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diff --git a/arm/linux/Conventions.v b/arm/linux/Conventions.v new file mode 100644 index 0000000..0342521 --- /dev/null +++ b/arm/linux/Conventions.v @@ -0,0 +1,858 @@ +(* *********************************************************************) +(* *) +(* 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 Locations. + +(** * 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 :: nil. + +Definition float_caller_save_regs := + F0 :: F1 :: nil. + +Definition int_callee_save_regs := + R4 :: R5 :: R6 :: R7 :: R8 :: R9 :: R11 :: nil. + +Definition float_callee_save_regs := + F4 :: F5 :: F6 :: F7 :: nil. + +Definition destroyed_at_call_regs := + int_caller_save_regs ++ float_caller_save_regs. + +Definition destroyed_at_call := + List.map R destroyed_at_call_regs. + +Definition int_temporaries := IT1 :: IT2 :: nil. + +Definition float_temporaries := FT1 :: FT2 :: nil. + +Definition temporaries := + R IT1 :: R IT2 :: R FT1 :: R FT2 :: nil. + +(** 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 | R11 => 6 + | _ => -1 + end. + +Definition index_float_callee_save (r: mreg) := + match r with + | F4 => 0 | F5 => 1 | F6 => 2 | F7 => 3 + | _ => -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 r) temporaries \/ In (R r) destroyed_at_call) \/ + (In r int_callee_save_regs \/ In r float_callee_save_regs). +Proof. + destruct r; + try (left; left; simpl; OrEq); + try (left; right; simpl; OrEq); + try (right; left; simpl; OrEq); + try (right; right; simpl; OrEq). +Qed. + +Lemma int_callee_save_not_destroyed: + forall r, + In (R r) temporaries \/ In (R r) destroyed_at_call -> + ~(In r int_callee_save_regs). +Proof. + intros; red; intros. elim H. + generalize H0. simpl; ElimOrEq; NotOrEq. + generalize H0. simpl; ElimOrEq; NotOrEq. +Qed. + +Lemma float_callee_save_not_destroyed: + forall r, + In (R r) temporaries \/ In (R r) destroyed_at_call -> + ~(In r float_callee_save_regs). +Proof. + intros; red; intros. elim H. + generalize H0. simpl; ElimOrEq; NotOrEq. + generalize H0. simpl; ElimOrEq; NotOrEq. +Qed. + +Lemma int_callee_save_type: + forall r, In r int_callee_save_regs -> mreg_type r = Tint. +Proof. + intro. simpl; ElimOrEq; reflexivity. +Qed. + +Lemma float_callee_save_type: + forall r, In r float_callee_save_regs -> mreg_type r = Tfloat. +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. + +(** * Acceptable locations for register allocation *) + +(** The following predicate describes the locations that can be assigned + to an RTL pseudo-register during register allocation: a non-temporary + machine register or a [Local] stack slot are acceptable. *) + +Definition loc_acceptable (l: loc) : Prop := + match l with + | R r => ~(In l temporaries) + | S (Local ofs ty) => ofs >= 0 + | S (Incoming _ _) => False + | S (Outgoing _ _) => False + end. + +Definition locs_acceptable (ll: list loc) : Prop := + forall l, In l ll -> loc_acceptable l. + +Lemma temporaries_not_acceptable: + forall l, loc_acceptable l -> Loc.notin l temporaries. +Proof. + unfold loc_acceptable; destruct l. + simpl. intuition congruence. + destruct s; try contradiction. + intro. simpl. tauto. +Qed. +Hint Resolve temporaries_not_acceptable: locs. + +Lemma locs_acceptable_disj_temporaries: + forall ll, locs_acceptable ll -> Loc.disjoint ll temporaries. +Proof. + intros. apply Loc.notin_disjoint. intros. + apply temporaries_not_acceptable. auto. +Qed. + +Lemma loc_acceptable_noteq_diff: + forall l1 l2, + loc_acceptable