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diff --git a/arm/Conventions1.v b/arm/Conventions1.v new file mode 100644 index 0000000..1689c77 --- /dev/null +++ b/arm/Conventions1.v @@ -0,0 +1,770 @@ +(* *********************************************************************) +(* *) +(* The Compcert verified compiler *) +(* *) +(* Xavier Leroy, INRIA Paris-Rocquencourt *) +(* *) +(* Copyright Institut National de Recherche en Informatique et en *) +(* Automatique. All rights reserved. This file is distributed *) +(* under the terms of the INRIA Non-Commercial License Agreement. *) +(* *) +(* *********************************************************************) + +(** Function calling conventions and other conventions regarding the use of + machine registers and stack slots. *) + +Require Import Coqlib. +Require Import AST. +Require Import Events. +Require Import Locations. +Require Archi. + +(** * Classification of machine registers *) + +(** Machine registers (type [mreg] in module [Locations]) are divided in + the following groups: +- Temporaries used for spilling, reloading, and parallel move operations. +- Allocatable registers, that can be assigned to RTL pseudo-registers. + These are further divided into: +-- 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 PowerPC application binary interface (ABI) in our choice + of callee- and caller-save registers. +*) + +Definition int_caller_save_regs := + R0 :: R1 :: R2 :: R3 :: R12 :: nil. + +Definition float_caller_save_regs := + F0 :: F1 :: F2 :: F3 :: F4 :: F5 :: F6 :: F7 :: nil. + +Definition int_callee_save_regs := + R4 :: R5 :: R6 :: R7 :: R8 :: R9 :: R10 :: R11 :: nil. + +Definition float_callee_save_regs := + F8 :: F9 :: F10 :: F11 :: F12 :: F13 :: F14 :: F15 :: nil. + +Definition destroyed_at_call := + int_caller_save_regs ++ float_caller_save_regs. + +Definition dummy_int_reg := R0. (**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 + | R4 => 0 | R5 => 1 | R6 => 2 | R7 => 3 + | R8 => 4 | R9 => 5 | R10 => 6 | R11 => 7 + | _ => -1 + end. + +Definition index_float_callee_save (r: mreg) := + match r with + | F8 => 0 | F9 => 1 | F10 => 2 | F11 => 3 + | F12 => 4 | F13 => 5 | F14 => 6 | F15 => 7 + | _ => -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 + 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. *) + +(** ** Location of function result *) + +(** The result value of a function is passed back to the caller in + registers [R0] or [F0] or [R0,R1], depending on the type of the + returned value. We treat a function without result as a function + with one integer result. + + For the "softfloat" convention, results of FP types should be passed + in [R0] or [R0,R1]. This doesn't fit the CompCert register model, + so we have code in [arm/PrintAsm.ml] that inserts additional moves + to/from [F0]. *) + +Definition loc_result (s: signature) : list mreg := + match s.