(* *********************************************************************)
(*                                                                     *)
(*              The Compcert verified compiler                         *)
(*                                                                     *)
(*                Xavier Leroy, INRIA Paris                            *)
(*                                                                     *)
(*  Copyright Institut National de Recherche en Informatique et en     *)
(*  Automatique.  All rights reserved.  This file is distributed       *)
(*  under the terms of the INRIA Non-Commercial License Agreement.     *)
(*                                                                     *)
(* *********************************************************************)

(** Function calling conventions and other conventions regarding the use of
    machine registers and stack slots. *)

Require Import Coqlib Decidableplus.
Require Import AST Machregs Locations.

(** * Classification of machine registers *)

(** Machine registers (type [mreg] in module [Locations]) are divided in
  the following groups:
- Callee-save registers, whose value is preserved across a function call.
- Caller-save registers that can be modified during a function call.

  We follow the x86-32 and x86-64 application binary interfaces (ABI)
  in our choice of callee- and caller-save registers.
*)

Definition is_callee_save (r: mreg) : bool :=
  match r with
  | AX | CX | DX => false
  | BX | BP => true
  | SI | DI => negb Archi.ptr64 (**r callee-save in 32 bits but not in 64 bits *)
  | R8 | R9 | R10 | R11 => false
  | R12 | R13 | R14 | R15 => true
  | X0 | X1 | X2 | X3 | X4 | X5 | X6 | X7 => false
  | X8 | X9 | X10 | X11 | X12 | X13 | X14 | X15 => false
  | FP0 => false
  end.

Definition int_caller_save_regs :=
  if Archi.ptr64
  then AX :: CX :: DX :: SI :: DI :: R8 :: R9 :: R10 :: R11 :: nil
  else AX :: CX :: DX :: nil.

Definition float_caller_save_regs :=
  if Archi.ptr64
  then X0 :: X1 :: X2 :: X3 :: X4 :: X5 :: X6 :: X7 ::
       X8 :: X9 :: X10 :: X11 :: X12 :: X13 :: X14 :: X15 :: nil
  else X0 :: X1 :: X2 :: X3 :: X4 :: X5 :: X6 :: X7 :: nil.

Definition int_callee_save_regs :=
  if Archi.ptr64
  then BX :: BP :: R12 :: R13 :: R14 :: R15 :: nil
  else BX :: SI :: DI :: BP :: nil.

Definition float_callee_save_regs : list mreg := nil.

Definition destroyed_at_call :=
  List.filter (fun r => negb (is_callee_save r)) all_mregs.

Definition dummy_int_reg := AX.     (**r Used in [Regalloc]. *)
Definition dummy_float_reg := X0.   (**r Used in [Regalloc]. *)

Definition callee_save_type := mreg_type.
  
Definition is_float_reg (r: mreg) :=
  match r with
  | AX | BX | CX | DX | SI | DI | BP
  | R8 | R9 | R10 | R11 | R12 | R13 | R14 | R15 => false
  | X0 | X1 | X2 | X3 | X4 | X5 | X6 | X7
  | X8 | X9 | X10 | X11 | X12 | X13 | X14 | X15 | FP0 => true
  end.

(** * Function calling conventions *)

(** The functions in this section determine the locations (machine registers
  and stack slots) used to communicate arguments and results between the
  caller and the callee during function calls.  These locations are functions
  of the signature of the function and of the call instruction.
  Agreement between the caller and the callee on the locations to use
  is guaranteed by our dynamic semantics for Cminor and RTL, which demand
  that the signature of the call instruction is identical to that of the
  called function.

  Calling conventions are largely arbitrary: they must respect the properties
  proved in this section (such as no overlapping between the locations
  of function arguments), but this leaves much liberty in choosing actual
  locations.  To ensure binary interoperability of code generated by our
  compiler with libraries compiled by another compiler, we
  implement the standard x86-32 and x86-64 conventions. *)

(** ** Location of function result *)

(** In 32 bit mode, the result value of a function is passed back to the
  caller in registers [AX] or [DX:AX] or [FP0], depending on the type
  of the returned value.  We treat a function without result as a
  function with one integer result. *)

Definition loc_result_32 (s: signature) : rpair mreg :=
  match s.(sig_res) with
  | None => One AX
  | Some (Tint | Tany32) => One AX
  | Some (Tfloat | Tsingle) => One FP0
  | Some Tany64 => One X0
  | Some Tlong => Twolong DX AX
  end.

(** In 64 bit mode, he result value of a function is passed back to
  the caller in registers [AX] or [X0]. *)

Definition loc_result_64 (s: signature) : rpair mreg :=
  match s.(sig_res) with
  | None => One AX
  | Some (Tint | Tlong | Tany32 | Tany64) => One AX
  | Some (Tfloat | Tsingle) => One X0
  end.

Definition loc_result :=
  if Archi.ptr64 then loc_result_64 else loc_result_32.

(** The result registers have types compatible with that given in the signature. *)

Lemma loc_result_type:
  forall sig,
  subtype (proj_sig_res sig) (typ_rpair mreg_type (loc_result sig)) = true.
Proof.
(* Goal: forall (ty0 : typ) (_ : @In loc (S Outgoing ofs ty) (@regs_of_rpairs loc match @list_nth_z mreg float_param_regs fr with | Some ireg => @cons (rpair loc) (@One loc (R ireg)) (loc_arguments_64 tyl ir (Z.add fr (Zpos xH)) ofs0) | None => @cons (rpair loc) (@One loc (S Outgoing ofs0 ty0)) (loc_arguments_64 tyl ir fr (Z.add ofs0 (Zpos (xO xH)))) end)), Z.le (Z.add ofs (typesize ty)) match @list_nth_z mreg float_param_regs fr with | Some m => size_arguments_64 tyl ir (Z.add fr (Zpos xH)) ofs0 | None => size_arguments_64 tyl ir fr (Z.add ofs0 (Zpos (xO xH))) end *)
  intros. unfold proj_sig_res, loc_result, loc_result_32, loc_result_64, mreg_type;
  destruct Archi.ptr64; destruct (sig_res sig) as [[]|]; auto.
Qed.

(** The result locations are caller-save registers *)

Lemma loc_result_caller_save:
  forall (s: signature),
  forall_rpair (fun r => is_callee_save r = false) (loc_result s).
Proof.
(* Goal: forall (ty0 : typ) (_ : @In loc (S Outgoing ofs ty) (@regs_of_rpairs loc match @list_nth_z mreg float_param_regs fr with | Some ireg => @cons (rpair loc) (@One loc (R ireg)) (loc_arguments_64 tyl ir (Z.add fr (Zpos xH)) ofs0) | None => @cons (rpair loc) (@One loc (S Outgoing ofs0 ty0)) (loc_arguments_64 tyl ir fr (Z.add ofs0 (Zpos (xO xH)))) end)), Z.le (Z.add ofs (typesize ty)) match @list_nth_z mreg float_param_regs fr with | Some m => size_arguments_64 tyl ir (Z.add fr (Zpos xH)) ofs0 | None => size_arguments_64 tyl ir fr (Z.add ofs0 (Zpos (xO xH))) end *)
  intros. unfold loc_result, loc_result_32, loc_result_64, is_callee_save;
  destruct Archi.ptr64; destruct (sig_res s) as [[]|]; simpl; auto.
Qed.

(** If the result is in a pair of registers, those registers are distinct and have type [Tint] at least. *)

Lemma loc_result_pair:
  forall sg,
  match loc_result sg with
  | One _ => True
  | Twolong r1 r2 =>
       r1 <> r2 /\ sg.(sig_res) = Some Tlong
    /\ subtype Tint (mreg_type r1) = true /\ subtype Tint (mreg_type r2) = true
    /\ Archi.ptr64 = false
  end.
Proof.
(* Goal: forall (ty0 : typ) (_ : @In loc (S Outgoing ofs ty) (@regs_of_rpairs loc match @list_nth_z mreg float_param_regs fr with | Some ireg => @cons (rpair loc) (@One loc (R ireg)) (loc_arguments_64 tyl ir (Z.add fr (Zpos xH)) ofs0) | None => @cons (rpair loc) (@One loc (S Outgoing ofs0 ty0)) (loc_arguments_64 tyl ir fr (Z.add ofs0 (Zpos (xO xH)))) end)), Z.le (Z.add ofs (typesize ty)) match @list_nth_z mreg float_param_regs fr with | Some m => size_arguments_64 tyl ir (Z.add fr (Zpos xH)) ofs0 | None => size_arguments_64 tyl ir fr (Z.add ofs0 (Zpos (xO xH))) end *)
  intros. 
  unfold loc_result, loc_result_32, loc_result_64, mreg_type;
  destruct Archi.ptr64; destruct (sig_res sg) as [[]|]; auto.
(* Goal: and (Z.ge ofs Z0) (Z.divide (typealign ty) ofs) *)
  split; auto. congruence.
Qed.

