Revise the Stacking pass and its proof to make it easier to adapt to 64-bit architectures

The original Stacking pass and its proof hard-wire assumptions about the processor and the register allocation, namely that integer registers are 32 bit wide and that all stack slots have natural alignment 4, which precludes having stack slots of type Tlong.  Those assumptions become false if the target processor has 64-bit integer registers.

This commit makes minimal adjustments to the Stacking pass so as to lift these assumptions:
- Stack slots of type Tlong (or more generally of natural alignment 8) are supported.  For slots produced by register allocation, the alignment is validated a posteriori in Lineartyping.  For slots produced by the calling conventions, alignment is proved as part of the "loc_argument_acceptable" property in Conventions1.
- The code generated by Stacking to save and restore used callee-save registers no longer assumes 32-bit integer registers.  Actually, it supports any combination of sizes for registers.
- To support the new save/restore code, Bounds was changed to record the set of all callee-save registers used, rather than just the max index of callee-save registers used.

On CompCert's current 32-bit target architectures, the new Stacking pass should generate pretty much the same code as the old one, modulo minor differences in the layout of the stack frame.  (E.g. padding could be introduced at different places.)

The bulk of this big commit is related to the proof of the Stacking phase.  The old proof strategy was painful and not obviously adaptable to the new Stacking phase, so I rewrote Stackingproof entirely, using an approach inspired by separation logic.  The new library common/Separation.v defines assertions about memory states that can be composed using a separating conjunction, just like pre- and post-conditions in separation logic.  Those assertions are used in Stackingproof to describe the contents of the stack frames during the execution of the generated Mach code, and relate them with the Linear location maps.

As a further simplification, the callee-save/caller-save distinction is now defined in Conventions1 by a function is_callee_save: mreg -> bool, instead of lists of registers of either kind as before.  This eliminates many boring classification lemmas from Conventions1.  LTL and Lineartyping were adapted accordingly.

Finally, this commit introduces a new library called Decidableplus to prove some propositions by reflection as Boolean computations.  It is used to further simplify the proofs in Conventions1.

3 files changed, 152 insertions, 268 deletions

diff --git a/powerpc/Conventions1.v b/powerpc/Conventions1.v
index 4ee25a32..e78395bf 100644
--- a/powerpc/Conventions1.v
+++ b/powerpc/Conventions1.v
@@ -14,6 +14,7 @@
     machine registers and stack slots. *)
 
 Require Import Coqlib.
+Require Import Decidableplus.
 Require Import AST.
 Require Import Events.
 Require Import Locations.
@@ -29,6 +30,17 @@ Require Import Locations.
   of callee- and caller-save registers.
 *)
 
+Definition is_callee_save (r: mreg): bool :=
+  match r with
+  | R3  | R4  | R5  | R6  | R7  | R8  | R9  | R10  | R11 | R12 => false
+  | R14 | R15 | R16 | R17 | R18 | R19 | R20 | R21 | R22 | R23 | R24
+  | R25 | R26 | R27 | R28 | R29 | R30 | R31 => true
+  | F0  | F1  | F2  | F3  | F4  | F5  | F6  | F7
+  | F8  | F9  | F10 | F11 | F12 | F13 => false
+  | F14 | F15 | F16 | F17 | F18 | F19 | F20 | F21 | F22 | F23
+  | F24 | F25 | F26 | F27 | F28 | F29 | F30 | F31 => true
+  end.
+
 Definition int_caller_save_regs :=
   R3 :: R4 :: R5 :: R6 :: R7 :: R8 :: R9 :: R10 :: R11 :: R12 :: nil.
 
@@ -44,174 +56,11 @@ Definition float_callee_save_regs :=
   F22 :: F21 :: F20 :: F19 :: F18 :: F17 :: F16 :: F15 :: F14 :: nil.
 
 Definition destroyed_at_call :=
-  int_caller_save_regs ++ float_caller_save_regs.
+  List.filter (fun r => negb (is_callee_save r)) all_mregs.
 