l1 -> l1 <> l2 -> Loc.diff l1 l2. +Proof. + unfold loc_acceptable, Loc.diff; destruct l1; destruct l2; + try (destruct s); try (destruct s0); intros; auto; try congruence. + case (zeq z z0); intro. + compare t t0; intro. + subst z0; subst t0; tauto. + tauto. tauto. + contradiction. contradiction. +Qed. + +Lemma loc_acceptable_notin_notin: + forall r ll, + loc_acceptable r -> + ~(In r ll) -> Loc.notin r ll. +Proof. + induction ll; simpl; intros. + auto. + split. apply loc_acceptable_noteq_diff. assumption. + apply sym_not_equal. tauto. + apply IHll. assumption. tauto. +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. To ensure binary interoperability of code generated by our + compiler with libraries compiled by another PowerPC compiler, we + implement the standard conventions defined in the PowerPC application + binary interface. *) + +(** ** Location of function result *) + +(** The result value of a function is passed back to the caller in + registers [R0] or [F0], depending on the type of the returned value. + We treat a function without result as a function with one integer result. *) + +Definition loc_result (s: signature) : mreg := + match s.(sig_res) with + | None => R0 + | Some Tint => R0 + | Some Tfloat => F0 + end. + +(** The result location has the type stated in the signature. *) + +Lemma loc_result_type: + forall sig, + mreg_type (loc_result sig) = + match sig.(sig_res) with None => Tint | Some ty => ty end. +Proof. + intros; unfold loc_result. + destruct (sig_res sig). + destruct t; reflexivity. + reflexivity. +Qed. + +(** The result location is acceptable. *) + +Lemma loc_result_acceptable: + forall sig, loc_acceptable (R (loc_result sig)). +Proof. + intros. unfold loc_acceptable. red. + unfold loc_result. destruct (sig_res sig). + destruct t; simpl; NotOrEq. + simpl; NotOrEq. +Qed. + +(** The result location is a caller-save register. *) + +Lemma loc_result_caller_save: + forall (s: signature), In (R (loc_result s)) destroyed_at_call. +Proof. + intros; unfold loc_result. + destruct (sig_res s). + destruct t; simpl; OrEq. + simpl; OrEq. +Qed. + +(** The result location is not a callee-save register. *) + +Lemma loc_result_not_callee_save: + forall (s: signature), + ~(In (loc_result s) int_callee_save_regs \/ In (loc_result s) float_callee_save_regs). +Proof. + intros. generalize (loc_result_caller_save s). + generalize (int_callee_save_not_destroyed (loc_result s)). + generalize (float_callee_save_not_destroyed (loc_result s)). + tauto. +Qed. + +(** ** Location of function arguments *) + +(** We use the following calling conventions, adapted from the ARM ABI: +- The first 4 integer arguments are passed in registers [R0] to [R3]. +- The first 2 float arguments are passed in registers [F0] and [F1]. +- Each float argument passed in a float register ``consumes'' two + integer arguments. +- Extra arguments are passed on the stack, in [Outgoing] slots, consecutively + assigned (1 word for an integer argument, 2 words for a float), + starting at word offset 0. + +These conventions are somewhat baroque, but they are mandated by the ABI. +*) + +Fixpoint loc_arguments_rec + (tyl: list typ) (iregl: list mreg) (fregl: list mreg) + (ofs: Z) {struct tyl} : list loc := + match tyl with + | nil => nil + | Tint :: tys => + match iregl with + | nil => + S (Outgoing ofs Tint) :: loc_arguments_rec tys nil fregl (ofs + 1) + | ireg :: iregs => + R ireg :: loc_arguments_rec tys iregs fregl ofs + end + | Tfloat :: tys => + match