(** The location of the result depends only on the result part of the signature *)

Lemma loc_result_exten:
  forall s1 s2, s1.(sig_res) = s2.(sig_res) -> loc_result s1 = loc_result s2.
Proof.
(* Goal: forall (ty0 : typ) (_ : @In loc (S Outgoing ofs ty) (@regs_of_rpairs loc match @list_nth_z mreg float_param_regs fr with | Some ireg => @cons (rpair loc) (@One loc (R ireg)) (loc_arguments_64 tyl ir (Z.add fr (Zpos xH)) ofs0) | None => @cons (rpair loc) (@One loc (S Outgoing ofs0 ty0)) (loc_arguments_64 tyl ir fr (Z.add ofs0 (Zpos (xO xH)))) end)), Z.le (Z.add ofs (typesize ty)) match @list_nth_z mreg float_param_regs fr with | Some m => size_arguments_64 tyl ir (Z.add fr (Zpos xH)) ofs0 | None => size_arguments_64 tyl ir fr (Z.add ofs0 (Zpos (xO xH))) end *)
  intros. unfold loc_result, loc_result_32, loc_result_64.
(* Goal: @eq (rpair mreg) ((if Archi.ptr64 then fun s : signature => match sig_res s with | Some (Tint as t) => @One mreg AX | Some (Tfloat as t) => @One mreg X0 | Some (Tlong as t) => @One mreg AX | Some (Tsingle as t) => @One mreg X0 | Some (Tany32 as t) => @One mreg AX | Some (Tany64 as t) => @One mreg AX | None => @One mreg AX end else fun s : signature => match sig_res s with | Some (Tint as t) => @One mreg AX | Some (Tfloat as t) => @One mreg FP0 | Some (Tlong as t) => @Twolong mreg DX AX | Some (Tsingle as t) => @One mreg FP0 | Some (Tany32 as t) => @One mreg AX | Some (Tany64 as t) => @One mreg X0 | None => @One mreg AX end) s1) ((if Archi.ptr64 then fun s : signature => match sig_res s with | Some (Tint as t) => @One mreg AX | Some (Tfloat as t) => @One mreg X0 | Some (Tlong as t) => @One mreg AX | Some (Tsingle as t) => @One mreg X0 | Some (Tany32 as t) => @One mreg AX | Some (Tany64 as t) => @One mreg AX | None => @One mreg AX end else fun s : signature => match sig_res s with | Some (Tint as t) => @One mreg AX | Some (Tfloat as t) => @One mreg FP0 | Some (Tlong as t) => @Twolong mreg DX AX | Some (Tsingle as t) => @One mreg FP0 | Some (Tany32 as t) => @One mreg AX | Some (Tany64 as t) => @One mreg X0 | None => @One mreg AX end) s2) *)
  destruct Archi.ptr64; rewrite H; auto.
Qed.

(** ** Location of function arguments *)

(** In the x86-32 ABI, all arguments are passed on stack. (Snif.) *)

Fixpoint loc_arguments_32
(* Goal: forall (ty0 : typ) (_ : @In loc (S Outgoing ofs ty) (@regs_of_rpairs loc match @list_nth_z mreg float_param_regs fr with | Some ireg => @cons (rpair loc) (@One loc (R ireg)) (loc_arguments_64 tyl ir (Z.add fr (Zpos xH)) ofs0) | None => @cons (rpair loc) (@One loc (S Outgoing ofs0 ty0)) (loc_arguments_64 tyl ir fr (Z.add ofs0 (Zpos (xO xH)))) end)), Z.le (Z.add ofs (typesize ty)) match @list_nth_z mreg float_param_regs fr with | Some m => size_arguments_64 tyl ir (Z.add fr (Zpos xH)) ofs0 | None => size_arguments_64 tyl ir fr (Z.add ofs0 (Zpos (xO xH))) end *)
(* Goal: Z.le (Z.add ofs (typesize ty)) match a with | Tint => match @list_nth_z mreg int_param_regs ir with | Some ireg => size_arguments_64 tyl (Z.add ir (Zpos xH)) fr ofs0 | None => size_arguments_64 tyl ir fr (Z.add ofs0 (Zpos (xO xH))) end | Tfloat => match @list_nth_z mreg float_param_regs fr with | Some freg => size_arguments_64 tyl ir (Z.add fr (Zpos xH)) ofs0 | None => size_arguments_64 tyl ir fr (Z.add ofs0 (Zpos (xO xH))) end | Tlong => match @list_nth_z mreg int_param_regs ir with | Some ireg => size_arguments_64 tyl (Z.add ir (Zpos xH)) fr ofs0 | None => size_arguments_64 tyl ir fr (Z.add ofs0 (Zpos (xO xH))) end | Tsingle => match @list_nth_z mreg float_param_regs fr with | Some freg => size_arguments_64 tyl ir (Z.add fr (Zpos xH)) ofs0 | None => size_arguments_64 tyl ir fr (Z.add ofs0 (Zpos (xO xH))) end | Tany32 => match @list_nth_z mreg int_param_regs ir with | Some ireg => size_arguments_64 tyl (Z.add ir (Zpos xH)) fr ofs0 | None => size_arguments_64 tyl ir fr (Z.add ofs0 (Zpos (xO xH))) end | Tany64 => match @list_nth_z mreg int_param_regs ir with | Some ireg => size_arguments_64 tyl (Z.add ir (Zpos xH)) fr ofs0 | None => size_arguments_64 tyl ir fr (Z.add ofs0 (Zpos (xO xH))) end end *)
    (tyl: list typ) (ofs: Z) {struct tyl} : list (rpair loc) :=
  match tyl with
  | nil => nil
  | ty :: tys =>
      match ty with
      | Tlong => Twolong (S Outgoing (ofs + 1) Tint) (S Outgoing ofs Tint)
      | _     => One (S Outgoing ofs ty)
      end
      :: loc_arguments_32 tys (ofs + typesize ty)
  end.

(** In the x86-64 ABI:
- The first 6 integer arguments are passed in registers [DI], [SI], [DX], [CX], [R8], [R9].
- The first 8 floating-point arguments are passed in registers [X0] to [X7].
- Extra arguments are passed on the stack, in [Outgoing] slots.
  Consecutive stack slots are separated by 8 bytes, even if only 4 bytes
  of data is used in a slot.
*)

Definition int_param_regs := DI :: SI :: DX :: CX :: R8 :: R9 :: nil.
Definition float_param_regs := X0 :: X1 :: X2 :: X3 :: X4 :: X5 :: X6 :: X7 :: nil.

Fixpoint loc_arguments_64
(* Goal: forall (ty0 : typ) (_ : @In loc (S Outgoing ofs ty) (@regs_of_rpairs loc match @list_nth_z mreg float_param_regs fr with | Some ireg => @cons (rpair loc) (@One loc (R ireg)) (loc_arguments_64 tyl ir (Z.add fr (Zpos xH)) ofs0) | None => @cons (rpair loc) (@One loc (S Outgoing ofs0 ty0)) (loc_arguments_64 tyl ir fr (Z.add ofs0 (Zpos (xO xH)))) end)), Z.le (Z.add ofs (typesize ty)) match @list_nth_z mreg float_param_regs fr with | Some m => size_arguments_64 tyl ir (Z.add fr (Zpos xH)) ofs0 | None => size_arguments_64 tyl ir fr (Z.add ofs0 (Zpos (xO xH))) end *)
(* Goal: Z.le (Z.add ofs (typesize ty)) match a with | Tint => match @list_nth_z mreg int_param_regs ir with | Some ireg => size_arguments_64 tyl (Z.add ir (Zpos xH)) fr ofs0 | None => size_arguments_64 tyl ir fr (Z.add ofs0 (Zpos (xO xH))) end | Tfloat => match @list_nth_z mreg float_param_regs fr with | Some freg => size_arguments_64 tyl ir (Z.add fr (Zpos xH)) ofs0 | None => size_arguments_64 tyl ir fr (Z.add ofs0 (Zpos (xO xH))) end | Tlong => match @list_nth_z mreg int_param_regs ir with | Some ireg => size_arguments_64 tyl (Z.add ir (Zpos xH)) fr ofs0 | None => size_arguments_64 tyl ir fr (Z.add ofs0 (Zpos (xO xH))) end | Tsingle => match @list_nth_z mreg float_param_regs fr with | Some freg => size_arguments_64 tyl ir (Z.add fr (Zpos xH)) ofs0 | None => size_arguments_64 tyl ir fr (Z.add ofs0 (Zpos (xO xH))) end | Tany32 => match @list_nth_z mreg int_param_regs ir with | Some ireg => size_arguments_64 tyl (Z.add ir (Zpos xH)) fr ofs0 | None => size_arguments_64 tyl ir fr (Z.add ofs0 (Zpos (xO xH))) end | Tany64 => match @list_nth_z mreg int_param_regs ir with | Some ireg => size_arguments_64 tyl (Z.add ir (Zpos xH)) fr ofs0 | None => size_arguments_64 tyl ir fr (Z.add ofs0 (Zpos (xO xH))) end end *)
    (tyl: list typ) (ir fr ofs: Z) {struct tyl} : list (rpair loc) :=
  match tyl with
  | nil => nil
  | (Tint | Tlong | Tany32 | Tany64) as ty :: tys =>
      match list_nth_z int_param_regs ir with
      | None =>
          One (S Outgoing ofs ty) :: loc_arguments_64 tys ir fr (ofs + 2)
      | Some ireg =>
          One (R ireg) :: loc_arguments_64 tys (ir + 1) fr ofs
      end
  | (Tfloat | Tsingle) as ty :: tys =>
      match list_nth_z float_param_regs fr with
      | None =>
          One (S Outgoing ofs ty) :: loc_arguments_64 tys ir fr (ofs + 2)
      | Some freg =>
          One (R freg) :: loc_arguments_64 tys ir (fr + 1) ofs
      end
  end.

(** [loc_arguments s] returns the list of locations where to store arguments
  when calling a function with signature [s].  *)

Definition loc_arguments (s: signature) : list (rpair loc) :=
  if Archi.ptr64
  then loc_arguments_64 s.(sig_args) 0 0 0
  else loc_arguments_32 s.(sig_args) 0.