 Definition dummy_int_reg := R3.     (**r Used in [Coloring]. *)
 Definition dummy_float_reg := F0.   (**r Used in [Coloring]. *)
 
-(** The [index_int_callee_save] and [index_float_callee_save] associate
-  a unique positive integer to callee-save registers.  This integer is
-  used in [Stacking] to determine where to save these registers in
-  the activation record if they are used by the current function. *)
-
-Definition index_int_callee_save (r: mreg) :=
-  match r with
-  | R14 => 17 | R15 => 16 | R16 => 15 | R17 => 14
-  | R18 => 13 | R19 => 12 | R20 => 11 | R21 => 10
-  | R22 => 9  | R23 => 8  | R24 => 7  | R25 => 6
-  | R26 => 5  | R27 => 4  | R28 => 3  | R29 => 2
-  | R30 => 1  | R31 => 0  | _ => -1
-  end.
-
-Definition index_float_callee_save (r: mreg) :=
-  match r with
-  | F14 => 17 | F15 => 16 | F16 => 15 | F17 => 14
-  | F18 => 13 | F19 => 12 | F20 => 11 | F21 => 10
-  | F22 => 9  | F23 => 8  | F24 => 7  | F25 => 6
-  | F26 => 5  | F27 => 4  | F28 => 3  | F29 => 2
-  | F30 => 1  | F31 => 0  | _ => -1
-  end.
-
-Ltac ElimOrEq :=
-  match goal with
-  |  |- (?x = ?y) \/ _ -> _ =>
-       let H := fresh in
-       (intro H; elim H; clear H;
-        [intro H; rewrite <- H; clear H | ElimOrEq])
-  |  |- False -> _ =>
-       let H := fresh in (intro H; contradiction)
-  end.
-
-Ltac OrEq :=
-  match goal with
-  | |- (?x = ?x) \/ _ => left; reflexivity
-  | |- (?x = ?y) \/ _ => right; OrEq
-  | |- False => fail
-  end.
-
-Ltac NotOrEq :=
-  match goal with
-  | |- (?x = ?y) \/ _ -> False =>
-       let H := fresh in (
-       intro H; elim H; clear H; [intro; discriminate | NotOrEq])
-  | |- False -> False =>
-       contradiction
-  end.
-
-Lemma index_int_callee_save_pos:
-  forall r, In r int_callee_save_regs -> index_int_callee_save r >= 0.
-Proof.
-  intro r. simpl; ElimOrEq; unfold index_int_callee_save; omega.
-Qed.
-
-Lemma index_float_callee_save_pos:
-  forall r, In r float_callee_save_regs -> index_float_callee_save r >= 0.
-Proof.
-  intro r. simpl; ElimOrEq; unfold index_float_callee_save; omega.
-Qed.
-
-Lemma index_int_callee_save_pos2:
-  forall r, index_int_callee_save r >= 0 -> In r int_callee_save_regs.
-Proof.
-  destruct r; simpl; intro; omegaContradiction || OrEq.
-Qed.
-
-Lemma index_float_callee_save_pos2:
-  forall r, index_float_callee_save r >= 0 -> In r float_callee_save_regs.
-Proof.
-  destruct r; simpl; intro; omegaContradiction || OrEq.
-Qed.
-
-Lemma index_int_callee_save_inj:
-  forall r1 r2,
-  In r1 int_callee_save_regs ->
-  In r2 int_callee_save_regs ->
-  r1 <> r2 ->
-  index_int_callee_save r1 <> index_int_callee_save r2.
-Proof.
-  intros r1 r2.
-  simpl; ElimOrEq; ElimOrEq; unfold index_int_callee_save;
-  intros; congruence.
-Qed.
-
-Lemma index_float_callee_save_inj:
-  forall r1 r2,
-  In r1 float_callee_save_regs ->
-  In r2 float_callee_save_regs ->
-  r1 <> r2 ->
-  index_float_callee_save r1 <> index_float_callee_save r2.
-Proof.
-  intros r1 r2.
-  simpl; ElimOrEq; ElimOrEq; unfold index_float_callee_save;
-  intros; congruence.
-Qed.
-
-(** The following lemmas show that
-    (temporaries, destroyed at call, integer callee-save, float callee-save)
-    is a partition of the set of machine registers. *)
-
-Lemma int_float_callee_save_disjoint:
-  list_disjoint int_callee_save_regs float_callee_save_regs.
-Proof.
-  red; intros r1 r2. simpl; ElimOrEq; ElimOrEq; discriminate.
-Qed.
-
-Lemma register_classification:
-  forall r,
-  In r destroyed_at_call \/ In r int_callee_save_regs \/ In r float_callee_save_regs.
-Proof.
-  destruct r;
-  try (left; simpl; OrEq);
-  try (right; left; simpl; OrEq);
-  try (right; right; simpl; OrEq).
-Qed.
-
-Lemma int_callee_save_not_destroyed:
-  forall r,
-    In r destroyed_at_call -> In r int_callee_save_regs -> False.
-Proof.
-  intros. revert H0 H. simpl. ElimOrEq; NotOrEq.
-Qed.
-
-Lemma float_callee_save_not_destroyed:
-  forall r,
-    In r destroyed_at_call -> In r float_callee_save_regs -> False.
-Proof.
-  intros. revert H0 H. simpl. ElimOrEq; NotOrEq.
-Qed.
-
-Lemma int_callee_save_type:
-  forall r, In r int_callee_save_regs -> mreg_type r = Tany32.
-Proof.
-  intro. simpl; ElimOrEq; reflexivity.
-Qed.
-
-Lemma float_callee_save_type:
-  forall r, In r float_callee_save_regs -> mreg_type r = Tany64.
-Proof.
-  intro. simpl; ElimOrEq; reflexivity.
-Qed.
-
-Ltac NoRepet :=
-  match goal with
-  | |- list_norepet nil =>
-      apply list_norepet_nil
-  | |- list_norepet (?a :: ?b) =>
-      apply list_norepet_cons; [simpl; intuition discriminate | NoRepet]
-  end.
-
-Lemma int_callee_save_norepet:
-  list_norepet int_callee_save_regs.
-Proof.
-  unfold int_callee_save_regs; NoRepet.
-Qed.
-
-Lemma float_callee_save_norepet:
-  list_norepet float_callee_save_regs.
-Proof.
-  unfold float_callee_save_regs; NoRepet.
-Qed.
-
 (** * Function calling conventions *)
 