fregl with + | nil => + S (Outgoing ofs Tfloat) :: + loc_arguments_rec tys iregl nil (ofs + 2) + | freg :: fregs => + match iregl with + | nil => + S (Outgoing ofs Tfloat) :: + loc_arguments_rec tys nil fregl (ofs + 2) + | ireg :: nil => + R freg :: + loc_arguments_rec tys nil fregs (ofs + 1) + | ireg1 :: ireg2 :: iregs => + R freg :: + loc_arguments_rec tys iregs fregs ofs + end + end + end. + +Definition int_param_regs := + R0 :: R1 :: R2 :: R3 :: nil. +Definition float_param_regs := + F0 :: F1 :: nil. + +(** [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 := + loc_arguments_rec s.(sig_args) int_param_regs float_param_regs 0. + +(** [size_arguments s] returns the number of [Outgoing] slots used + to call a function with signature [s]. *) + +Fixpoint size_arguments_rec + (tyl: list typ) (iregl: list mreg) (fregl: list mreg) + (ofs: Z) {struct tyl} : Z := + match tyl with + | nil => ofs + | Tint :: tys => + match iregl with + | nil => size_arguments_rec tys nil fregl (ofs + 1) + | ireg :: iregs => size_arguments_rec tys iregs fregl ofs + end + | Tfloat :: tys => + match fregl with + | nil => + size_arguments_rec tys iregl nil (ofs + 2) + | freg :: fregs => + match iregl with + | nil => + size_arguments_rec tys nil fregl (ofs + 2) + | ireg :: nil => + size_arguments_rec tys nil fregs (ofs + 1) + | ireg1 :: ireg2 :: iregs => + size_arguments_rec tys iregs fregs ofs + end + end + end. + +Definition size_arguments (s: signature) : Z := + size_arguments_rec s.(sig_args) int_param_regs float_param_regs 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 non-temporary registers or [Outgoing] + stack slots at nonnegative offsets. *) + +Definition loc_argument_acceptable (l: loc) : Prop := + match l with + | R r => ~(In l temporaries) + | S (Outgoing ofs ty) => ofs >= 0 + | _ => False + end. + +Remark loc_arguments_rec_charact: + forall tyl iregl fregl ofs l, + In l (loc_arguments_rec tyl iregl fregl ofs) -> + match l with + | R r => In r iregl \/ In r fregl + | S (Outgoing ofs' ty) => ofs' >= ofs + | S _ => False + end. +Proof. + induction tyl; simpl loc_arguments_rec; intros. + elim H. + destruct a. + destruct iregl; elim H; intro. + subst l. omega. + generalize (IHtyl _ _ _ _ H0). destruct l; auto. destruct s; auto. omega. + subst l. auto with coqlib. + generalize (IHtyl _ _ _ _ H0). destruct l; auto. simpl; intuition. + destruct fregl. + elim H; intro. + subst l. omega. + generalize (IHtyl _ _ _ _ H0). destruct l; auto. destruct s; auto. omega. + destruct iregl. + elim H; intro. + subst l. omega. + generalize (IHtyl _ _ _ _ H0). destruct l; auto. destruct s; auto. omega. + destruct iregl. + elim H; intro. + subst l. auto with coqlib. + generalize (IHtyl _ _ _ _ H0). destruct l; auto. + intros [A|B]. elim A. auto with coqlib. + destruct s; auto. omega. + elim H; intro. + subst l. auto with coqlib. + generalize (IHtyl _ _ _ _ H0). destruct l; auto. + intros [A|B]; auto with coqlib. +Qed. + +Lemma loc_arguments_acceptable: + forall (s: signature) (r: loc), + In r (loc_arguments s) -> loc_argument_acceptable r. +Proof. + unfold loc_arguments; intros. + generalize (loc_arguments_rec_charact _ _ _ _ _ H). + destruct r. + intro H0; elim H0. simpl. unfold not. ElimOrEq; NotOrEq. + simpl. unfold not. ElimOrEq; NotOrEq. + destruct s0; try contradiction. + simpl. omega. +Qed. +Hint Resolve loc_arguments_acceptable: locs. + +(** Arguments are parwise disjoint (in the sense of [Loc.norepet]). *) + +Remark loc_arguments_rec_notin_reg: + forall tyl iregl fregl ofs r, + ~(In r iregl) -> ~(In r fregl) -> + Loc.notin (R r) (loc_arguments_rec tyl iregl fregl ofs). +Proof. + induction tyl; simpl; intros. + auto. + destruct a. + destruct iregl; simpl. auto. + simpl in H. split. apply sym_not_equal. tauto. + apply IHtyl. tauto. tauto. + destruct fregl; simpl. auto. simpl in H0. + destruct iregl; simpl. auto. + destruct iregl; simpl. + split. apply sym_not_equal. tauto. apply IHtyl. hnf. tauto. tauto. + split. apply sym_not_equal. tauto. apply IHtyl. + red; intro. apply H. auto with coqlib. tauto. +Qed. + +Remark loc_arguments_rec_notin_local: + forall tyl iregl fregl ofs ofs0 ty0, + Loc.notin (S (Local ofs0 ty0)) (loc_arguments_rec tyl iregl fregl ofs). +Proof. + induction tyl; simpl; intros. + auto. + destruct a. + destruct iregl; simpl; auto. + destruct fregl; simpl; auto. + destruct iregl; simpl; auto. + destruct iregl; simpl; auto. +Qed. + +Remark loc_arguments_rec_notin_outgoing: + forall tyl iregl fregl ofs ofs0 ty0, + ofs0 + typesize ty0 <= ofs -> + Loc.notin (S (Outgoing ofs0 ty0)) (loc_arguments_rec tyl iregl fregl ofs). +Proof. + induction tyl; simpl; intros. + auto. + destruct a. + destruct iregl; simpl. + split. omega. eapply IHtyl. omega. + auto. + destruct fregl; simpl. + split. omega. eapply IHtyl. omega. + destruct iregl; simpl. + split. omega. eapply IHtyl. omega. + destruct iregl; simpl. + split; auto. eapply IHtyl. omega. + split; auto. +Qed. + +Lemma loc_arguments_norepet: + forall (s: signature), Loc.norepet (loc_arguments s). +Proof. + assert (forall tyl iregl fregl ofs, + list_norepet iregl -> + list_norepet fregl -> + list_disjoint iregl fregl -> + Loc.norepet (loc_arguments_rec tyl iregl fregl ofs)). + induction tyl; simpl; intros. + constructor. + destruct a. + destruct iregl; constructor. + apply loc_arguments_rec_notin_outgoing. simpl; omega. auto. + apply loc_arguments_rec_notin_reg. inversion H. auto. + apply list_disjoint_notin with (m :: iregl); auto with coqlib. + apply IHtyl. inv H; auto. auto. + eapply list_disjoint_cons_left; eauto. + + destruct fregl. constructor. + apply loc_arguments_rec_notin_outgoing. simpl; omega. auto. + destruct iregl. constructor. + apply loc_arguments_rec_notin_outgoing. simpl; omega. auto. + destruct iregl; constructor. + apply loc_arguments_rec_notin_reg. + red; intro. apply (H1 m m). auto with coqlib. auto with coqlib. auto. + inv H0; auto. + apply IHtyl. constructor. inv H0; auto. + red; intros. elim H2. + apply loc_arguments_rec_notin_reg. + red; intros. elim (H1 m m); auto with coqlib. + inv H0; auto. + apply IHtyl. inv H. inv H5. auto. inv H0; auto. + red; intros. apply H1; auto with coqlib. + + intro. unfold loc_arguments. apply H. + unfold int_param_regs. NoRepet. + unfold float_param_regs. NoRepet. + red; intros x y; simpl. ElimOrEq; ElimOrEq; discriminate. +Qed. + +(** The offsets of [Outgoing] arguments are below [size_arguments s]. *) + +Remark size_arguments_rec_above: + forall tyl iregl fregl ofs0, + ofs0 <= size_arguments_rec tyl iregl fregl ofs0. +Proof. + induction tyl; simpl; intros. + omega. + destruct a. + destruct iregl. apply Zle_trans with (ofs0 + 1); auto; omega. auto. + destruct fregl. apply Zle_trans with (ofs0 + 2); auto; omega. + destruct iregl. apply Zle_trans with (ofs0 + 2); auto; omega. + destruct iregl. apply Zle_trans with (ofs0 + 1); auto; omega. + auto. +Qed. + +Lemma size_arguments_above: + forall s, size_arguments s >= 0. +Proof. + intros; unfold size_arguments. apply