(** [size_arguments s] returns the number of [Outgoing] slots used
  to call a function with signature [s]. *)

Fixpoint size_arguments_32
(* Goal: forall (ty0 : typ) (_ : @In loc (S Outgoing ofs ty) (@regs_of_rpairs loc match @list_nth_z mreg float_param_regs fr with | Some ireg => @cons (rpair loc) (@One loc (R ireg)) (loc_arguments_64 tyl ir (Z.add fr (Zpos xH)) ofs0) | None => @cons (rpair loc) (@One loc (S Outgoing ofs0 ty0)) (loc_arguments_64 tyl ir fr (Z.add ofs0 (Zpos (xO xH)))) end)), Z.le (Z.add ofs (typesize ty)) match @list_nth_z mreg float_param_regs fr with | Some m => size_arguments_64 tyl ir (Z.add fr (Zpos xH)) ofs0 | None => size_arguments_64 tyl ir fr (Z.add ofs0 (Zpos (xO xH))) end *)
(* Goal: Z.le (Z.add ofs (typesize ty)) match a with | Tint => match @list_nth_z mreg int_param_regs ir with | Some ireg => size_arguments_64 tyl (Z.add ir (Zpos xH)) fr ofs0 | None => size_arguments_64 tyl ir fr (Z.add ofs0 (Zpos (xO xH))) end | Tfloat => match @list_nth_z mreg float_param_regs fr with | Some freg => size_arguments_64 tyl ir (Z.add fr (Zpos xH)) ofs0 | None => size_arguments_64 tyl ir fr (Z.add ofs0 (Zpos (xO xH))) end | Tlong => match @list_nth_z mreg int_param_regs ir with | Some ireg => size_arguments_64 tyl (Z.add ir (Zpos xH)) fr ofs0 | None => size_arguments_64 tyl ir fr (Z.add ofs0 (Zpos (xO xH))) end | Tsingle => match @list_nth_z mreg float_param_regs fr with | Some freg => size_arguments_64 tyl ir (Z.add fr (Zpos xH)) ofs0 | None => size_arguments_64 tyl ir fr (Z.add ofs0 (Zpos (xO xH))) end | Tany32 => match @list_nth_z mreg int_param_regs ir with | Some ireg => size_arguments_64 tyl (Z.add ir (Zpos xH)) fr ofs0 | None => size_arguments_64 tyl ir fr (Z.add ofs0 (Zpos (xO xH))) end | Tany64 => match @list_nth_z mreg int_param_regs ir with | Some ireg => size_arguments_64 tyl (Z.add ir (Zpos xH)) fr ofs0 | None => size_arguments_64 tyl ir fr (Z.add ofs0 (Zpos (xO xH))) end end *)
    (tyl: list typ) (ofs: Z) {struct tyl} : Z :=
  match tyl with
  | nil => ofs
  | ty :: tys => size_arguments_32 tys (ofs + typesize ty)
  end.

(* Goal: forall (ty0 : typ) (_ : @In loc (S Outgoing ofs ty) (@regs_of_rpairs loc match @list_nth_z mreg float_param_regs fr with | Some ireg => @cons (rpair loc) (@One loc (R ireg)) (loc_arguments_64 tyl ir (Z.add fr (Zpos xH)) ofs0) | None => @cons (rpair loc) (@One loc (S Outgoing ofs0 ty0)) (loc_arguments_64 tyl ir fr (Z.add ofs0 (Zpos (xO xH)))) end)), Z.le (Z.add ofs (typesize ty)) match @list_nth_z mreg float_param_regs fr with | Some m => size_arguments_64 tyl ir (Z.add fr (Zpos xH)) ofs0 | None => size_arguments_64 tyl ir fr (Z.add ofs0 (Zpos (xO xH))) end *)
(* Goal: Z.le (Z.add ofs (typesize ty)) match a with | Tint => match @list_nth_z mreg int_param_regs ir with | Some ireg => size_arguments_64 tyl (Z.add ir (Zpos xH)) fr ofs0 | None => size_arguments_64 tyl ir fr (Z.add ofs0 (Zpos (xO xH))) end | Tfloat => match @list_nth_z mreg float_param_regs fr with | Some freg => size_arguments_64 tyl ir (Z.add fr (Zpos xH)) ofs0 | None => size_arguments_64 tyl ir fr (Z.add ofs0 (Zpos (xO xH))) end | Tlong => match @list_nth_z mreg int_param_regs ir with | Some ireg => size_arguments_64 tyl (Z.add ir (Zpos xH)) fr ofs0 | None => size_arguments_64 tyl ir fr (Z.add ofs0 (Zpos (xO xH))) end | Tsingle => match @list_nth_z mreg float_param_regs fr with | Some freg => size_arguments_64 tyl ir (Z.add fr (Zpos xH)) ofs0 | None => size_arguments_64 tyl ir fr (Z.add ofs0 (Zpos (xO xH))) end | Tany32 => match @list_nth_z mreg int_param_regs ir with | Some ireg => size_arguments_64 tyl (Z.add ir (Zpos xH)) fr ofs0 | None => size_arguments_64 tyl ir fr (Z.add ofs0 (Zpos (xO xH))) end | Tany64 => match @list_nth_z mreg int_param_regs ir with | Some ireg => size_arguments_64 tyl (Z.add ir (Zpos xH)) fr ofs0 | None => size_arguments_64 tyl ir fr (Z.add ofs0 (Zpos (xO xH))) end end *)
Fixpoint size_arguments_64 (tyl: list typ) (ir fr ofs: Z) {struct tyl} : Z :=
  match tyl with
  | nil => ofs
  | (Tint | Tlong | Tany32 | Tany64) :: tys =>
      match list_nth_z int_param_regs ir with
      | None => size_arguments_64 tys ir fr (ofs + 2)
      | Some ireg => size_arguments_64 tys (ir + 1) fr ofs
      end
  | (Tfloat | Tsingle) :: tys =>
      match list_nth_z float_param_regs fr with
      | None => size_arguments_64 tys ir fr (ofs + 2)
      | Some freg => size_arguments_64 tys ir (fr + 1) ofs
      end
  end.

Definition size_arguments (s: signature) : Z :=
  if Archi.ptr64
  then size_arguments_64 s.(sig_args) 0 0 0
  else size_arguments_32 s.(sig_args) 0.

(** Argument locations are either caller-save registers or [Outgoing]
  stack slots at nonnegative offsets. *)

Definition loc_argument_acceptable (l: loc) : Prop :=
  match l with
  | R r => is_callee_save r = false
  | S Outgoing ofs ty => ofs >= 0 /\ (typealign ty | ofs)
  | _ => False
  end.

Definition loc_argument_32_charact (ofs: Z) (l: loc) : Prop :=
  match l with
  | S Outgoing ofs' ty => ofs' >= ofs /\ typealign ty = 1
  | _ => False
  end.

Definition loc_argument_64_charact (ofs: Z) (l: loc) : Prop :=
  match l with
  | R r => In r int_param_regs \/ In r float_param_regs
  | S Outgoing ofs' ty => ofs' >= ofs /\ (2 | ofs')
  | _ => False
  end.

Remark loc_arguments_32_charact:
  forall tyl ofs p,
  In p (loc_arguments_32 tyl ofs) -> forall_rpair (loc_argument_32_charact ofs) p.
Proof.
(* Goal: forall (tyl : list typ) (ofs : Z) (p : rpair loc) (_ : @In (rpair loc) p (loc_arguments_32 tyl ofs)), @forall_rpair loc (loc_argument_32_charact ofs) p *)
  assert (X: forall ofs1 ofs2 l, loc_argument_32_charact ofs2 l -> ofs1 <= ofs2 -> loc_argument_32_charact ofs1 l).
(* Goal: forall (ty0 : typ) (_ : @In loc (S Outgoing ofs ty) (@regs_of_rpairs loc match @list_nth_z mreg float_param_regs fr with | Some ireg => @cons (rpair loc) (@One loc (R ireg)) (loc_arguments_64 tyl ir (Z.add fr (Zpos xH)) ofs0) | None => @cons (rpair loc) (@One loc (S Outgoing ofs0 ty0)) (loc_arguments_64 tyl ir fr (Z.add ofs0 (Zpos (xO xH)))) end)), Z.le (Z.add ofs (typesize ty)) match @list_nth_z mreg float_param_regs fr with | Some m => size_arguments_64 tyl ir (Z.add fr (Zpos xH)) ofs0 | None => size_arguments_64 tyl ir fr (Z.add ofs0 (Zpos (xO xH))) end *)
(* Goal: Z.le (Z.add ofs (typesize ty)) match a with | Tint => match @list_nth_z mreg int_param_regs ir with | Some ireg => size_arguments_64 tyl (Z.add ir (Zpos xH)) fr ofs0 | None => size_arguments_64 tyl ir fr (Z.add ofs0 (Zpos (xO xH))) end | Tfloat => match @list_nth_z mreg float_param_regs fr with | Some freg => size_arguments_64 tyl ir (Z.add fr (Zpos xH)) ofs0 | None => size_arguments_64 tyl ir fr (Z.add ofs0 (Zpos (xO xH))) end | Tlong => match @list_nth_z mreg int_param_regs ir with | Some ireg => size_arguments_64 tyl (Z.add ir (Zpos xH)) fr ofs0 | None => size_arguments_64 tyl ir fr (Z.add ofs0 (Zpos (xO xH))) end | Tsingle => match @list_nth_z mreg float_param_regs fr with | Some freg => size_arguments_64 tyl ir (Z.add fr (Zpos xH)) ofs0 | None => size_arguments_64 tyl ir fr (Z.add ofs0 (Zpos (xO xH))) end | Tany32 => match @list_nth_z mreg int_param_regs ir with | Some ireg => size_arguments_64 tyl (Z.add ir (Zpos xH)) fr ofs0 | None => size_arguments_64 tyl ir fr (Z.add ofs0 (Zpos (xO xH))) end | Tany64 => match @list_nth_z mreg int_param_regs ir with | Some ireg => size_arguments_64 tyl (Z.add ir (Zpos xH)) fr ofs0 | None => size_arguments_64 tyl ir fr (Z.add ofs0 (Zpos (xO xH))) end end *)
  { destruct l; simpl; intros; auto. destruct sl; auto. intuition omega. }
(* Goal: forall (ty0 : typ) (_ : @In loc (S Outgoing ofs ty) (@regs_of_rpairs loc match @list_nth_z mreg float_param_regs fr with | Some ireg => @cons (rpair loc) (@One loc (R ireg)) (loc_arguments_64 tyl ir (Z.add fr (Zpos xH)) ofs0) | None => @cons (rpair loc) (@One loc (S Outgoing ofs0 ty0)) (loc_arguments_64 tyl ir fr (Z.add ofs0 (Zpos (xO xH)))) end)), Z.le (Z.add ofs (typesize ty)) match @list_nth_z mreg float_param_regs fr with | Some m => size_arguments_64 tyl ir (Z.add fr (Zpos xH)) ofs0 | None => size_arguments_64 tyl ir fr (Z.add ofs0 (Zpos (xO xH))) end *)
  induction tyl as [ | ty tyl]; simpl loc_arguments_32; intros.
- contradiction.
- destruct H.
+ destruct ty; subst p; simpl; omega.
(* Goal: forall (ty0 : typ) (_ : @In loc (S Outgoing ofs ty) (@regs_of_rpairs loc match @list_nth_z mreg float_param_regs fr with | Some ireg => @cons (rpair loc) (@One loc (R ireg)) (loc_arguments_64 tyl ir (Z.add fr (Zpos xH)) ofs0) | None => @cons (rpair loc) (@One loc (S Outgoing ofs0 ty0)) (loc_arguments_64 tyl ir fr (Z.add ofs0 (Zpos (xO xH)))) end)), Z.le (Z.add ofs (typesize ty)) match @list_nth_z mreg float_param_regs fr with | Some m => size_arguments_64 tyl ir (Z.add fr (Zpos xH)) ofs0 | None => size_arguments_64 tyl ir fr (Z.add ofs0 (Zpos (xO xH))) end *)
+ apply IHtyl in H. generalize (typesize_pos ty); intros. destruct p; simpl in *.
* eapply X; eauto; omega.
* destruct H; split; eapply X; eauto; omega.
Qed.