(sig_res) with + | None => R0 :: nil + | Some (Tint | Tany32) => R0 :: nil + | Some (Tfloat | Tsingle | Tany64) => F0 :: nil + | Some Tlong => R1 :: R0 :: nil + end. + +(** The result registers have types compatible with that given in the signature. *) + +Lemma loc_result_type: + forall sig, + subtype_list (proj_sig_res' sig) (map mreg_type (loc_result sig)) = true. +Proof. + intros. unfold proj_sig_res', loc_result. destruct (sig_res sig) as [[]|]; auto. +Qed. + +(** The result locations are caller-save registers *) + +Lemma loc_result_caller_save: + forall (s: signature) (r: mreg), + In r (loc_result s) -> In r destroyed_at_call. +Proof. + intros. + assert (r = R0 \/ r = R1 \/ r = F0). + 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. +Qed. + +(** ** Location of function arguments *) + +(** For the "hardfloat" configuration, we use the following calling conventions, + adapted from the ARM EABI-HF: +- The first 4 integer arguments are passed in registers [R0] to [R3]. +- The first 2 long integer arguments are passed in an aligned pair of + two integer registers. +- The first 8 single- and double-precision float arguments are passed + in registers [F0...F7] +- Extra arguments are passed on the stack, in [Outgoing] slots, consecutively + assigned (1 word for an integer or single argument, 2 words for a float + or a long), starting at word offset 0. + +This convention is not quite that of the ARM EABI-HF, whereas single float +arguments are passed in 32-bit float registers. Unfortunately, +this does not fit the data model of CompCert. In [PrintAsm.ml] +we insert additional code around function calls that moves +data appropriately. *) + +Definition int_param_regs := + R0 :: R1 :: R2 :: R3 :: nil. + +Definition float_param_regs := + F0 :: F1 :: F2 :: F3 :: F4 :: F5 :: F6 :: F7 :: nil. + +Definition ireg_param (n: Z) : mreg := + match list_nth_z int_param_regs n with Some r => r | None => R0 end. + +Definition freg_param (n: Z) : mreg := + match list_nth_z float_param_regs n with Some r => r | None => F0 end. + +Fixpoint loc_arguments_hf + (tyl: list typ) (ir fr ofs: Z) {struct tyl} : list loc := + match tyl with + | nil => nil + | (Tint | Tany32) as ty :: tys => + if zlt ir 4 + then R (ireg_param ir) :: loc_arguments_hf tys (ir + 1) fr ofs + else S Outgoing ofs ty :: loc_arguments_hf tys ir fr (ofs + 1) + | (Tfloat | Tany64) as ty :: tys => + if zlt fr 8 + then R (freg_param fr) :: loc_arguments_hf tys ir (fr + 1) ofs + else let ofs := align ofs 2 in + S Outgoing ofs ty :: loc_arguments_hf tys ir fr (ofs + 2) + | Tsingle :: tys => + if zlt fr 8 + then R (freg_param fr) :: loc_arguments_hf tys ir (fr + 1) ofs + else S Outgoing ofs Tsingle :: loc_arguments_hf tys ir fr (ofs + 1) + | Tlong :: tys => + let ir := align ir 2 in + if zlt ir 4 + then R (ireg_param (ir + 1)) :: R (ireg_param ir) :: loc_arguments_hf tys (ir + 2) fr ofs + else let ofs := align ofs 2 in + S Outgoing (ofs + 1) Tint :: S Outgoing ofs Tint :: loc_arguments_hf tys ir fr (ofs + 2) + end. + +(** For the "softfloat" configuration, as well as for variable-argument functions + in the "hardfloat" configuration, we use the default ARM EABI (not HF) + calling conventions: +- The first 4 integer arguments are passed in registers [R0] to [R3]. +- The first 2 long integer arguments are passed in an aligned pair of + two integer registers. +- The first 2 double-precision float arguments are passed in [F0] or [F2] +- The first 4 single-precision float arguments are passed in [F0...F3] +- Integer arguments and float arguments are kept in sync so that + they can all be mapped back to [R0...R3] in [PrintAsm.ml]. +- Extra arguments are passed on the stack, in [Outgoing] slots, consecutively + assigned (1 word for an integer or single argument, 2 words for a float + or a