 (** The functions in this section determine the locations (machine registers
@@ -258,12 +107,12 @@ Qed.
 
 Lemma loc_result_caller_save:
   forall (s: signature) (r: mreg),
-  In r (loc_result s) -> In r destroyed_at_call.
+  In r (loc_result s) -> is_callee_save r = false.
 Proof.
   intros.
   assert (r = R3 \/ r = R4 \/ r = F1).
     unfold loc_result in H. destruct (sig_res s); [destruct t|idtac]; simpl in H; intuition.
-  destruct H0 as [A | [A | A]]; subst r; simpl; OrEq.
+  destruct H0 as [A | [A | A]]; subst r; reflexivity.
 Qed.
 
 (** ** Location of function arguments *)
@@ -347,20 +196,13 @@ Fixpoint size_arguments_rec (tyl: list typ) (ir fr ofs: Z) {struct tyl} : Z :=
 Definition size_arguments (s: signature) : Z :=
   size_arguments_rec s.(sig_args) 0 0 0.
 
-(** A tail-call is possible for a signature if the corresponding
-    arguments are all passed in registers. *)
-
-Definition tailcall_possible (s: signature) : Prop :=
-  forall l, In l (loc_arguments s) ->
-  match l with R _ => True | S _ _ _ => False end.
-
 (** 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 => In r destroyed_at_call
-  | S Outgoing ofs ty => ofs >= 0 /\ ty <> Tlong
+  | R r => is_callee_save r = false
+  | S Outgoing ofs ty => ofs >= 0 /\ (typealign ty | ofs)
   | _ => False
   end.
 
