Locations are a refinement of RTL pseudo-registers, used to reflect the results of register allocation (file Allocation).

Require Import Coqlib.
Require Import Maps.
Require Import AST.
Require Import Values.
Require Export Machregs.

Representation of locations

A location is either a processor register or (an abstract designation of) a slot in the activation record of the current function.

Processor registers

Processor registers usable for register allocation are defined in module Machregs.

Slots in activation records

A slot in an activation record is designated abstractly by a kind, a type and an integer offset. Three kinds are considered:

Local: these are the slots used by register allocation for pseudo-registers that cannot be assigned a hardware register.
Incoming: used to store the parameters of the current function that cannot reside in hardware registers, as determined by the calling conventions.
Outgoing: used to store arguments to called functions that cannot reside in hardware registers, as determined by the calling conventions.

Inductive slot: Type :=
  | Local: Z -> typ -> slot
  | Incoming: Z -> typ -> slot
  | Outgoing: Z -> typ -> slot.

Morally, the Incoming slots of a function are the Outgoing slots of its caller function. The type of a slot indicates how it will be accessed later once mapped to actual memory locations inside a memory-allocated activation record: as 32-bit integers/pointers (type Tint) or as 64-bit floats (type Tfloat). The offset of a slot, combined with its type and its kind, identifies uniquely the slot and will determine later where it resides within the memory-allocated activation record. Offsets are always positive. Conceptually, slots will be mapped to four non-overlapping memory areas within activation records:

The area for Local slots of type Tint. The offset is interpreted as a 4-byte word index.
The area for Local slots of type Tfloat. The offset is interpreted as a 8-byte word index. Thus, two Local slots always refer either to the same memory chunk (if they have the same types and offsets) or to non-overlapping memory chunks (if the types or offsets differ).
The area for Outgoing slots. The offset is a 4-byte word index. Unlike Local slots, the PowerPC calling conventions demand that integer and float Outgoing slots reside in the same memory area. Therefore, Outgoing Tint 0 and Outgoing Tfloat 0 refer to overlapping memory chunks and cannot be used simultaneously: one will lose its value when the other is assigned. We will reflect this overlapping behaviour in the environments mapping locations to values defined later in this file.
The area for Incoming slots. Same structure as the Outgoing slots.

Definition slot_type (s: slot): typ :=
  match s with
  | Local ofs ty => ty
  | Incoming ofs ty => ty
  | Outgoing ofs ty => ty
  end.

Lemma slot_eq: forall (p q: slot), {p = q} + {p <> q}.

Proof.

  assert (typ_eq: forall (t1 t2: typ), {t1 = t2} + {t1 <> t2}).
  decide equality.
  generalize zeq; intro.
  decide equality.
Qed.

Open Scope Z_scope.

Definition typesize (ty: typ) : Z :=
match ty with Tint => 1 | Tfloat => 2 end.

Lemma typesize_pos:
forall (ty: typ), typesize ty > 0.

Proof.

destruct ty; compute; auto.
Qed.

Locations

Locations are just the disjoint union of machine registers and activation record slots.

Inductive loc : Type :=
  | R: mreg -> loc
  | S: slot -> loc.

Module Loc.

  Definition type (l: loc) : typ :=
    match l with
    | R r => mreg_type r
    | S s => slot_type s
    end.

  Lemma eq: forall (p q: loc), {p = q} + {p <> q}.

Proof.

decide equality. apply mreg_eq. apply slot_eq.
Qed.

As mentioned previously, two locations can be different (in the sense of the <> mathematical disequality), yet denote overlapping memory chunks within the activation record. Given two locations, three cases are possible:

They are equal (in the sense of the = equality)
They are different and non-overlapping.
They are different but overlapping.

The second case (different and non-overlapping) is characterized by the following Loc.diff predicate.

  Definition diff (l1 l2: loc) : Prop :=
    match l1, l2 with
    | R r1, R r2 => r1 <> r2
    | S (Local d1 t1), S (Local d2 t2) =>
        d1 <> d2 \/ t1 <> t2
    | S (Incoming d1 t1), S (Incoming d2 t2) =>
        d1 + typesize t1 <= d2 \/ d2 + typesize t2 <= d1
    | S (Outgoing d1 t1), S (Outgoing d2 t2) =>
        d1 + typesize t1 <= d2 \/ d2 + typesize t2 <= d1
    | _, _ =>
        True
    end.

  Lemma same_not_diff:
    forall l, ~(diff l l).

Proof.

    destruct l; unfold diff; try tauto.
    destruct s.
    tauto.
    generalize (typesize_pos t); omega.
    generalize (typesize_pos t); omega.
  Qed.

Lemma diff_not_eq:
forall l1 l2, diff l1 l2 -> l1 <> l2.

Proof.

unfold not; intros. subst l2. elim (same_not_diff l1 H).
Qed.

Lemma diff_sym:
forall l1 l2, diff l1 l2 -> diff l2 l1.

Proof.

    destruct l1; destruct l2; unfold diff; auto.
    destruct s; auto.
    destruct s; destruct s0; intuition auto.
  Qed.