long), starting at word offset 0. + +This convention is not quite that of the ARM EABI, whereas every float +argument are passed in one or two integer registers. Unfortunately, +this does not fit the data model of CompCert. In [PrintAsm.ml] +we insert additional code around function calls and returns that moves +data appropriately. *) + +Fixpoint loc_arguments_sf + (tyl: list typ) (ofs: Z) {struct tyl} : list loc := + match tyl with + | nil => nil + | (Tint|Tany32) as ty :: tys => + (if zlt ofs 0 then R (ireg_param (ofs + 4)) else S Outgoing ofs ty) + :: loc_arguments_sf tys (ofs + 1) + | (Tfloat|Tany64) as ty :: tys => + let ofs := align ofs 2 in + (if zlt ofs 0 then R (freg_param (ofs + 4)) else S Outgoing ofs ty) + :: loc_arguments_sf tys (ofs + 2) + | Tsingle :: tys => + (if zlt ofs 0 then R (freg_param (ofs + 4)) else S Outgoing ofs Tsingle) + :: loc_arguments_sf tys (ofs + 1) + | Tlong :: tys => + let ofs := align ofs 2 in + (if zlt ofs 0 then R (ireg_param (ofs+1+4)) else S Outgoing (ofs+1) Tint) + :: (if zlt ofs 0 then R (ireg_param (ofs+4)) else S Outgoing ofs Tint) + :: loc_arguments_sf tys (ofs + 2) + 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 loc := + match Archi.abi with + | Archi.Softfloat => + loc_arguments_sf s.(sig_args) (-4) + | Archi.Hardfloat => + if s.(sig_cc).(cc_vararg) + then loc_arguments_sf s.(sig_args) (-4) + else loc_arguments_hf s.(sig_args) 0 0 0 + end. + +(** [size_arguments s] returns the number of [Outgoing] slots used + to call a function with signature [s]. *) + +Fixpoint size_arguments_hf (tyl: list typ) (ir fr ofs: Z) {struct tyl} : Z := + match tyl with + | nil => ofs + | (Tint|Tany32) :: tys => + if zlt ir 4 + then size_arguments_hf tys (ir + 1) fr ofs + else size_arguments_hf tys ir fr (ofs + 1) + | (Tfloat|Tany64) :: tys => + if zlt fr 8 + then size_arguments_hf tys ir (fr + 1) ofs + else size_arguments_hf tys ir fr (align ofs 2 + 2) + | Tsingle :: tys => + if zlt fr 8 + then size_arguments_hf tys ir (fr + 1) ofs + else size_arguments_hf tys ir fr (ofs + 1) + | Tlong :: tys => + let ir := align ir 2 in + if zlt ir 4 + then size_arguments_hf tys (ir + 2) fr ofs + else size_arguments_hf tys ir fr (align ofs 2 + 2) + end. + +Fixpoint size_arguments_sf (tyl: list typ) (ofs: Z) {struct tyl} : Z := + match tyl with + | nil => Zmax 0 ofs + | (Tint | Tsingle | Tany32) :: tys => size_arguments_sf tys (ofs + 1) + | (Tfloat | Tlong | Tany64) :: tys => size_arguments_sf tys (align ofs 2 + 2) + end. + +Definition size_arguments (s: signature) : Z := + match Archi.abi with + | Archi.Softfloat => + size_arguments_sf s.(sig_args) (-4) + | Archi.Hardfloat => + if s.(sig_cc).(cc_vararg) + then size_arguments_sf s.(sig_args) (-4) + else size_arguments_hf s.(sig_args) 0 0 0 + end. + +(** Argument locations are either non-temporary 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 + | _ => False + end. + +Remark ireg_param_in_params: forall n, In (ireg_param n) int_param_regs. +Proof. + unfold ireg_param; intros. + destruct (list_nth_z int_param_regs n) as [r|] eqn:NTH. + eapply list_nth_z_in; eauto. + simpl; auto. +Qed. + +Remark freg_param_in_params: forall n, In (freg_param n) float_param_regs. +Proof. + unfold freg_param; intros. + destruct (list_nth_z float_param_regs n) as [r|] eqn:NTH. + eapply list_nth_z_in; eauto. + simpl; auto. +Qed. + +Remark