Zle_ge. + apply size_arguments_rec_above. +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. + intros. + assert (forall tyl iregl fregl ofs0, + In (S (Outgoing ofs ty)) (loc_arguments_rec tyl iregl fregl ofs0) -> + ofs + typesize ty <= size_arguments_rec tyl iregl fregl ofs0). + induction tyl; simpl; intros. + elim H0. + destruct a. destruct iregl; elim H0; intro. + inv H1. simpl. apply size_arguments_rec_above. auto. + discriminate. auto. + destruct fregl. elim H0; intro. + inv H1. simpl. apply size_arguments_rec_above. auto. + destruct iregl. elim H0; intro. + inv H1. simpl. apply size_arguments_rec_above. auto. + destruct iregl. + elim H0; intro. inv H1. auto. + elim H0; intro. inv H1. auto. + + unfold size_arguments. eapply H0. unfold loc_arguments in H. eauto. +Qed. + +(** Temporary registers do not overlap with argument locations. *) + +Lemma loc_arguments_not_temporaries: + forall sig, Loc.disjoint (loc_arguments sig) temporaries. +Proof. + intros; red; intros x1 x2 H. + generalize (loc_arguments_rec_charact _ _ _ _ _ H). + destruct x1. + intro H0; elim H0; simpl; (ElimOrEq; ElimOrEq; congruence). + destruct s; try contradiction. intro. + simpl; ElimOrEq; auto. +Qed. +Hint Resolve loc_arguments_not_temporaries: locs. + +(** Argument registers are caller-save. *) + +Lemma arguments_caller_save: + forall sig r, + In (R r) (loc_arguments sig) -> In (R r) destroyed_at_call. +Proof. + unfold loc_arguments; intros. + elim (loc_arguments_rec_charact _ _ _ _ _ H); simpl. + ElimOrEq; intuition. + ElimOrEq; intuition. +Qed. + +(** Callee-save registers do not overlap with argument locations. *) + +Lemma arguments_not_preserved: + forall sig l, + Loc.notin l destroyed_at_call -> loc_acceptable l -> + Loc.notin l (loc_arguments sig). +Proof. + intros. unfold loc_arguments. destruct l. + apply loc_arguments_rec_notin_reg. + generalize (Loc.notin_not_in _ _ H). intro; red; intro. + apply H1. generalize H2. simpl. ElimOrEq; OrEq. + generalize (Loc.notin_not_in _ _ H). intro; red; intro. + apply H1. generalize H2. simpl. ElimOrEq; OrEq. + destruct s; simpl in H0; try contradiction. + apply loc_arguments_rec_notin_local. +Qed. +Hint Resolve arguments_not_preserved: locs. + +(** Argument locations agree in number with the function signature. *) + +Lemma loc_arguments_length: + forall sig, + List.length (loc_arguments sig) = List.length sig.(sig_args). +Proof. + assert (forall tyl iregl fregl ofs, + List.length (loc_arguments_rec tyl iregl fregl ofs) = List.length tyl). + induction tyl; simpl; intros. + auto. + destruct a. + destruct iregl; simpl; decEq; auto. + destruct fregl; simpl; decEq; auto. + destruct iregl; simpl. decEq; auto. + destruct iregl; simpl; decEq; auto. + + intros. unfold loc_arguments. auto. +Qed. + +(** Argument locations agree in types with the function signature. *) + +Lemma loc_arguments_type: + forall sig, List.map Loc.type (loc_arguments sig) = sig.(sig_args). +Proof. + assert (forall tyl iregl fregl ofs, + (forall r, In r iregl -> mreg_type r = Tint) -> + (forall r, In r fregl -> mreg_type r = Tfloat) -> + List.map Loc.type (loc_arguments_rec tyl iregl fregl ofs) = tyl). + induction tyl; simpl; intros. + auto. + destruct a. + destruct iregl; simpl; f_equal; eauto with coqlib. + destruct fregl; simpl. + f_equal; eauto with coqlib. + destruct iregl; simpl. + f_equal; eauto with coqlib. + destruct iregl; simpl; f_equal; eauto with coqlib. + apply IHtyl. simpl; tauto. auto with coqlib. + apply IHtyl. auto with