@@ -369,7 +211,7 @@ Remark loc_arguments_rec_charact:
   In l (loc_arguments_rec tyl ir fr ofs) ->
   match l with
   | R r => In r int_param_regs \/ In r float_param_regs
-  | S Outgoing ofs' ty => ofs' >= ofs /\ ty <> Tlong
+  | S Outgoing ofs' ty => ofs' >= ofs /\ (typealign ty | ofs')
   | S _ _ _ => False
   end.
 Proof.
@@ -381,13 +223,13 @@ Opaque list_nth_z.
   destruct (list_nth_z int_param_regs ir) as [r|] eqn:E; destruct H.
   subst. left. eapply list_nth_z_in; eauto.
   eapply IHtyl; eauto.
-  subst. split. omega. congruence.
+  subst. split. omega. apply Z.divide_1_l.
   exploit IHtyl; eauto. destruct l; auto. destruct sl; auto. intuition omega.
 - (* float *)
   destruct (list_nth_z float_param_regs fr) as [r|] eqn:E; destruct H.
   subst. right. eapply list_nth_z_in; eauto.
   eapply IHtyl; eauto.
-  subst. split. apply Zle_ge. apply align_le. omega. congruence.
+  subst. split. apply Zle_ge. apply align_le. omega. apply Z.divide_1_l.
   exploit IHtyl; eauto. destruct l; auto. destruct sl; auto.
   assert (ofs <= align ofs 2) by (apply align_le; omega).
   intuition omega.
@@ -399,18 +241,18 @@ Opaque list_nth_z.
   destruct H. subst; left; eapply list_nth_z_in; eauto.
   eapply IHtyl; eauto.
   assert (ofs <= align ofs 2) by (apply align_le; omega).
-  destruct H. subst. split. omega. congruence.
-  destruct H. subst. split. omega. congruence.
+  destruct H. subst. split. omega. apply Z.divide_1_l.
+  destruct H. subst. split. omega. apply Z.divide_1_l.
   exploit IHtyl; eauto. destruct l; auto. destruct sl; auto. intuition omega.
   assert (ofs <= align ofs 2) by (apply align_le; omega).
-  destruct H. subst. split. omega. congruence.
-  destruct H. subst. split. omega. congruence.
+  destruct H. subst. split. omega. apply Z.divide_1_l.
+  destruct H. subst. split. omega. apply Z.divide_1_l.
   exploit IHtyl; eauto. destruct l; auto. destruct sl; auto. intuition omega.
 - (* single *)
   destruct (list_nth_z float_param_regs fr) as [r|] eqn:E; destruct H.
   subst. right. eapply list_nth_z_in; eauto.
   eapply IHtyl; eauto.
-  subst. split. apply Zle_ge. apply align_le. omega. congruence.
+  subst. split. apply Zle_ge. apply align_le. omega. apply Z.divide_1_l.
   exploit IHtyl; eauto. destruct l; auto. destruct sl; auto.
   assert (ofs <= align ofs 2) by (apply align_le; omega).
   intuition omega.
@@ -418,13 +260,13 @@ Opaque list_nth_z.
   destruct (list_nth_z int_param_regs ir) as [r|] eqn:E; destruct H.
   subst. left. eapply list_nth_z_in; eauto.
   eapply IHtyl; eauto.
-  subst. split. omega. congruence.
+  subst. split. omega. apply Z.divide_1_l.
   exploit IHtyl; eauto. destruct l; auto. destruct sl; auto. intuition omega.
 - (* any64 *)
   destruct (list_nth_z float_param_regs fr) as [r|] eqn:E; destruct H.
   subst. right. eapply list_nth_z_in; eauto.
   eapply IHtyl; eauto.
-  subst. split. apply Zle_ge. apply align_le. omega. congruence.
+  subst. split. apply Zle_ge. apply align_le. omega. apply Z.divide_1_l.
   exploit IHtyl; eauto. destruct l; auto. destruct sl; auto.
   assert (ofs <= align ofs 2) by (apply align_le; omega).
   intuition omega.
@@ -435,10 +277,12 @@ Lemma loc_arguments_acceptable:
   In l (loc_arguments s) -> loc_argument_acceptable l.
 Proof.
   unfold loc_arguments; intros.
+  assert (A: forall r, In r int_param_regs -> is_callee_save r = false) by decide_goal.
+  assert (B: forall r, In r float_param_regs -> is_callee_save r = false) by decide_goal.
   generalize (loc_arguments_rec_charact _ _ _ _ _ H).
   destruct l.
-  intro H0; elim H0; simpl; ElimOrEq; OrEq.
-  destruct sl; try contradiction. simpl. intuition omega.
+  intros [C|C]; simpl; auto.
+  destruct sl; try contradiction. simpl; auto.
 Qed.
 Hint Resolve loc_arguments_acceptable: locs.
 
diff --git a/powerpc/Machregs.v b/powerpc/Machregs.v
index 4ee6493c..24065254 100644
--- a/powerpc/Machregs.v
+++ b/powerpc/Machregs.v
@@ -12,6 +12,7 @@
 
 Require Import String.
 Require Import Coqlib.
+Require Import Decidableplus.
 Require Import Maps.
 Require Import AST.
 Require Import Op.
@@ -53,6 +54,34 @@ Lemma mreg_eq: forall (r1 r2: mreg), {r1 = r2} + {r1 <> r2}.
 Proof. decide equality. Defined.
 Global Opaque mreg_eq.
 