loc_arguments_hf_charact: + forall tyl ir fr ofs l, + In l (loc_arguments_hf 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 _ _ _ => False + end. +Proof. + assert (INCR: forall l ofs1 ofs2, + match l with + | R r => In r int_param_regs \/ In r float_param_regs + | S Outgoing ofs' ty => ofs' >= ofs2 /\ ty <> Tlong + | S _ _ _ => False + end -> + ofs1 <= ofs2 -> + match l with + | R r => In r int_param_regs \/ In r float_param_regs + | S Outgoing ofs' ty => ofs' >= ofs1 /\ ty <> Tlong + | S _ _ _ => False + end). + { + intros. destruct l; auto. destruct sl; auto. intuition omega. + } + induction tyl; simpl loc_arguments_hf; intros. + elim H. + destruct a. +- (* int *) + destruct (zlt ir 4); destruct H. + subst. left; apply ireg_param_in_params. + eapply IHtyl; eauto. + subst. split; [omega | congruence]. + eapply INCR. eapply IHtyl; eauto. omega. +- (* float *) + destruct (zlt fr 8); destruct H. + subst. right; apply freg_param_in_params. + eapply IHtyl; eauto. + subst. split. apply Zle_ge. apply align_le. omega. congruence. + eapply INCR. eapply IHtyl; eauto. + apply Zle_trans with (align ofs 2). apply align_le; omega. omega. +- (* long *) + set (ir' := align ir 2) in *. + assert (ofs <= align ofs 2) by (apply align_le; omega). + destruct (zlt ir' 4). + destruct H. subst l; left; apply ireg_param_in_params. + destruct H. subst l; left; apply ireg_param_in_params. + eapply IHtyl; eauto. + destruct H. subst l; split; [ omega | congruence ]. + destruct H. subst l; split; [ omega | congruence ]. + eapply INCR. eapply IHtyl; eauto. omega. +- (* single *) + destruct (zlt fr 8); destruct H. + subst. right; apply freg_param_in_params. + eapply IHtyl; eauto. + subst. split; [omega | congruence]. + eapply INCR. eapply IHtyl; eauto. omega. +- (* any32 *) + destruct (zlt ir 4); destruct H. + subst. left; apply ireg_param_in_params. + eapply IHtyl; eauto. + subst. split; [omega | congruence]. + eapply INCR. eapply IHtyl; eauto. omega. +- (* any64 *) + destruct (zlt fr 8); destruct H. + subst. right; apply freg_param_in_params. + eapply IHtyl; eauto. + subst. split. apply Zle_ge. apply align_le. omega. congruence. + eapply INCR. eapply IHtyl; eauto. + apply Zle_trans with (align ofs 2). apply align_le; omega. omega. +Qed. + +Remark loc_arguments_sf_charact: + forall tyl ofs l, + In l (loc_arguments_sf tyl ofs) -> + match l with + | R r => In r int_param_regs \/ In r float_param_regs + | S Outgoing ofs' ty => ofs' >= Zmax 0 ofs /\ ty <> Tlong + | S _ _ _ => False + end. +Proof. + assert (INCR: forall l ofs1 ofs2, + match l with + | R r => In r int_param_regs \/ In r float_param_regs + | S Outgoing ofs' ty => ofs' >= Zmax 0 ofs2 /\ ty <> Tlong + | S _ _ _ => False + end -> + ofs1 <= ofs2 -> + match l with + | R r => In r int_param_regs \/ In r float_param_regs + | S Outgoing ofs' ty => ofs' >= Zmax 0 ofs1 /\ ty <> Tlong + | S _ _ _ => False + end). + { + intros. destruct l; auto. destruct sl; auto. intuition xomega. + } + induction tyl; simpl loc_arguments_sf; intros. + elim H. + destruct a. +- (* int *) + destruct H. + destruct (zlt ofs 0); subst l. + left; apply ireg_param_in_params. + split. xomega. congruence. + eapply INCR. eapply IHtyl; eauto. omega. +- (* float *) + set (ofs' := align ofs 2) in *. + assert (ofs <= ofs') by (apply align_le; omega). + destruct H. + destruct (zlt ofs' 