Lemma diff_reg_right:
forall l r, l <> R r -> diff (R r) l.

Proof.

intros. simpl. destruct l. congruence. auto.
Qed.

Lemma diff_reg_left:
forall l r, l <> R r -> diff l (R r).

Proof.

intros. apply diff_sym. apply diff_reg_right. auto.
Qed.

Loc.overlap l1 l2 returns false if l1 and l2 are different and non-overlapping, and true otherwise: either l1 = l2 or they partially overlap.

  Definition overlap_aux (t1: typ) (d1 d2: Z) : bool :=
    if zeq d1 d2 then true else
    match t1 with
    | Tint => false
    | Tfloat => if zeq (d1 + 1) d2 then true else false
    end.

  Definition overlap (l1 l2: loc) : bool :=
    match l1, l2 with
    | S (Incoming d1 t1), S (Incoming d2 t2) =>
        overlap_aux t1 d1 d2 || overlap_aux t2 d2 d1
    | S (Outgoing d1 t1), S (Outgoing d2 t2) =>
        overlap_aux t1 d1 d2 || overlap_aux t2 d2 d1
    | _, _ => false
    end.

  Lemma overlap_aux_true_1:
    forall d1 t1 d2 t2,
    overlap_aux t1 d1 d2 = true ->
    ~(d1 + typesize t1 <= d2 \/ d2 + typesize t2 <= d1).

Proof.

    intros until t2.
    generalize (typesize_pos t1); intro.
    generalize (typesize_pos t2); intro.
    unfold overlap_aux.
    case (zeq d1 d2).
    intros. omega.
    case t1. intros; discriminate.
    case (zeq (d1 + 1) d2); intros.
    subst d2. simpl. omega.
    discriminate.
  Qed.

  Lemma overlap_aux_true_2:
    forall d1 t1 d2 t2,
    overlap_aux t2 d2 d1 = true ->
    ~(d1 + typesize t1 <= d2 \/ d2 + typesize t2 <= d1).

Proof.

    intros. generalize (overlap_aux_true_1 d2 t2 d1 t1 H).
    tauto.
  Qed.

Lemma overlap_not_diff:
forall l1 l2, overlap l1 l2 = true -> ~(diff l1 l2).

Proof.

    unfold overlap, diff; destruct l1; destruct l2; intros; try discriminate.
    destruct s; discriminate.
    destruct s; destruct s0; try discriminate.
    elim (orb_true_elim _ _ H); intro.
    apply overlap_aux_true_1; auto.
    apply overlap_aux_true_2; auto.
    elim (orb_true_elim _ _ H); intro.
    apply overlap_aux_true_1; auto.
    apply overlap_aux_true_2; auto.
  Qed.

  Lemma overlap_aux_false_1:
    forall t1 d1 t2 d2,
    overlap_aux t1 d1 d2 || overlap_aux t2 d2 d1 = false ->
    d1 + typesize t1 <= d2 \/ d2 + typesize t2 <= d1.

Proof.

    intros until d2. intro OV.
    generalize (orb_false_elim _ _ OV). intro OV'. elim OV'.
    unfold overlap_aux.
    case (zeq d1 d2); intro.
    intros; discriminate.
    case (zeq d2 d1); intro.
    intros; discriminate.
    case t1; case t2; simpl.
    intros; omega.
    case (zeq (d2 + 1) d1); intros. discriminate. omega.
    case (zeq (d1 + 1) d2); intros. discriminate. omega.
    case (zeq (d1 + 1) d2); intros H1 H2. discriminate.
    case (zeq (d2 + 1) d1); intros. discriminate. omega.
  Qed.

Lemma non_overlap_diff:
forall l1 l2, l1 <> l2 -> overlap l1 l2 = false -> diff l1 l2.

Proof.

    intros. unfold diff; destruct l1; destruct l2.
    congruence.
    auto.
    destruct s; auto.
    destruct s; destruct s0; auto.
    case (zeq z z0); intro.
    compare t t0; intro.
    congruence. tauto. tauto.
    apply overlap_aux_false_1. exact H0.
    apply overlap_aux_false_1. exact H0.
  Qed.

Definition diff_dec (l1 l2: loc) : { Loc.diff l1 l2 } + {~Loc.diff l1 l2}.

Proof.

    intros. case (eq l1 l2); intros.
    right. rewrite e. apply same_not_diff.
    case_eq (overlap l1 l2); intros.
    right. apply overlap_not_diff; auto.
    left. apply non_overlap_diff; auto.
  Qed.

We now redefine some standard notions over lists, using the Loc.diff predicate instead of standard disequality <>. Loc.notin l ll holds if the location l is different from all locations in the list ll.

  Fixpoint notin (l: loc) (ll: list loc) {struct ll} : Prop :=
    match ll with
    | nil => True
    | l1 :: ls => diff l l1 /\ notin l ls
    end.

  Lemma notin_iff:
    forall l ll, notin l ll <-> (forall l', In l' ll -> Loc.diff l l').