+Definition all_mregs :=
+     R3  :: R4  :: R5  :: R6  :: R7  :: R8  :: R9  :: R10
+  :: R11 :: R12 :: R14 :: R15 :: R16 :: R17 :: R18 :: R19 :: R20
+  :: R21 :: R22 :: R23 :: R24 :: R25 :: R26 :: R27 :: R28
+  :: R29 :: R30 :: R31
+  :: F0  :: F1  :: F2  :: F3  :: F4
+  :: F5  :: F6  :: F7  :: F8
+  :: F9  :: F10 :: F11 :: F12
+  :: F13 :: F14 :: F15
+  :: F16 :: F17 :: F18 :: F19
+  :: F20 :: F21 :: F22 :: F23
+  :: F24 :: F25 :: F26 :: F27
+  :: F28 :: F29 :: F30 :: F31 :: nil.
+
+Lemma all_mregs_complete:
+  forall (r: mreg), In r all_mregs.
+Proof.
+  assert (forall r, proj_sumbool (In_dec mreg_eq r all_mregs) = true) by (destruct r; reflexivity).
+  intros. specialize (H r). InvBooleans. auto.
+Qed.
+
+Instance Decidable_eq_mreg : forall (x y: mreg), Decidable (eq x y) := Decidable_eq mreg_eq.
+  
+Instance Finite_mreg : Finite mreg := {
+  Finite_elements := all_mregs;
+  Finite_elements_spec := all_mregs_complete
+}.
+
 Definition mreg_type (r: mreg): typ :=
   match r with
   | R3  | R4  | R5  | R6  | R7  | R8  | R9  | R10  | R11 | R12
@@ -92,7 +121,7 @@ Module IndexedMreg <: INDEXED_TYPE.
   Lemma index_inj:
     forall r1 r2, index r1 = index r2 -> r1 = r2.
   Proof.
-    destruct r1; destruct r2; simpl; intro; discriminate || reflexivity.
+    decide_goal.
   Qed.
 End IndexedMreg.
 
diff --git a/powerpc/Stacklayout.v b/powerpc/Stacklayout.v
index a751fd98..2b78fd11 100644
--- a/powerpc/Stacklayout.v
+++ b/powerpc/Stacklayout.v
@@ -13,6 +13,7 @@
 (** Machine- and ABI-dependent layout information for activation records. *)
 
 Require Import Coqlib.
+Require Import Memory Separation.
 Require Import Bounds.
 
 (** In the PowerPC/EABI application binary interface,
@@ -25,8 +26,7 @@ Require Import Bounds.
   frame, we will not use these 4 bytes, and just reserve them.
 - Space for outgoing arguments to function calls.
 - Local stack slots.
-- Saved values of integer callee-save registers used by the function.
-- Saved values of float callee-save registers used by the function.
+- Saved values of callee-save registers used by the function.
 - Space for the stack-allocated data declared in Cminor.
 
 The [frame_env] compilation environment records the positions of
@@ -35,100 +35,111 @@ the boundaries between areas in the frame part.
 
 Definition fe_ofs_arg := 8.
 
-Record frame_env : Type := mk_frame_env {
-  fe_size: Z;
-  fe_ofs_link: Z;
-  fe_ofs_retaddr: Z;
-  fe_ofs_local: Z;
-  fe_ofs_int_callee_save: Z;
-  fe_num_int_callee_save: Z;
-  fe_ofs_float_callee_save: Z;
-  fe_num_float_callee_save: Z;
-  fe_stack_data: Z
-}.
-
 (** Computation of the frame environment from the bounds of the current
   function. *)
 
 Definition make_env (b: bounds) :=
   let ol := align (8 + 4 * b.(bound_outgoing)) 8 in    (* locals *)
   let ora := ol + 4 * b.(bound_local) in (* saved return address *)
-  let oics := ora + 4 in            (* integer callee-saves *)
-  let oendi := oics + 4 * b.(bound_int_callee_save) in
-  let ofcs := align oendi 8 in       (* float callee-saves *)
-  let ostkdata := ofcs + 8 * b.(bound_float_callee_save) in (* stack data *)
+  let ocs := ora + 4 in            (* callee-saves *)
+  let oendcs := size_callee_save_area b ocs in
+  let ostkdata := align oendcs 8 in (* stack data *)
   let sz := align (ostkdata + b.(bound_stack_data)) 16 in
-  mk_frame_env sz 0 ora
-                  ol
-                  oics b.(bound_int_callee_save)
-                  ofcs b.(bound_float_callee_save)
-                  ostkdata.
+  {| fe_size := sz;
+     fe_ofs_link := 0;
+     fe_ofs_retaddr := ora;
+     fe_ofs_local := ol;
+     fe_ofs_callee_save := ocs;
+     fe_stack_data := ostkdata;
+     fe_used_callee_save := b.(used_callee_save) |}.
 
