.Definition parent_sp (s: list stackframe) : val :=
  match s with
  | nil => Vnullptr
  | Stackframe f sp ra c :: s' => sp
  end.Definition parent_ra (s: list stackframe) : val :=
  match s with
  | nil => Vnullptr
  | Stackframe f sp ra c :: s' => ra
  end.Inductive initial_state (p: program): state -> Prop :=
  | initial_state_intro: forall fb m0,
      let ge := Genv.globalenv p in
      Genv.init_mem p = Some m0 ->
      Genv.find_symbol ge p.(prog_main) = Some fb ->
      initial_state p (Callstate nil fb (Regmap.init Vundef) m0).Inductive initial_state (p: program): state -> Prop :=
  | initial_state_intro: forall fb m0,
      let ge := Genv.globalenv p in
      Genv.init_mem p = Some m0 ->
      Genv.find_symbol ge p.(prog_main) = Some fb ->
      initial_state p (Callstate nil fb (Regmap.init Vundef) m0).Inductive initial_state (p: program): state -> Prop :=
  | initial_state_intro: forall fb m0,
      let ge := Genv.globalenv p in
      Genv.init_mem p = Some m0 ->
      Genv.find_symbol ge p.(prog_main) = Some fb ->
      initial_state p (Callstate nil fb (Regmap.init Vundef) m0).Inductive final_state: state -> int -> Prop :=
  | final_state_intro: forall rs m r retcode,
      loc_result signature_main = One r ->
      rs r = Vint retcode ->
      final_state (Returnstate nil rs m) retcode.Inductive final_state: state -> int -> Prop :=
  | final_state_intro: forall rs m r retcode,
      loc_result signature_main = One r ->
      rs r = Vint retcode ->
      final_state (Returnstate nil rs m) retcode.Definition semantics (rao: function -> code -> ptrofs -> Prop) (p: program) :=
  Semantics (step rao) (initial_state p) final_state (Genv.globalenv p).(** * Leaf functions *)

(** A leaf function is a function that contains no [Mcall] instruction. *)

Definition is_leaf_function (f: function) : bool :=
  List.forallb
    (fun i => match i with Mcall _ _ => false | _ => true end)
    f.(fn_code).Inductive wf_frame: stackframe -> Prop :=
  | wf_stackframe_intro: forall fb sp ra c f
        (CODE: Genv.find_funct_ptr ge fb = Some (Internal f))
        (LEAF: is_leaf_function f = false)
        (TAIL: is_tail c f.(fn_code)),
      wf_frame (Stackframe fb sp ra c).Inductive wf_frame: stackframe -> Prop :=
  | wf_stackframe_intro: forall fb sp ra c f
        (CODE: Genv.find_funct_ptr ge fb = Some (Internal f))
        (LEAF: is_leaf_function f = false)
        (TAIL: is_tail c f.(fn_code)),
      wf_frame (Stackframe fb sp ra c).Inductive wf_state: state -> Prop :=
  | wf_normal_state: forall s fb sp c rs m f
        (STACK: Forall wf_frame s)
        (CODE: Genv.find_funct_ptr ge fb = Some (Internal f))
        (TAIL: is_tail c f.(fn_code)),
      wf_state (State s fb sp c rs m)
  | wf_call_state: forall s fb rs m
        (STACK: Forall wf_frame s),
      wf_state (Callstate s fb rs m)
  | wf_return_state: forall s rs m
        (STACK: Forall wf_frame s),
      wf_state (Returnstate s rs m).Inductive wf_state: state -> Prop :=
  | wf_normal_state: forall s fb sp c rs m f
        (STACK: Forall wf_frame s)
        (CODE: Genv.find_funct_ptr ge fb = Some (Internal f))
        (TAIL: is_tail c f.(fn_code)),
      wf_state (State s fb sp c rs m)
  | wf_call_state: forall s fb rs m
        (STACK: Forall wf_frame s),
      wf_state (Callstate s fb rs m)
  | wf_return_state: forall s rs m
        (STACK: Forall wf_frame s),
      wf_state (Returnstate s rs m).Inductive wf_state: state -> Prop :=
  | wf_normal_state: forall s fb sp c rs m f
        (STACK: Forall wf_frame s)
        (CODE: Genv.find_funct_ptr ge fb = Some (Internal f))
        (TAIL: is_tail c f.(fn_code)),
      wf_state (State s fb sp c rs m)
  | wf_call_state: forall s fb rs m
        (STACK: Forall wf_frame s),
      wf_state (Callstate s fb rs m)
  | wf_return_state: forall s rs m
        (STACK: Forall wf_frame s),
      wf_state (Returnstate s rs m).Inductive wf_state: state -> Prop :=
  | wf_normal_state: forall s fb sp c rs m f
        (STACK: Forall wf_frame s)
        (CODE: Genv.find_funct_ptr ge fb = Some (Internal f))
        (TAIL: is_tail c f.(fn_code)),
      wf_state (State s fb sp c rs m)
  | wf_call_state: forall s fb rs m
        (STACK: Forall wf_frame s),
      wf_state (Callstate s fb rs m)
  | wf_return_state: forall s rs m
        (STACK: Forall wf_frame s),
      wf_state (Returnstate s rs m).

Lemma wf_initial: forall p S, initial_state p S -> wf_state (Genv.globalenv p) S.