0); subst l. + right; apply freg_param_in_params. + split. xomega. congruence. + eapply INCR. eapply IHtyl; eauto. omega. +- (* long *) + set (ofs' := align ofs 2) in *. + assert (ofs <= ofs') by (apply align_le; omega). + destruct H. + destruct (zlt ofs' 0); subst l. + left; apply ireg_param_in_params. + split. xomega. congruence. + destruct H. + destruct (zlt ofs' 0); subst l. + left; apply ireg_param_in_params. + split. xomega. congruence. + eapply INCR. eapply IHtyl; eauto. omega. +- (* single *) + destruct H. + destruct (zlt ofs 0); subst l. + right; apply freg_param_in_params. + split. xomega. congruence. + eapply INCR. eapply IHtyl; eauto. omega. +- (* any32 *) + destruct H. + destruct (zlt ofs 0); subst l. + left; apply ireg_param_in_params. + split. xomega. congruence. + eapply INCR. eapply IHtyl; eauto. omega. +- (* any64 *) + set (ofs' := align ofs 2) in *. + assert (ofs <= ofs') by (apply align_le; omega). + destruct H. + destruct (zlt ofs' 0); subst l. + right; apply freg_param_in_params. + split. xomega. congruence. + eapply INCR. eapply IHtyl; eauto. omega. +Qed. + +Lemma loc_arguments_acceptable: + forall (s: signature) (l: loc), + In l (loc_arguments s) -> loc_argument_acceptable l. +Proof. + unfold loc_arguments; intros. + assert (forall r, In r int_param_regs \/ In r float_param_regs -> In r destroyed_at_call). + { + intros. elim H0; simpl; ElimOrEq; OrEq. + } + assert (In l (loc_arguments_sf (sig_args s) (-4)) -> loc_argument_acceptable l). + { intros. red. exploit loc_arguments_sf_charact; eauto. destruct l; auto. } + assert (In l (loc_arguments_hf (sig_args s) 0 0 0) -> loc_argument_acceptable l). + { intros. red. exploit loc_arguments_hf_charact; eauto. destruct l; auto. } + destruct Archi.abi; [ | destruct (cc_vararg (sig_cc s)) ]; auto. +Qed. + +Hint Resolve loc_arguments_acceptable: locs. + +(** The offsets of [Outgoing] arguments are below [size_arguments s]. *) + +Remark size_arguments_hf_above: + forall tyl ir fr ofs0, + ofs0 <= size_arguments_hf tyl ir fr ofs0. +Proof. + induction tyl; simpl; intros. + omega. + destruct a. + destruct (zlt ir 4); eauto. apply Zle_trans with (ofs0 + 1); auto; omega. + destruct (zlt fr 8); eauto. + apply Zle_trans with (align ofs0 2). apply align_le; omega. + apply Zle_trans with (align ofs0 2 + 2); auto; omega. + set (ir' := align ir 2). + destruct (zlt ir' 4); eauto. + apply Zle_trans with (align ofs0 2). apply align_le; omega. + apply Zle_trans with (align ofs0 2 + 2); auto; omega. + destruct (zlt fr 8); eauto. + apply Zle_trans with (ofs0 + 1); eauto. omega. + destruct (zlt ir 4); eauto. apply Zle_trans with (ofs0 + 1); auto; omega. + destruct (zlt fr 8); eauto. + apply Zle_trans with (align ofs0 2). apply align_le; omega. + apply Zle_trans with (align ofs0 2 + 2); auto; omega. +Qed. + +Remark size_arguments_sf_above: + forall tyl ofs0, + Zmax 0 ofs0 <= size_arguments_sf tyl ofs0. +Proof. + induction tyl; simpl; intros. + omega. + destruct a; (eapply Zle_trans; [idtac|eauto]). + xomega. + assert (ofs0 <= align ofs0 2) by (apply align_le; omega). xomega. + assert (ofs0 <= align ofs0 2) by (apply align_le; omega). xomega. + xomega. + xomega. + assert (ofs0 <= align ofs0 2) by (apply align_le; omega). xomega. +Qed. + +Lemma size_arguments_above: + forall s, size_arguments s >= 0. +Proof. + intros; unfold size_arguments. apply