 (** Separation property *)
 
-Remark frame_env_separated:
-  forall b,
+Local Open Scope sep_scope.
+
+Lemma frame_env_separated:
+  forall b sp m P,
   let fe := make_env b in
-  0 <= fe.(fe_ofs_link)
-  /\ fe.(fe_ofs_link) + 4 <= fe_ofs_arg
-  /\ fe_ofs_arg + 4 * b.(bound_outgoing) <= fe.(fe_ofs_local)
-  /\ fe.(fe_ofs_local) + 4 * b.(bound_local) <= fe.(fe_ofs_retaddr)
-  /\ fe.(fe_ofs_retaddr) + 4 <= fe.(fe_ofs_int_callee_save)
-  /\ fe.(fe_ofs_int_callee_save) + 4 * b.(bound_int_callee_save) <= fe.(fe_ofs_float_callee_save)
-  /\ fe.(fe_ofs_float_callee_save) + 8 * b.(bound_float_callee_save) <= fe.(fe_stack_data)
-  /\ fe.(fe_stack_data) + b.(bound_stack_data) <= fe.(fe_size)
-  /\ fe.(fe_ofs_retaddr) + 4 <= fe.(fe_size).
+  m |= range sp 0 (fe_stack_data fe) ** range sp (fe_stack_data fe + bound_stack_data b) (fe_size fe) ** P ->
+  m |= range sp (fe_ofs_local fe) (fe_ofs_local fe + 4 * bound_local b)
+       ** range sp fe_ofs_arg (fe_ofs_arg + 4 * bound_outgoing b)
+       ** range sp (fe_ofs_link fe) (fe_ofs_link fe + 4)
+       ** range sp (fe_ofs_retaddr fe) (fe_ofs_retaddr fe + 4)
+       ** range sp (fe_ofs_callee_save fe) (size_callee_save_area b (fe_ofs_callee_save fe))
+       ** P.
 Proof.
-  intros.
-  generalize (align_le (8 + 4 * b.(bound_outgoing)) 8 (refl_equal _)).
-  generalize (align_le (fe.(fe_ofs_int_callee_save) + 4 * b.(bound_int_callee_save)) 8 (refl_equal _)).
-  generalize (align_le (fe.(fe_stack_data) + b.(bound_stack_data)) 16 (refl_equal _)).
-  unfold fe, make_env, fe_size, fe_ofs_link, fe_ofs_retaddr,
-    fe_ofs_local, fe_ofs_int_callee_save,
-    fe_num_int_callee_save,
-    fe_ofs_float_callee_save, fe_num_float_callee_save,
-    fe_stack_data, fe_ofs_arg.
-  intros.
-  generalize (bound_local_pos b); intro;
-  generalize (bound_int_callee_save_pos b); intro;
-  generalize (bound_float_callee_save_pos b); intro;
-  generalize (bound_outgoing_pos b); intro;
-  generalize (bound_stack_data_pos b); intro.
-  omega.
+Local Opaque Z.add Z.mul sepconj range.
+  intros; simpl.
+  set (ol := align (8 + 4 * b.(bound_outgoing)) 8).
+  set (ora := ol + 4 * b.(bound_local)).
+  set (ocs := ora + 4).
+  set (oendcs := size_callee_save_area b ocs).
+  set (ostkdata := align oendcs 8).
+  generalize b.(bound_local_pos) b.(bound_outgoing_pos) b.(bound_stack_data_pos); intros.
+  unfold fe_ofs_arg.
+  assert (8 + 4 * b.(bound_outgoing) <= ol) by (apply align_le; omega).
+  assert (ol <= ora) by (unfold ora; omega).
+  assert (ora <= ocs) by (unfold ocs; omega).
+  assert (ocs <= oendcs) by (apply size_callee_save_area_incr).
+  assert (oendcs <= ostkdata) by (apply align_le; omega).
+(* Reorder as:
+     back link
+     outgoing
+     locals
+     retaddr
+     callee-save *)
+  rewrite sep_swap3.
+(* Apply range_split and range_split2 repeatedly *)
+  apply range_drop_right with 8. omega.
+  apply range_split. omega.
+  apply range_split_2. fold ol; omega. omega.
+  apply range_split. omega. 
+  apply range_split. omega.
+  apply range_drop_right with ostkdata. omega.
+  eapply sep_drop2. eexact H.
 Qed.
 