Remark loc_arguments_64_charact:
  forall tyl ir fr ofs p,
  In p (loc_arguments_64 tyl ir fr ofs) -> (2 | ofs) -> forall_rpair (loc_argument_64_charact ofs) p.
Proof.
  assert (X: forall ofs1 ofs2 l, loc_argument_64_charact ofs2 l -> ofs1 <= ofs2 -> loc_argument_64_charact ofs1 l).
(* Goal: forall (ty0 : typ) (_ : @In loc (S Outgoing ofs ty) (@regs_of_rpairs loc match @list_nth_z mreg float_param_regs fr with | Some ireg => @cons (rpair loc) (@One loc (R ireg)) (loc_arguments_64 tyl ir (Z.add fr (Zpos xH)) ofs0) | None => @cons (rpair loc) (@One loc (S Outgoing ofs0 ty0)) (loc_arguments_64 tyl ir fr (Z.add ofs0 (Zpos (xO xH)))) end)), Z.le (Z.add ofs (typesize ty)) match @list_nth_z mreg float_param_regs fr with | Some m => size_arguments_64 tyl ir (Z.add fr (Zpos xH)) ofs0 | None => size_arguments_64 tyl ir fr (Z.add ofs0 (Zpos (xO xH))) end *)
(* Goal: Z.le (Z.add ofs (typesize ty)) match a with | Tint => match @list_nth_z mreg int_param_regs ir with | Some ireg => size_arguments_64 tyl (Z.add ir (Zpos xH)) fr ofs0 | None => size_arguments_64 tyl ir fr (Z.add ofs0 (Zpos (xO xH))) end | Tfloat => match @list_nth_z mreg float_param_regs fr with | Some freg => size_arguments_64 tyl ir (Z.add fr (Zpos xH)) ofs0 | None => size_arguments_64 tyl ir fr (Z.add ofs0 (Zpos (xO xH))) end | Tlong => match @list_nth_z mreg int_param_regs ir with | Some ireg => size_arguments_64 tyl (Z.add ir (Zpos xH)) fr ofs0 | None => size_arguments_64 tyl ir fr (Z.add ofs0 (Zpos (xO xH))) end | Tsingle => match @list_nth_z mreg float_param_regs fr with | Some freg => size_arguments_64 tyl ir (Z.add fr (Zpos xH)) ofs0 | None => size_arguments_64 tyl ir fr (Z.add ofs0 (Zpos (xO xH))) end | Tany32 => match @list_nth_z mreg int_param_regs ir with | Some ireg => size_arguments_64 tyl (Z.add ir (Zpos xH)) fr ofs0 | None => size_arguments_64 tyl ir fr (Z.add ofs0 (Zpos (xO xH))) end | Tany64 => match @list_nth_z mreg int_param_regs ir with | Some ireg => size_arguments_64 tyl (Z.add ir (Zpos xH)) fr ofs0 | None => size_arguments_64 tyl ir fr (Z.add ofs0 (Zpos (xO xH))) end end *)
  { destruct l; simpl; intros; auto. destruct sl; auto. intuition omega. }
  assert (Y: forall ofs1 ofs2 p, forall_rpair (loc_argument_64_charact ofs2) p -> ofs1 <= ofs2 -> forall_rpair (loc_argument_64_charact ofs1) p).
(* Goal: forall (ty0 : typ) (_ : @In loc (S Outgoing ofs ty) (@regs_of_rpairs loc match @list_nth_z mreg float_param_regs fr with | Some ireg => @cons (rpair loc) (@One loc (R ireg)) (loc_arguments_64 tyl ir (Z.add fr (Zpos xH)) ofs0) | None => @cons (rpair loc) (@One loc (S Outgoing ofs0 ty0)) (loc_arguments_64 tyl ir fr (Z.add ofs0 (Zpos (xO xH)))) end)), Z.le (Z.add ofs (typesize ty)) match @list_nth_z mreg float_param_regs fr with | Some m => size_arguments_64 tyl ir (Z.add fr (Zpos xH)) ofs0 | None => size_arguments_64 tyl ir fr (Z.add ofs0 (Zpos (xO xH))) end *)
(* Goal: Z.le (Z.add ofs (typesize ty)) match a with | Tint => match @list_nth_z mreg int_param_regs ir with | Some ireg => size_arguments_64 tyl (Z.add ir (Zpos xH)) fr ofs0 | None => size_arguments_64 tyl ir fr (Z.add ofs0 (Zpos (xO xH))) end | Tfloat => match @list_nth_z mreg float_param_regs fr with | Some freg => size_arguments_64 tyl ir (Z.add fr (Zpos xH)) ofs0 | None => size_arguments_64 tyl ir fr (Z.add ofs0 (Zpos (xO xH))) end | Tlong => match @list_nth_z mreg int_param_regs ir with | Some ireg => size_arguments_64 tyl (Z.add ir (Zpos xH)) fr ofs0 | None => size_arguments_64 tyl ir fr (Z.add ofs0 (Zpos (xO xH))) end | Tsingle => match @list_nth_z mreg float_param_regs fr with | Some freg => size_arguments_64 tyl ir (Z.add fr (Zpos xH)) ofs0 | None => size_arguments_64 tyl ir fr (Z.add ofs0 (Zpos (xO xH))) end | Tany32 => match @list_nth_z mreg int_param_regs ir with | Some ireg => size_arguments_64 tyl (Z.add ir (Zpos xH)) fr ofs0 | None => size_arguments_64 tyl ir fr (Z.add ofs0 (Zpos (xO xH))) end | Tany64 => match @list_nth_z mreg int_param_regs ir with | Some ireg => size_arguments_64 tyl (Z.add ir (Zpos xH)) fr ofs0 | None => size_arguments_64 tyl ir fr (Z.add ofs0 (Zpos (xO xH))) end end *)
  { destruct p; simpl; intuition eauto. }
  assert (Z: forall ofs, (2 | ofs) -> (2 | ofs + 2)).
(* Goal: forall (ty0 : typ) (_ : @In loc (S Outgoing ofs ty) (@regs_of_rpairs loc match @list_nth_z mreg float_param_regs fr with | Some ireg => @cons (rpair loc) (@One loc (R ireg)) (loc_arguments_64 tyl ir (Z.add fr (Zpos xH)) ofs0) | None => @cons (rpair loc) (@One loc (S Outgoing ofs0 ty0)) (loc_arguments_64 tyl ir fr (Z.add ofs0 (Zpos (xO xH)))) end)), Z.le (Z.add ofs (typesize ty)) match @list_nth_z mreg float_param_regs fr with | Some m => size_arguments_64 tyl ir (Z.add fr (Zpos xH)) ofs0 | None => size_arguments_64 tyl ir fr (Z.add ofs0 (Zpos (xO xH))) end *)
  { intros. apply Z.divide_add_r; auto. apply Z.divide_refl. }
Opaque list_nth_z.
(* Goal: forall (ty0 : typ) (_ : @In loc (S Outgoing ofs ty) (@regs_of_rpairs loc match @list_nth_z mreg float_param_regs fr with | Some ireg => @cons (rpair loc) (@One loc (R ireg)) (loc_arguments_64 tyl ir (Z.add fr (Zpos xH)) ofs0) | None => @cons (rpair loc) (@One loc (S Outgoing ofs0 ty0)) (loc_arguments_64 tyl ir fr (Z.add ofs0 (Zpos (xO xH)))) end)), Z.le (Z.add ofs (typesize ty)) match @list_nth_z mreg float_param_regs fr with | Some m => size_arguments_64 tyl ir (Z.add fr (Zpos xH)) ofs0 | None => size_arguments_64 tyl ir fr (Z.add ofs0 (Zpos (xO xH))) end *)
  induction tyl; simpl loc_arguments_64; intros.
  elim H.
  assert (A: forall ty, In p
      match list_nth_z int_param_regs ir with
      | Some ireg => One (R ireg) :: loc_arguments_64 tyl (ir + 1) fr ofs
      | None => One (S Outgoing ofs ty) :: loc_arguments_64 tyl ir fr (ofs + 2)
      end ->
      forall_rpair (loc_argument_64_charact ofs) p).
(* Goal: forall (ty0 : typ) (_ : @In loc (S Outgoing ofs ty) (@regs_of_rpairs loc match @list_nth_z mreg float_param_regs fr with | Some ireg => @cons (rpair loc) (@One loc (R ireg)) (loc_arguments_64 tyl ir (Z.add fr (Zpos xH)) ofs0) | None => @cons (rpair loc) (@One loc (S Outgoing ofs0 ty0)) (loc_arguments_64 tyl ir fr (Z.add ofs0 (Zpos (xO xH)))) end)), Z.le (Z.add ofs (typesize ty)) match @list_nth_z mreg float_param_regs fr with | Some m => size_arguments_64 tyl ir (Z.add fr (Zpos xH)) ofs0 | None => size_arguments_64 tyl ir fr (Z.add ofs0 (Zpos (xO xH))) end *)
  { intros. destruct (list_nth_z int_param_regs ir) as [r|] eqn:E; destruct H1.
    subst. left. eapply list_nth_z_in; eauto.
(* Goal: Z.le (Z.add ofs (typesize ty)) (size_arguments_64 tyl ir fr (Z.add ofs0 (Zpos (xO xH)))) *)
    eapply IHtyl; eauto.
    subst. split. omega. assumption.
    eapply Y; eauto. omega. }
  assert (B: forall ty, In p
      match list_nth_z float_param_regs fr with
      | Some ireg => One (R ireg) :: loc_arguments_64 tyl ir (fr + 1) ofs
      | None => One (S Outgoing ofs ty) :: loc_arguments_64 tyl ir fr (ofs + 2)
      end ->
      forall_rpair (loc_argument_64_charact ofs) p).
(* Goal: forall (ty0 : typ) (_ : @In loc (S Outgoing ofs ty) (@regs_of_rpairs loc match @list_nth_z mreg float_param_regs fr with | Some ireg => @cons (rpair loc) (@One loc (R ireg)) (loc_arguments_64 tyl ir (Z.add fr (Zpos xH)) ofs0) | None => @cons (rpair loc) (@One loc (S Outgoing ofs0 ty0)) (loc_arguments_64 tyl ir fr (Z.add ofs0 (Zpos (xO xH)))) end)), Z.le (Z.add ofs (typesize ty)) match @list_nth_z mreg float_param_regs fr with | Some m => size_arguments_64 tyl ir (Z.add fr (Zpos xH)) ofs0 | None => size_arguments_64 tyl ir fr (Z.add ofs0 (Zpos (xO xH))) end *)
  { intros. destruct (list_nth_z float_param_regs fr) as [r|] eqn:E; destruct H1.
    subst. right. eapply list_nth_z_in; eauto.
(* Goal: Z.le (Z.add ofs (typesize ty)) (size_arguments_64 tyl ir fr (Z.add ofs0 (Zpos (xO xH)))) *)
    eapply IHtyl; eauto.
    subst. split. omega. assumption.
    eapply Y; eauto. omega. }
(* Goal: Z.le (Z.add ofs (typesize ty)) match a with | Tint => match @list_nth_z mreg int_param_regs ir with | Some ireg => size_arguments_64 tyl (Z.add ir (Zpos xH)) fr ofs0 | None => size_arguments_64 tyl ir fr (Z.add ofs0 (Zpos (xO xH))) end | Tfloat => match @list_nth_z mreg float_param_regs fr with | Some freg => size_arguments_64 tyl ir (Z.add fr (Zpos xH)) ofs0 | None => size_arguments_64 tyl ir fr (Z.add ofs0 (Zpos (xO xH))) end | Tlong => match @list_nth_z mreg int_param_regs ir with | Some ireg => size_arguments_64 tyl (Z.add ir (Zpos xH)) fr ofs0 | None => size_arguments_64 tyl ir fr (Z.add ofs0 (Zpos (xO xH))) end | Tsingle => match @list_nth_z mreg float_param_regs fr with | Some freg => size_arguments_64 tyl ir (Z.add fr (Zpos xH)) ofs0 | None => size_arguments_64 tyl ir fr (Z.add ofs0 (Zpos (xO xH))) end | Tany32 => match @list_nth_z mreg int_param_regs ir with | Some ireg => size_arguments_64 tyl (Z.add ir (Zpos xH)) fr ofs0 | None => size_arguments_64 tyl ir fr (Z.add ofs0 (Zpos (xO xH))) end | Tany64 => match @list_nth_z mreg int_param_regs ir with | Some ireg => size_arguments_64 tyl (Z.add ir (Zpos xH)) fr ofs0 | None => size_arguments_64 tyl ir fr (Z.add ofs0 (Zpos (xO xH))) end end *)
  destruct a; eauto.
Qed.