Proof.

    induction ll; simpl.
    tauto.
    rewrite IHll. intuition. subst a. auto.
  Qed.

Lemma notin_not_in:
forall l ll, notin l ll -> ~(In l ll).

Proof.

    intros; red; intros. rewrite notin_iff in H.
    elim (diff_not_eq l l); auto.
  Qed.

Lemma reg_notin:
forall r ll, ~(In (R r) ll) -> notin (R r) ll.

Proof.

    intros. rewrite notin_iff; intros.
    destruct l'; simpl. congruence. auto.
  Qed.

Loc.disjoint l1 l2 is true if the locations in list l1 are different from all locations in list l2.

  Definition disjoint (l1 l2: list loc) : Prop :=
    forall x1 x2, In x1 l1 -> In x2 l2 -> diff x1 x2.

  Lemma disjoint_cons_left:
    forall a l1 l2,
    disjoint (a :: l1) l2 -> disjoint l1 l2.

Proof.

unfold disjoint; intros. auto with coqlib.
Qed.

  Lemma disjoint_cons_right:
    forall a l1 l2,
    disjoint l1 (a :: l2) -> disjoint l1 l2.

Proof.

unfold disjoint; intros. auto with coqlib.
Qed.

Lemma disjoint_sym:
forall l1 l2, disjoint l1 l2 -> disjoint l2 l1.

Proof.

unfold disjoint; intros. apply diff_sym; auto.
Qed.

Lemma in_notin_diff:
forall l1 l2 ll, notin l1 ll -> In l2 ll -> diff l1 l2.

Proof.

intros. rewrite notin_iff in H. auto.
Qed.

  Lemma notin_disjoint:
    forall l1 l2,
    (forall x, In x l1 -> notin x l2) -> disjoint l1 l2.

Proof.

intros; red; intros. exploit H; eauto. rewrite notin_iff; intros. auto.
Qed.

Lemma disjoint_notin:
forall l1 l2 x, disjoint l1 l2 -> In x l1 -> notin x l2.

Proof.

intros; rewrite notin_iff; intros. red in H. auto.
Qed.

Loc.norepet ll holds if the locations in list ll are pairwise different.

  Inductive norepet : list loc -> Prop :=
  | norepet_nil:
      norepet nil
  | norepet_cons:
      forall hd tl, notin hd tl -> norepet tl -> norepet (hd :: tl).

Loc.no_overlap l1 l2 holds if elements of l1 never overlap partially with elements of l2.

Definition no_overlap (l1 l2 : list loc) :=
forall r, In r l1 -> forall s, In s l2 -> r = s \/ Loc.diff r s.

End Loc.

Mappings from locations to values

The Locmap module defines mappings from locations to values, used as evaluation environments for the semantics of the LTL and LTLin intermediate languages.

Set Implicit Arguments.

Module Locmap.

  Definition t := loc -> val.

  Definition init (x: val) : t := fun (_: loc) => x.

  Definition get (l: loc) (m: t) : val := m l.

The set operation over location mappings reflects the overlapping properties of locations: changing the value of a location l invalidates (sets to Vundef) the locations that partially overlap with l. In other terms, the result of set l v m maps location l to value v, locations that overlap with l to Vundef, and locations that are different (and non-overlapping) from l to their previous values in m. This is apparent in the ``good variables'' properties Locmap.gss and Locmap.gso.

  Definition set (l: loc) (v: val) (m: t) : t :=
    fun (p: loc) =>
      if Loc.eq l p then v else if Loc.overlap l p then Vundef else m p.

  Lemma gss: forall l v m, (set l v m) l = v.

Proof.

intros. unfold set. case (Loc.eq l l); tauto.
Qed.

Lemma gso: forall l v m p, Loc.diff l p -> (set l v m) p = m p.

Proof.

    intros. unfold set. case (Loc.eq l p); intro.
    subst p. elim (Loc.same_not_diff _ H).
    caseEq (Loc.overlap l p); intro.
    elim (Loc.overlap_not_diff _ _ H0 H).
    auto.
  Qed.

  Fixpoint undef (ll: list loc) (m: t) {struct ll} : t :=
    match ll with
    | nil => m
    | l1 :: ll' => undef ll' (set l1 Vundef m)
    end.

  Lemma guo: forall ll l m, Loc.notin l ll -> (undef ll m) l = m l.

Proof.

    induction ll; simpl; intros. auto.
    destruct H. rewrite IHll; auto. apply gso. apply Loc.diff_sym; auto.
  Qed.

Lemma gus: forall ll l m, In l ll -> (undef ll m) l = Vundef.

Proof.

    assert (P: forall ll l m, m l = Vundef -> (undef ll m) l = Vundef).
      induction ll; simpl; intros. auto. apply IHll.
      unfold set. destruct (Loc.eq a l); auto.
      destruct (Loc.overlap a l); auto.
    induction ll; simpl; intros. contradiction.
    destruct H. apply P. subst a. apply gss.
    auto.
  Qed.

End Locmap.

Module Locations

Representation of locations

Processor registers

Slots in activation records

Locations

Mappings from locations to values