coqlib. auto with coqlib. + + intros. unfold loc_arguments. apply H. + intro; simpl. ElimOrEq; reflexivity. + intro; simpl. ElimOrEq; reflexivity. +Qed. + +(** There is no partial overlap between an argument location and an + acceptable location: they are either identical or disjoint. *) + +Lemma no_overlap_arguments: + forall args sg, + locs_acceptable args -> + Loc.no_overlap args (loc_arguments sg). +Proof. + unfold Loc.no_overlap; intros. + generalize (H r H0). + generalize (loc_arguments_acceptable _ _ H1). + destruct s; destruct r; simpl. + intros. case (mreg_eq m0 m); intro. left; congruence. tauto. + intros. right; destruct s; auto. + intros. right. auto. + destruct s; try tauto. destruct s0; tauto. +Qed. + +(** Decide whether a tailcall is possible. *) + +Definition tailcall_is_possible (sg: signature) : bool := + let fix tcisp (l: list loc) := + match l with + | nil => true + | R _ :: l' => tcisp l' + | S _ :: l' => false + end + in tcisp (loc_arguments sg). + +Lemma tailcall_is_possible_correct: + forall s, tailcall_is_possible s = true -> tailcall_possible s. +Proof. + intro s. unfold tailcall_is_possible, tailcall_possible. + generalize (loc_arguments s). induction l; simpl; intros. + elim H0. + destruct a. + destruct H0. subst l0. auto. apply IHl. auto. auto. discriminate. +Qed. + +(** ** Location of function parameters *) + +(** A function finds the values of its parameter in the same locations + where its caller stored them, except that the stack-allocated arguments, + viewed as [Outgoing] slots by the caller, are accessed via [Incoming] + slots (at the same offsets and types) in the callee. *) + +Definition parameter_of_argument (l: loc) : loc := + match l with + | S (Outgoing n ty) => S (Incoming n ty) + | _ => l + end. + +Definition loc_parameters (s: signature) := + List.map parameter_of_argument (loc_arguments s). + +Lemma loc_parameters_type: + forall sig, List.map Loc.type (loc_parameters sig) = sig.(sig_args). +Proof. + intros. unfold loc_parameters. + rewrite list_map_compose. + rewrite <- loc_arguments_type. + apply list_map_exten. + intros. destruct x; simpl. auto. + destruct s; reflexivity. +Qed. + +Lemma loc_parameters_length: + forall sg, List.length (loc_parameters sg) = List.length sg.(sig_args). +Proof. + intros. unfold loc_parameters. rewrite list_length_map. + apply loc_arguments_length. +Qed. + +Lemma loc_parameters_not_temporaries: + forall sig, Loc.disjoint (loc_parameters sig) temporaries. +Proof. + intro; red; intros. + unfold loc_parameters in H. + elim (list_in_map_inv _ _ _ H). intros y [EQ IN]. + generalize (loc_arguments_not_temporaries sig y x2 IN H0). + subst x1. destruct x2. + destruct y; simpl. auto. destruct s; auto. + byContradiction. generalize H0. simpl. NotOrEq. +Qed. + +Lemma no_overlap_parameters: + forall params sg, + locs_acceptable params -> + Loc.no_overlap (loc_parameters sg) params. +Proof. + unfold Loc.no_overlap; intros. + unfold loc_parameters in H0. + elim (list_in_map_inv _ _ _ H0). intros t [EQ IN]. + rewrite EQ. + generalize (loc_arguments_acceptable _ _ IN). + generalize (H s H1). + destruct s; destruct t; simpl. + intros. case (mreg_eq m0 m); intro. left; congruence. tauto. + intros. right; destruct s; simpl; auto. + intros; right; auto. + destruct s; try tauto. destruct s0; try tauto. + intros; simpl. tauto. +Qed. + +(** ** Location of argument and result for dynamic memory allocation *) + +Definition loc_alloc_argument := R0. +Definition loc_alloc_result := R0. |