Lemma loc_arguments_acceptable:
  forall (s: signature) (p: rpair loc),
  In p (loc_arguments s) -> forall_rpair loc_argument_acceptable p.
Proof.
(* Goal: forall (ty0 : typ) (_ : @In loc (S Outgoing ofs ty) (@regs_of_rpairs loc match @list_nth_z mreg float_param_regs fr with | Some ireg => @cons (rpair loc) (@One loc (R ireg)) (loc_arguments_64 tyl ir (Z.add fr (Zpos xH)) ofs0) | None => @cons (rpair loc) (@One loc (S Outgoing ofs0 ty0)) (loc_arguments_64 tyl ir fr (Z.add ofs0 (Zpos (xO xH)))) end)), Z.le (Z.add ofs (typesize ty)) match @list_nth_z mreg float_param_regs fr with | Some m => size_arguments_64 tyl ir (Z.add fr (Zpos xH)) ofs0 | None => size_arguments_64 tyl ir fr (Z.add ofs0 (Zpos (xO xH))) end *)
  unfold loc_arguments; intros. destruct Archi.ptr64 eqn:SF.
- (* 64 bits *)
  assert (A: forall r, In r int_param_regs -> is_callee_save r = false) by (unfold is_callee_save; rewrite SF; decide_goal).
  assert (B: forall r, In r float_param_regs -> is_callee_save r = false) by decide_goal.
  assert (X: forall l, loc_argument_64_charact 0 l -> loc_argument_acceptable l).
(* Goal: forall (ty0 : typ) (_ : @In loc (S Outgoing ofs ty) (@regs_of_rpairs loc match @list_nth_z mreg float_param_regs fr with | Some ireg => @cons (rpair loc) (@One loc (R ireg)) (loc_arguments_64 tyl ir (Z.add fr (Zpos xH)) ofs0) | None => @cons (rpair loc) (@One loc (S Outgoing ofs0 ty0)) (loc_arguments_64 tyl ir fr (Z.add ofs0 (Zpos (xO xH)))) end)), Z.le (Z.add ofs (typesize ty)) match @list_nth_z mreg float_param_regs fr with | Some m => size_arguments_64 tyl ir (Z.add fr (Zpos xH)) ofs0 | None => size_arguments_64 tyl ir fr (Z.add ofs0 (Zpos (xO xH))) end *)
(* Goal: Z.le (Z.add ofs (typesize ty)) match a with | Tint => match @list_nth_z mreg int_param_regs ir with | Some ireg => size_arguments_64 tyl (Z.add ir (Zpos xH)) fr ofs0 | None => size_arguments_64 tyl ir fr (Z.add ofs0 (Zpos (xO xH))) end | Tfloat => match @list_nth_z mreg float_param_regs fr with | Some freg => size_arguments_64 tyl ir (Z.add fr (Zpos xH)) ofs0 | None => size_arguments_64 tyl ir fr (Z.add ofs0 (Zpos (xO xH))) end | Tlong => match @list_nth_z mreg int_param_regs ir with | Some ireg => size_arguments_64 tyl (Z.add ir (Zpos xH)) fr ofs0 | None => size_arguments_64 tyl ir fr (Z.add ofs0 (Zpos (xO xH))) end | Tsingle => match @list_nth_z mreg float_param_regs fr with | Some freg => size_arguments_64 tyl ir (Z.add fr (Zpos xH)) ofs0 | None => size_arguments_64 tyl ir fr (Z.add ofs0 (Zpos (xO xH))) end | Tany32 => match @list_nth_z mreg int_param_regs ir with | Some ireg => size_arguments_64 tyl (Z.add ir (Zpos xH)) fr ofs0 | None => size_arguments_64 tyl ir fr (Z.add ofs0 (Zpos (xO xH))) end | Tany64 => match @list_nth_z mreg int_param_regs ir with | Some ireg => size_arguments_64 tyl (Z.add ir (Zpos xH)) fr ofs0 | None => size_arguments_64 tyl ir fr (Z.add ofs0 (Zpos (xO xH))) end end *)
  { unfold loc_argument_64_charact, loc_argument_acceptable.
(* Goal: forall (l : loc) (_ : match l with | R r => or (@In mreg r int_param_regs) (@In mreg r float_param_regs) | S (Local as sl) ofs' ty => False | S (Incoming as sl) ofs' ty => False | S (Outgoing as sl) ofs' ty => and (Z.ge ofs' Z0) (Z.divide (Zpos (xO xH)) ofs') end), match l with | R r => @eq bool (is_callee_save r) false | S (Local as sl) ofs ty => False | S (Incoming as sl) ofs ty => False | S (Outgoing as sl) ofs ty => and (Z.ge ofs Z0) (Z.divide (typealign ty) ofs) end *)
    destruct l as [r | [] ofs ty]; auto.  intros [C|C]; auto.
(* Goal: and (Z.ge ofs Z0) (Z.divide (typealign ty) ofs) *)
    intros [C D]. split; auto. apply Z.divide_trans with 2; auto.
(* Goal: Z.divide (typealign ty) (Zpos (xO xH)) *)
    exists (2 / typealign ty); destruct ty; reflexivity.
  }
(* Goal: @forall_rpair loc loc_argument_acceptable p *)
  exploit loc_arguments_64_charact; eauto using Z.divide_0_r.
(* Goal: forall _ : @forall_rpair loc (loc_argument_32_charact Z0) p, @forall_rpair loc loc_argument_acceptable p *)
  unfold forall_rpair; destruct p; intuition auto.
- (* 32 bits *)
  assert (X: forall l, loc_argument_32_charact 0 l -> loc_argument_acceptable l).
(* Goal: forall (ty0 : typ) (_ : @In loc (S Outgoing ofs ty) (@regs_of_rpairs loc match @list_nth_z mreg float_param_regs fr with | Some ireg => @cons (rpair loc) (@One loc (R ireg)) (loc_arguments_64 tyl ir (Z.add fr (Zpos xH)) ofs0) | None => @cons (rpair loc) (@One loc (S Outgoing ofs0 ty0)) (loc_arguments_64 tyl ir fr (Z.add ofs0 (Zpos (xO xH)))) end)), Z.le (Z.add ofs (typesize ty)) match @list_nth_z mreg float_param_regs fr with | Some m => size_arguments_64 tyl ir (Z.add fr (Zpos xH)) ofs0 | None => size_arguments_64 tyl ir fr (Z.add ofs0 (Zpos (xO xH))) end *)
(* Goal: Z.le (Z.add ofs (typesize ty)) match a with | Tint => match @list_nth_z mreg int_param_regs ir with | Some ireg => size_arguments_64 tyl (Z.add ir (Zpos xH)) fr ofs0 | None => size_arguments_64 tyl ir fr (Z.add ofs0 (Zpos (xO xH))) end | Tfloat => match @list_nth_z mreg float_param_regs fr with | Some freg => size_arguments_64 tyl ir (Z.add fr (Zpos xH)) ofs0 | None => size_arguments_64 tyl ir fr (Z.add ofs0 (Zpos (xO xH))) end | Tlong => match @list_nth_z mreg int_param_regs ir with | Some ireg => size_arguments_64 tyl (Z.add ir (Zpos xH)) fr ofs0 | None => size_arguments_64 tyl ir fr (Z.add ofs0 (Zpos (xO xH))) end | Tsingle => match @list_nth_z mreg float_param_regs fr with | Some freg => size_arguments_64 tyl ir (Z.add fr (Zpos xH)) ofs0 | None => size_arguments_64 tyl ir fr (Z.add ofs0 (Zpos (xO xH))) end | Tany32 => match @list_nth_z mreg int_param_regs ir with | Some ireg => size_arguments_64 tyl (Z.add ir (Zpos xH)) fr ofs0 | None => size_arguments_64 tyl ir fr (Z.add ofs0 (Zpos (xO xH))) end | Tany64 => match @list_nth_z mreg int_param_regs ir with | Some ireg => size_arguments_64 tyl (Z.add ir (Zpos xH)) fr ofs0 | None => size_arguments_64 tyl ir fr (Z.add ofs0 (Zpos (xO xH))) end end *)
  { destruct l as [r | [] ofs ty]; simpl; intuition auto. rewrite H2; apply Z.divide_1_l. }
(* Goal: @forall_rpair loc loc_argument_acceptable p *)
  exploit loc_arguments_32_charact; eauto.
(* Goal: forall _ : @forall_rpair loc (loc_argument_32_charact Z0) p, @forall_rpair loc loc_argument_acceptable p *)
  unfold forall_rpair; destruct p; intuition auto.
Qed.