-(** Alignment property *)
+Lemma frame_env_range:
+  forall b,
+  let fe := make_env b in
+  0 <= fe_stack_data fe /\ fe_stack_data fe + bound_stack_data b <= fe_size fe.
+Proof.
+  intros; simpl.
+  set (ol := align (8 + 4 * b.(bound_outgoing)) 8).
+  set (ora := ol + 4 * b.(bound_local)).
+  set (ocs := ora + 4).
+  set (oendcs := size_callee_save_area b ocs).
+  set (ostkdata := align oendcs 8).
+  generalize b.(bound_local_pos) b.(bound_outgoing_pos) b.(bound_stack_data_pos); intros.
+  unfold fe_ofs_arg.
+  assert (8 + 4 * b.(bound_outgoing) <= ol) by (apply align_le; omega).
+  assert (ol <= ora) by (unfold ora; omega).
+  assert (ora <= ocs) by (unfold ocs; omega).
+  assert (ocs <= oendcs) by (apply size_callee_save_area_incr).
+  assert (oendcs <= ostkdata) by (apply align_le; omega).
+  split. omega. apply align_le. omega.
+Qed.
 
-Remark frame_env_aligned:
+Lemma frame_env_aligned:
   forall b,
   let fe := make_env b in
-  (4 | fe.(fe_ofs_link))
-  /\ (8 | fe.(fe_ofs_local))
-  /\ (4 | fe.(fe_ofs_int_callee_save))
-  /\ (8 | fe.(fe_ofs_float_callee_save))
-  /\ (4 | fe.(fe_ofs_retaddr))
-  /\ (8 | fe.(fe_stack_data))
-  /\ (16 | fe.(fe_size)).
+     (8 | fe_ofs_arg)
+  /\ (8 | fe_ofs_local fe)
+  /\ (8 | fe_stack_data fe)
+  /\ (4 | fe_ofs_link fe)
+  /\ (4 | fe_ofs_retaddr fe).
 Proof.
-  intros.
-  unfold fe, make_env, fe_size, fe_ofs_link, fe_ofs_retaddr,
-    fe_ofs_local, fe_ofs_int_callee_save,
-    fe_num_int_callee_save,
-    fe_ofs_float_callee_save, fe_num_float_callee_save,
-    fe_stack_data.
-  set (x1 := align (8 + 4 * bound_outgoing b) 8).
-  assert (8 | x1). unfold x1; apply align_divides. omega.
-  set (x2 := x1 + 4 * bound_local b).
-  assert (4 | x2). unfold x2; apply Zdivide_plus_r; auto.
-  apply Zdivides_trans with 8. exists 2; auto. auto.
-  exists (bound_local b); ring.
-  set (x3 := x2 + 4).
-  assert (4 | x3). unfold x3; apply Zdivide_plus_r; auto. exists 1; auto.
-  set (x4 := align (x3 + 4 * bound_int_callee_save b) 8).
-  assert (8 | x4). unfold x4. apply align_divides. omega.
-  set (x5 := x4 + 8 * bound_float_callee_save b).
-  assert (8 | x5). unfold x5. apply Zdivide_plus_r; auto. exists (bound_float_callee_save b); ring.
-  set (x6 := align (x5 + bound_stack_data b) 16).
-  assert (16 | x6). unfold x6; apply align_divides. omega.
-  intuition.
+  intros; simpl.
+  set (ol := align (8 + 4 * b.(bound_outgoing)) 8).
+  set (ora := ol + 4 * b.(bound_local)).
+  set (ocs := ora + 4).
+  set (oendcs := size_callee_save_area b ocs).
+  set (ostkdata := align oendcs 8).
+  split. exists (fe_ofs_arg / 8); reflexivity.
+  split. apply align_divides; omega.
+  split. apply align_divides; omega.
+  split. apply Zdivide_0.
+  apply Z.divide_add_r.
+    apply Zdivide_trans with 8. exists 2; auto. apply align_divides; omega.
+    apply Z.divide_factor_l.
 Qed.