Zle_ge. + assert (0 <= size_arguments_sf (sig_args s) (-4)). + { change 0 with (Zmax 0 (-4)). apply size_arguments_sf_above. } + assert (0 <= size_arguments_hf (sig_args s) 0 0 0). + { apply size_arguments_hf_above. } + destruct Archi.abi; [ | destruct (cc_vararg (sig_cc s)) ]; auto. +Qed. + +Lemma loc_arguments_hf_bounded: + forall ofs ty tyl ir fr ofs0, + In (S Outgoing ofs ty) (loc_arguments_hf tyl ir fr ofs0) -> + ofs + typesize ty <= size_arguments_hf tyl ir fr ofs0. +Proof. + induction tyl; simpl; intros. + elim H. + destruct a. +- (* int *) + destruct (zlt ir 4); destruct H. + discriminate. + eauto. + inv H. apply size_arguments_hf_above. + eauto. +- (* float *) + destruct (zlt fr 8); destruct H. + discriminate. + eauto. + inv H. apply size_arguments_hf_above. + eauto. +- (* long *) + destruct (zlt (align ir 2) 4). + destruct H. discriminate. destruct H. discriminate. eauto. + destruct H. inv H. + rewrite <- Zplus_assoc. simpl. apply size_arguments_hf_above. + destruct H. inv H. + eapply Zle_trans. 2: apply size_arguments_hf_above. simpl; omega. + eauto. +- (* float *) + destruct (zlt fr 8); destruct H. + discriminate. + eauto. + inv H. apply size_arguments_hf_above. + eauto. +- (* any32 *) + destruct (zlt ir 4); destruct H. + discriminate. + eauto. + inv H. apply size_arguments_hf_above. + eauto. +- (* any64 *) + destruct (zlt fr 8); destruct H. + discriminate. + eauto. + inv H. apply size_arguments_hf_above. + eauto. +Qed. + +Lemma loc_arguments_sf_bounded: + forall ofs ty tyl ofs0, + In (S Outgoing ofs ty) (loc_arguments_sf tyl ofs0) -> + Zmax 0 (ofs + typesize ty) <= size_arguments_sf tyl ofs0. +Proof. + induction tyl; simpl; intros. + elim H. + destruct a. +- (* int *) + destruct H. + destruct (zlt ofs0 0); inv H. apply size_arguments_sf_above. + eauto. +- (* float *) + destruct H. + destruct (zlt (align ofs0 2) 0); inv H. apply size_arguments_sf_above. + eauto. +- (* long *) + destruct H. + destruct (zlt (align ofs0 2) 0); inv H. + rewrite <- Zplus_assoc. simpl. apply size_arguments_sf_above. + destruct H. + destruct (zlt (align ofs0 2) 0); inv H. + eapply Zle_trans. 2: apply size_arguments_sf_above. simpl; xomega. + eauto. +- (* float *) + destruct H. + destruct (zlt ofs0 0); inv H. apply size_arguments_sf_above. + eauto. +- (* any32 *) + destruct H. + destruct (zlt ofs0 0); inv H. apply size_arguments_sf_above. + eauto. +- (* any64 *) + destruct H. + destruct (zlt (align ofs0 2) 0); inv H. apply size_arguments_sf_above. + eauto. +Qed. + +Lemma loc_arguments_bounded: + forall (s: signature) (ofs: Z) (ty: typ), + In (S Outgoing ofs ty) (loc_arguments s) -> + ofs + typesize ty <= size_arguments s. +Proof. + unfold loc_arguments, size_arguments; intros. + assert (In (S Outgoing ofs ty) (loc_arguments_sf (sig_args s) (-4)) -> + ofs + typesize ty <= size_arguments_sf (sig_args s) (-4)). + { intros. eapply Zle_trans. 2: eapply loc_arguments_sf_bounded; eauto. xomega. } + assert (In (S Outgoing ofs ty) (loc_arguments_hf (sig_args s) 0 0 0) -> + ofs + typesize ty <= size_arguments_hf (sig_args s) 0 0 0). + { intros. eapply loc_arguments_hf_bounded; eauto. } + destruct Archi.abi; [ | destruct (cc_vararg (sig_cc s)) ]; eauto. +Qed. + +Lemma loc_arguments_main: + loc_arguments signature_main = nil. +Proof. + unfold loc_arguments. + destruct Archi.abi; reflexivity. +Qed. |