Hint Resolve loc_arguments_acceptable: locs.

(** The offsets of [Outgoing] arguments are below [size_arguments s]. *)

Remark size_arguments_32_above:
  forall tyl ofs0, ofs0 <= size_arguments_32 tyl ofs0.
Proof.
(* Goal: forall (ty0 : typ) (_ : @In loc (S Outgoing ofs ty) (@regs_of_rpairs loc match @list_nth_z mreg float_param_regs fr with | Some ireg => @cons (rpair loc) (@One loc (R ireg)) (loc_arguments_64 tyl ir (Z.add fr (Zpos xH)) ofs0) | None => @cons (rpair loc) (@One loc (S Outgoing ofs0 ty0)) (loc_arguments_64 tyl ir fr (Z.add ofs0 (Zpos (xO xH)))) end)), Z.le (Z.add ofs (typesize ty)) match @list_nth_z mreg float_param_regs fr with | Some m => size_arguments_64 tyl ir (Z.add fr (Zpos xH)) ofs0 | None => size_arguments_64 tyl ir fr (Z.add ofs0 (Zpos (xO xH))) end *)
  induction tyl; simpl; intros.
(* Goal: Z.le (Z.add ofs (typesize ty)) (Z.add ofs (Zpos (xO xH))) *)
(* Goal: Z.le (Z.add ofs (Zpos (xO xH))) (size_arguments_64 tyl ir fr (Z.add ofs (Zpos (xO xH)))) *)
  omega.
  apply Z.le_trans with (ofs0 + typesize a); auto.
(* Goal: Z.le (Z.add ofs (typesize ty)) (Z.add ofs (Zpos (xO xH))) *)
(* Goal: Z.le (Z.add ofs (Zpos (xO xH))) (size_arguments_64 tyl ir fr (Z.add ofs (Zpos (xO xH)))) *)
  generalize (typesize_pos a); omega.
Qed.

Remark size_arguments_64_above:
  forall tyl ir fr ofs0,
  ofs0 <= size_arguments_64 tyl ir fr ofs0.
Proof.
(* Goal: forall (ty0 : typ) (_ : @In loc (S Outgoing ofs ty) (@regs_of_rpairs loc match @list_nth_z mreg float_param_regs fr with | Some ireg => @cons (rpair loc) (@One loc (R ireg)) (loc_arguments_64 tyl ir (Z.add fr (Zpos xH)) ofs0) | None => @cons (rpair loc) (@One loc (S Outgoing ofs0 ty0)) (loc_arguments_64 tyl ir fr (Z.add ofs0 (Zpos (xO xH)))) end)), Z.le (Z.add ofs (typesize ty)) match @list_nth_z mreg float_param_regs fr with | Some m => size_arguments_64 tyl ir (Z.add fr (Zpos xH)) ofs0 | None => size_arguments_64 tyl ir fr (Z.add ofs0 (Zpos (xO xH))) end *)
  induction tyl; simpl; intros.
(* Goal: Z.le (Z.add ofs (typesize ty)) (Z.add ofs (Zpos (xO xH))) *)
(* Goal: Z.le (Z.add ofs (Zpos (xO xH))) (size_arguments_64 tyl ir fr (Z.add ofs (Zpos (xO xH)))) *)
  omega.
  assert (A: ofs0 <=
    match list_nth_z int_param_regs ir with
    | Some _ => size_arguments_64 tyl (ir + 1) fr ofs0
    | None => size_arguments_64 tyl ir fr (ofs0 + 2)
    end).
(* Goal: forall (ty0 : typ) (_ : @In loc (S Outgoing ofs ty) (@regs_of_rpairs loc match @list_nth_z mreg float_param_regs fr with | Some ireg => @cons (rpair loc) (@One loc (R ireg)) (loc_arguments_64 tyl ir (Z.add fr (Zpos xH)) ofs0) | None => @cons (rpair loc) (@One loc (S Outgoing ofs0 ty0)) (loc_arguments_64 tyl ir fr (Z.add ofs0 (Zpos (xO xH)))) end)), Z.le (Z.add ofs (typesize ty)) match @list_nth_z mreg float_param_regs fr with | Some m => size_arguments_64 tyl ir (Z.add fr (Zpos xH)) ofs0 | None => size_arguments_64 tyl ir fr (Z.add ofs0 (Zpos (xO xH))) end *)
(* Goal: Z.le (Z.add ofs (typesize ty)) match a with | Tint => match @list_nth_z mreg int_param_regs ir with | Some ireg => size_arguments_64 tyl (Z.add ir (Zpos xH)) fr ofs0 | None => size_arguments_64 tyl ir fr (Z.add ofs0 (Zpos (xO xH))) end | Tfloat => match @list_nth_z mreg float_param_regs fr with | Some freg => size_arguments_64 tyl ir (Z.add fr (Zpos xH)) ofs0 | None => size_arguments_64 tyl ir fr (Z.add ofs0 (Zpos (xO xH))) end | Tlong => match @list_nth_z mreg int_param_regs ir with | Some ireg => size_arguments_64 tyl (Z.add ir (Zpos xH)) fr ofs0 | None => size_arguments_64 tyl ir fr (Z.add ofs0 (Zpos (xO xH))) end | Tsingle => match @list_nth_z mreg float_param_regs fr with | Some freg => size_arguments_64 tyl ir (Z.add fr (Zpos xH)) ofs0 | None => size_arguments_64 tyl ir fr (Z.add ofs0 (Zpos (xO xH))) end | Tany32 => match @list_nth_z mreg int_param_regs ir with | Some ireg => size_arguments_64 tyl (Z.add ir (Zpos xH)) fr ofs0 | None => size_arguments_64 tyl ir fr (Z.add ofs0 (Zpos (xO xH))) end | Tany64 => match @list_nth_z mreg int_param_regs ir with | Some ireg => size_arguments_64 tyl (Z.add ir (Zpos xH)) fr ofs0 | None => size_arguments_64 tyl ir fr (Z.add ofs0 (Zpos (xO xH))) end end *)
  { destruct (list_nth_z int_param_regs ir); eauto.
    apply Z.le_trans with (ofs0 + 2); auto. omega. }
  assert (B: ofs0 <=
    match list_nth_z float_param_regs fr with
    | Some _ => size_arguments_64 tyl ir (fr + 1) ofs0
    | None => size_arguments_64 tyl ir fr (ofs0 + 2)
    end).
(* Goal: forall (ty0 : typ) (_ : @In loc (S Outgoing ofs ty) (@regs_of_rpairs loc match @list_nth_z mreg float_param_regs fr with | Some ireg => @cons (rpair loc) (@One loc (R ireg)) (loc_arguments_64 tyl ir (Z.add fr (Zpos xH)) ofs0) | None => @cons (rpair loc) (@One loc (S Outgoing ofs0 ty0)) (loc_arguments_64 tyl ir fr (Z.add ofs0 (Zpos (xO xH)))) end)), Z.le (Z.add ofs (typesize ty)) match @list_nth_z mreg float_param_regs fr with | Some m => size_arguments_64 tyl ir (Z.add fr (Zpos xH)) ofs0 | None => size_arguments_64 tyl ir fr (Z.add ofs0 (Zpos (xO xH))) end *)
(* Goal: Z.le (Z.add ofs (typesize ty)) match a with | Tint => match @list_nth_z mreg int_param_regs ir with | Some ireg => size_arguments_64 tyl (Z.add ir (Zpos xH)) fr ofs0 | None => size_arguments_64 tyl ir fr (Z.add ofs0 (Zpos (xO xH))) end | Tfloat => match @list_nth_z mreg float_param_regs fr with | Some freg => size_arguments_64 tyl ir (Z.add fr (Zpos xH)) ofs0 | None => size_arguments_64 tyl ir fr (Z.add ofs0 (Zpos (xO xH))) end | Tlong => match @list_nth_z mreg int_param_regs ir with | Some ireg => size_arguments_64 tyl (Z.add ir (Zpos xH)) fr ofs0 | None => size_arguments_64 tyl ir fr (Z.add ofs0 (Zpos (xO xH))) end | Tsingle => match @list_nth_z mreg float_param_regs fr with | Some freg => size_arguments_64 tyl ir (Z.add fr (Zpos xH)) ofs0 | None => size_arguments_64 tyl ir fr (Z.add ofs0 (Zpos (xO xH))) end | Tany32 => match @list_nth_z mreg int_param_regs ir with | Some ireg => size_arguments_64 tyl (Z.add ir (Zpos xH)) fr ofs0 | None => size_arguments_64 tyl ir fr (Z.add ofs0 (Zpos (xO xH))) end | Tany64 => match @list_nth_z mreg int_param_regs ir with | Some ireg => size_arguments_64 tyl (Z.add ir (Zpos xH)) fr ofs0 | None => size_arguments_64 tyl ir fr (Z.add ofs0 (Zpos (xO xH))) end end *)
  { destruct (list_nth_z float_param_regs fr); eauto.
    apply Z.le_trans with (ofs0 + 2); auto. omega. }
(* Goal: Z.le ofs0 match a with | Tint => match @list_nth_z mreg int_param_regs ir with | Some ireg => size_arguments_64 tyl (Z.add ir (Zpos xH)) fr ofs0 | None => size_arguments_64 tyl ir fr (Z.add ofs0 (Zpos (xO xH))) end | Tfloat => match @list_nth_z mreg float_param_regs fr with | Some freg => size_arguments_64 tyl ir (Z.add fr (Zpos xH)) ofs0 | None => size_arguments_64 tyl ir fr (Z.add ofs0 (Zpos (xO xH))) end | Tlong => match @list_nth_z mreg int_param_regs ir with | Some ireg => size_arguments_64 tyl (Z.add ir (Zpos xH)) fr ofs0 | None => size_arguments_64 tyl ir fr (Z.add ofs0 (Zpos (xO xH))) end | Tsingle => match @list_nth_z mreg float_param_regs fr with | Some freg => size_arguments_64 tyl ir (Z.add fr (Zpos xH)) ofs0 | None => size_arguments_64 tyl ir fr (Z.add ofs0 (Zpos (xO xH))) end | Tany32 => match @list_nth_z mreg int_param_regs ir with | Some ireg => size_arguments_64 tyl (Z.add ir (Zpos xH)) fr ofs0 | None => size_arguments_64 tyl ir fr (Z.add ofs0 (Zpos (xO xH))) end | Tany64 => match @list_nth_z mreg int_param_regs ir with | Some ireg => size_arguments_64 tyl (Z.add ir (Zpos xH)) fr ofs0 | None => size_arguments_64 tyl ir fr (Z.add ofs0 (Zpos (xO xH))) end end *)
  destruct a; auto.
Qed.

Lemma size_arguments_above:
  forall s, size_arguments s >= 0.
Proof.
(* Goal: forall s : signature, Z.ge (size_arguments s) Z0 *)
  intros; unfold size_arguments. apply Z.le_ge.
(* Goal: Z.le Z0 (if Archi.ptr64 then size_arguments_64 (sig_args s) Z0 Z0 Z0 else size_arguments_32 (sig_args s) Z0) *)
  destruct Archi.ptr64; [apply size_arguments_64_above|apply size_arguments_32_above].
Qed.

Lemma loc_arguments_32_bounded:
  forall ofs ty tyl ofs0,
  In (S Outgoing ofs ty) (regs_of_rpairs (loc_arguments_32 tyl ofs0)) ->
  ofs + typesize ty <= size_arguments_32 tyl ofs0.
Proof.
(* Goal: forall (ofs : Z) (ty : typ) (tyl : list typ) (ofs0 : Z) (_ : @In loc (S Outgoing ofs ty) (@regs_of_rpairs loc (loc_arguments_32 tyl ofs0))), Z.le (Z.add ofs (typesize ty)) (size_arguments_32 tyl ofs0) *)
  induction tyl as [ | t l]; simpl; intros x IN.
- contradiction.
- rewrite in_app_iff in IN; destruct IN as [IN|IN].
+ apply Z.le_trans with (x + typesize t); [|apply size_arguments_32_above].
  Ltac decomp :=
  match goal with
  | [ H: _ \/ _ |- _ ] => destruct H; decomp
  | [ H: S _ _ _ = S _ _ _ |- _ ] => inv H
  | [ H: False |- _ ] => contradiction
  end.
(* Goal: Z.le (Z.add ofs (typesize ty)) (Z.add ofs (Zpos (xO xH))) *)
(* Goal: Z.le (Z.add ofs (Zpos (xO xH))) (size_arguments_64 tyl ir fr (Z.add ofs (Zpos (xO xH)))) *)
  destruct t; simpl in IN; decomp; simpl; omega.
+ apply IHl; auto.
Qed.

Lemma loc_arguments_64_bounded:
  forall ofs ty tyl ir fr ofs0,
  In (S Outgoing ofs ty) (regs_of_rpairs (loc_arguments_64 tyl ir fr ofs0)) ->
  ofs + typesize ty <= size_arguments_64 tyl ir fr ofs0.
Proof.
(* Goal: forall (ty0 : typ) (_ : @In loc (S Outgoing ofs ty) (@regs_of_rpairs loc match @list_nth_z mreg float_param_regs fr with | Some ireg => @cons (rpair loc) (@One loc (R ireg)) (loc_arguments_64 tyl ir (Z.add fr (Zpos xH)) ofs0) | None => @cons (rpair loc) (@One loc (S Outgoing ofs0 ty0)) (loc_arguments_64 tyl ir fr (Z.add ofs0 (Zpos (xO xH)))) end)), Z.le (Z.add ofs (typesize ty)) match @list_nth_z mreg float_param_regs fr with | Some m => size_arguments_64 tyl ir (Z.add fr (Zpos xH)) ofs0 | None => size_arguments_64 tyl ir fr (Z.add ofs0 (Zpos (xO xH))) end *)
  induction tyl; simpl; intros.
(* Goal: Z.le (Z.add ofs (typesize ty)) ofs0 *)
(* Goal: Z.le (Z.add ofs (typesize ty)) match a with | Tint => match @list_nth_z mreg int_param_regs ir with | Some ireg => size_arguments_64 tyl (Z.add ir (Zpos xH)) fr ofs0 | None => size_arguments_64 tyl ir fr (Z.add ofs0 (Zpos (xO xH))) end | Tfloat => match @list_nth_z mreg float_param_regs fr with | Some freg => size_arguments_64 tyl ir (Z.add fr (Zpos xH)) ofs0 | None => size_arguments_64 tyl ir fr (Z.add ofs0 (Zpos (xO xH))) end | Tlong => match @list_nth_z mreg int_param_regs ir with | Some ireg => size_arguments_64 tyl (Z.add ir (Zpos xH)) fr ofs0 | None => size_arguments_64 tyl ir fr (Z.add ofs0 (Zpos (xO xH))) end | Tsingle => match @list_nth_z mreg float_param_regs fr with | Some freg => size_arguments_64 tyl ir (Z.add fr (Zpos xH)) ofs0 | None => size_arguments_64 tyl ir fr (Z.add ofs0 (Zpos (xO xH))) end | Tany32 => match @list_nth_z mreg int_param_regs ir with | Some ireg => size_arguments_64 tyl (Z.add ir (Zpos xH)) fr ofs0 | None => size_arguments_64 tyl ir fr (Z.add ofs0 (Zpos (xO xH))) end | Tany64 => match @list_nth_z mreg int_param_regs ir with | Some ireg => size_arguments_64 tyl (Z.add ir (Zpos xH)) fr ofs0 | None => size_arguments_64 tyl ir fr (Z.add ofs0 (Zpos (xO xH))) end end *)
  contradiction.
(* Goal: Z.le (Z.add ofs (typesize ty)) match a with | Tint => match @list_nth_z mreg int_param_regs ir with | Some ireg => size_arguments_64 tyl (Z.add ir (Zpos xH)) fr ofs0 | None => size_arguments_64 tyl ir fr (Z.add ofs0 (Zpos (xO xH))) end | Tfloat => match @list_nth_z mreg float_param_regs fr with | Some freg => size_arguments_64 tyl ir (Z.add fr (Zpos xH)) ofs0 | None => size_arguments_64 tyl ir fr (Z.add ofs0 (Zpos (xO xH))) end | Tlong => match @list_nth_z mreg int_param_regs ir with | Some ireg => size_arguments_64 tyl (Z.add ir (Zpos xH)) fr ofs0 | None => size_arguments_64 tyl ir fr (Z.add ofs0 (Zpos (xO xH))) end | Tsingle => match @list_nth_z mreg float_param_regs fr with | Some freg => size_arguments_64 tyl ir (Z.add fr (Zpos xH)) ofs0 | None => size_arguments_64 tyl ir fr (Z.add ofs0 (Zpos (xO xH))) end | Tany32 => match @list_nth_z mreg int_param_regs ir with | Some ireg => size_arguments_64 tyl (Z.add ir (Zpos xH)) fr ofs0 | None => size_arguments_64 tyl ir fr (Z.add ofs0 (Zpos (xO xH))) end | Tany64 => match @list_nth_z mreg int_param_regs ir with | Some ireg => size_arguments_64 tyl (Z.add ir (Zpos xH)) fr ofs0 | None => size_arguments_64 tyl ir fr (Z.add ofs0 (Zpos (xO xH))) end end *)
  assert (T: forall ty0, typesize ty0 <= 2).
(* Goal: forall (ty0 : typ) (_ : @In loc (S Outgoing ofs ty) (@regs_of_rpairs loc match @list_nth_z mreg float_param_regs fr with | Some ireg => @cons (rpair loc) (@One loc (R ireg)) (loc_arguments_64 tyl ir (Z.add fr (Zpos xH)) ofs0) | None => @cons (rpair loc) (@One loc (S Outgoing ofs0 ty0)) (loc_arguments_64 tyl ir fr (Z.add ofs0 (Zpos (xO xH)))) end)), Z.le (Z.add ofs (typesize ty)) match @list_nth_z mreg float_param_regs fr with | Some m => size_arguments_64 tyl ir (Z.add fr (Zpos xH)) ofs0 | None => size_arguments_64 tyl ir fr (Z.add ofs0 (Zpos (xO xH))) end *)
(* Goal: Z.le (Z.add ofs (typesize ty)) match a with | Tint => match @list_nth_z mreg int_param_regs ir with | Some ireg => size_arguments_64 tyl (Z.add ir (Zpos xH)) fr ofs0 | None => size_arguments_64 tyl ir fr (Z.add ofs0 (Zpos (xO xH))) end | Tfloat => match @list_nth_z mreg float_param_regs fr with | Some freg => size_arguments_64 tyl ir (Z.add fr (Zpos xH)) ofs0 | None => size_arguments_64 tyl ir fr (Z.add ofs0 (Zpos (xO xH))) end | Tlong => match @list_nth_z mreg int_param_regs ir with | Some ireg => size_arguments_64 tyl (Z.add ir (Zpos xH)) fr ofs0 | None => size_arguments_64 tyl ir fr (Z.add ofs0 (Zpos (xO xH))) end | Tsingle => match @list_nth_z mreg float_param_regs fr with | Some freg => size_arguments_64 tyl ir (Z.add fr (Zpos xH)) ofs0 | None => size_arguments_64 tyl ir fr (Z.add ofs0 (Zpos (xO xH))) end | Tany32 => match @list_nth_z mreg int_param_regs ir with | Some ireg => size_arguments_64 tyl (Z.add ir (Zpos xH)) fr ofs0 | None => size_arguments_64 tyl ir fr (Z.add ofs0 (Zpos (xO xH))) end | Tany64 => match @list_nth_z mreg int_param_regs ir with | Some ireg => size_arguments_64 tyl (Z.add ir (Zpos xH)) fr ofs0 | None => size_arguments_64 tyl ir fr (Z.add ofs0 (Zpos (xO xH))) end end *)
  { destruct ty0; simpl; omega. }
  assert (A: forall ty0,
             In (S Outgoing ofs ty) (regs_of_rpairs
              match list_nth_z int_param_regs ir with
              | Some ireg =>
                  One (R ireg) :: loc_arguments_64 tyl (ir + 1) fr ofs0
              | None => One (S Outgoing ofs0 ty0) :: loc_arguments_64 tyl ir fr (ofs0 + 2)
              end) ->
             ofs + typesize ty <=
             match list_nth_z int_param_regs ir with
             | Some _ => size_arguments_64 tyl (ir + 1) fr ofs0
             | None => size_arguments_64 tyl ir fr (ofs0 + 2)
             end).
(* Goal: forall (ty0 : typ) (_ : @In loc (S Outgoing ofs ty) (@regs_of_rpairs loc match @list_nth_z mreg float_param_regs fr with | Some ireg => @cons (rpair loc) (@One loc (R ireg)) (loc_arguments_64 tyl ir (Z.add fr (Zpos xH)) ofs0) | None => @cons (rpair loc) (@One loc (S Outgoing ofs0 ty0)) (loc_arguments_64 tyl ir fr (Z.add ofs0 (Zpos (xO xH)))) end)), Z.le (Z.add ofs (typesize ty)) match @list_nth_z mreg float_param_regs fr with | Some m => size_arguments_64 tyl ir (Z.add fr (Zpos xH)) ofs0 | None => size_arguments_64 tyl ir fr (Z.add ofs0 (Zpos (xO xH))) end *)
  { intros. destruct (list_nth_z int_param_regs ir); simpl in H0; destruct H0.
  - discriminate.
  - eapply IHtyl; eauto.
  - inv H0. apply Z.le_trans with (ofs + 2). specialize (T ty). omega. apply size_arguments_64_above.
  - eapply IHtyl; eauto. }
  assert (B: forall ty0,
             In (S Outgoing ofs ty) (regs_of_rpairs
              match list_nth_z float_param_regs fr with
              | Some ireg =>
                  One (R ireg) :: loc_arguments_64 tyl ir (fr + 1) ofs0
              | None => One (S Outgoing ofs0 ty0) :: loc_arguments_64 tyl ir fr (ofs0 + 2)
              end) ->
             ofs + typesize ty <=
             match list_nth_z float_param_regs fr with
             | Some _ => size_arguments_64 tyl ir (fr + 1) ofs0
             | None => size_arguments_64 tyl ir fr (ofs0 + 2)
             end).
(* Goal: forall (ty0 : typ) (_ : @In loc (S Outgoing ofs ty) (@regs_of_rpairs loc match @list_nth_z mreg float_param_regs fr with | Some ireg => @cons (rpair loc) (@One loc (R ireg)) (loc_arguments_64 tyl ir (Z.add fr (Zpos xH)) ofs0) | None => @cons (rpair loc) (@One loc (S Outgoing ofs0 ty0)) (loc_arguments_64 tyl ir fr (Z.add ofs0 (Zpos (xO xH)))) end)), Z.le (Z.add ofs (typesize ty)) match @list_nth_z mreg float_param_regs fr with | Some m => size_arguments_64 tyl ir (Z.add fr (Zpos xH)) ofs0 | None => size_arguments_64 tyl ir fr (Z.add ofs0 (Zpos (xO xH))) end *)
  { intros. destruct (list_nth_z float_param_regs fr); simpl in H0; destruct H0.
  - discriminate.
  - eapply IHtyl; eauto.
  - inv H0. apply Z.le_trans with (ofs + 2). specialize (T ty). omega. apply size_arguments_64_above.
  - eapply IHtyl; eauto. }
(* Goal: Z.le (Z.add ofs (typesize ty)) match a with | Tint => match @list_nth_z mreg int_param_regs ir with | Some ireg => size_arguments_64 tyl (Z.add ir (Zpos xH)) fr ofs0 | None => size_arguments_64 tyl ir fr (Z.add ofs0 (Zpos (xO xH))) end | Tfloat => match @list_nth_z mreg float_param_regs fr with | Some freg => size_arguments_64 tyl ir (Z.add fr (Zpos xH)) ofs0 | None => size_arguments_64 tyl ir fr (Z.add ofs0 (Zpos (xO xH))) end | Tlong => match @list_nth_z mreg int_param_regs ir with | Some ireg => size_arguments_64 tyl (Z.add ir (Zpos xH)) fr ofs0 | None => size_arguments_64 tyl ir fr (Z.add ofs0 (Zpos (xO xH))) end | Tsingle => match @list_nth_z mreg float_param_regs fr with | Some freg => size_arguments_64 tyl ir (Z.add fr (Zpos xH)) ofs0 | None => size_arguments_64 tyl ir fr (Z.add ofs0 (Zpos (xO xH))) end | Tany32 => match @list_nth_z mreg int_param_regs ir with | Some ireg => size_arguments_64 tyl (Z.add ir (Zpos xH)) fr ofs0 | None => size_arguments_64 tyl ir fr (Z.add ofs0 (Zpos (xO xH))) end | Tany64 => match @list_nth_z mreg int_param_regs ir with | Some ireg => size_arguments_64 tyl (Z.add ir (Zpos xH)) fr ofs0 | None => size_arguments_64 tyl ir fr (Z.add ofs0 (Zpos (xO xH))) end end *)
  destruct a; eauto.
Qed.

Lemma loc_arguments_bounded:
  forall (s: signature) (ofs: Z) (ty: typ),
  In (S Outgoing ofs ty) (regs_of_rpairs (loc_arguments s)) ->
  ofs + typesize ty <= size_arguments s.
Proof.
(* Goal: forall (ty0 : typ) (_ : @In loc (S Outgoing ofs ty) (@regs_of_rpairs loc match @list_nth_z mreg float_param_regs fr with | Some ireg => @cons (rpair loc) (@One loc (R ireg)) (loc_arguments_64 tyl ir (Z.add fr (Zpos xH)) ofs0) | None => @cons (rpair loc) (@One loc (S Outgoing ofs0 ty0)) (loc_arguments_64 tyl ir fr (Z.add ofs0 (Zpos (xO xH)))) end)), Z.le (Z.add ofs (typesize ty)) match @list_nth_z mreg float_param_regs fr with | Some m => size_arguments_64 tyl ir (Z.add fr (Zpos xH)) ofs0 | None => size_arguments_64 tyl ir fr (Z.add ofs0 (Zpos (xO xH))) end *)
  unfold loc_arguments, size_arguments; intros.
(* Goal: Z.le (Z.add ofs (typesize ty)) (if Archi.ptr64 then size_arguments_64 (sig_args s) Z0 Z0 Z0 else size_arguments_32 (sig_args s) Z0) *)
  destruct Archi.ptr64; eauto using loc_arguments_32_bounded, loc_arguments_64_bounded.
Qed.

Lemma loc_arguments_main:
  loc_arguments signature_main = nil.
Proof.
(* Goal: @eq (list (rpair loc)) (loc_arguments signature_main) (@nil (rpair loc)) *)
  unfold loc_arguments; destruct Archi.ptr64; reflexivity.
Qed.
