Module Parallelmove



Translation of parallel moves into sequences of individual moves. In this file, we adapt the generic "parallel move" algorithm (developed and proved correct in module Parmov) to the idiosyncraties of the LTLin and Linear intermediate languages. While the generic algorithm assumes that registers never overlap, the locations used in LTLin and Linear can overlap, and assigning one location can set the values of other, overlapping locations to Vundef. We address this issue in the remainder of this file.

Require Import Coqlib.
Require Parmov.
Require Import Values.
Require Import Events.
Require Import Ast.
Require Import Locations.
Require Import Conventions.

Instantiating the generic parallel move algorithm


The temporary location to use for a move is determined by the type of the data being moved: register IT2 for an integer datum, and register FT2 for a floating-point datum.

Definition temp_for (l: loc) : loc :=
  match Loc.type l with Tint => R IT2 | Tfloat => R FT2 end.

Definition parmove (srcs dsts: list loc) :=
  Parmov.parmove2 loc Loc.eq temp_for srcs dsts.

Definition moves := (list (loc * loc))%type.

exec_seq m gives semantics to a sequence of elementary moves. This semantics ignores the possibility of overlap: only the target locations are updated, but the locations they overlap with are not set to Vundef. See effect_seqmove below for a semantics that accounts for overlaps.

Definition exec_seq (m: moves) (e: Locmap.t) : Locmap.t :=
  Parmov.exec_seq loc Loc.eq val m e.
  
Lemma temp_for_charact:
  forall l, temp_for l = R IT2 \/ temp_for l = R FT2.
Proof.
  intro; unfold temp_for. destruct (Loc.type l); tauto.
Qed.

Lemma is_not_temp_charact:
  forall l,
  Parmov.is_not_temp loc temp_for l <-> l <> R IT2 /\ l <> R FT2.
Proof.
  intros. unfold Parmov.is_not_temp.
  destruct (Loc.eq l (R IT2)).
  subst l. intuition. apply (H (R IT2)). reflexivity. discriminate.
  destruct (Loc.eq l (R FT2)).
  subst l. intuition. apply (H (R FT2)). reflexivity.
  assert (forall d, l <> temp_for d).
    intro. elim (temp_for_charact d); congruence.
  intuition.
Qed.

Lemma disjoint_temp_not_temp:
  forall l, Loc.notin l temporaries -> Parmov.is_not_temp loc temp_for l.
Proof.
  intros. rewrite is_not_temp_charact.
  unfold temporaries in H; simpl in H.
  split; apply Loc.diff_not_eq; tauto.
Qed.

Lemma loc_norepet_norepet:
  forall l, Loc.norepet l -> NoDup l.
Proof.
  induction 1; constructor.
  apply Loc.notin_not_in; auto. auto.
Qed.

Instantiating the theorems proved in Parmov, we obtain the following properties of semantic correctness and well-typedness of the generated sequence of moves. Note that the semantic correctness result is stated in terms of the exec_seq semantics, and therefore does not account for overlap between locations.

Lemma parmove_prop_1:
  forall srcs dsts,
  List.length srcs = List.length dsts ->
  Loc.norepet dsts ->
  Loc.disjoint srcs temporaries ->
  Loc.disjoint dsts temporaries ->
  forall e,
  let e' := exec_seq (parmove srcs dsts) e in
  List.map e' dsts = List.map e srcs /\
  forall l, ~In l dsts -> l <> R IT2 -> l <> R FT2 -> e' l = e l.
Proof.
  intros.
  assert (NR: NoDup dsts) by (apply loc_norepet_norepet; auto).
  assert (NTS: forall r, In r srcs -> Parmov.is_not_temp loc temp_for r).
    intros. apply disjoint_temp_not_temp. apply Loc.disjoint_notin with srcs; auto.
  assert (NTD: forall r, In r dsts -> Parmov.is_not_temp loc temp_for r).
    intros. apply disjoint_temp_not_temp. apply Loc.disjoint_notin with dsts; auto.
  generalize (Parmov.parmove2_correctness loc Loc.eq temp_for val srcs dsts H NR NTS NTD e).
  change (Parmov.exec_seq loc Loc.eq val (Parmov.parmove2 loc Loc.eq temp_for srcs dsts) e) with e'.
  intros [A B].
  split. auto. intros. apply B. auto. rewrite is_not_temp_charact; auto.
Qed.

Lemma parmove_prop_2:
  forall srcs dsts s d,
  In (s, d) (parmove srcs dsts) ->
     (In s srcs \/ s = R IT2 \/ s = R FT2)
  /\ (In d dsts \/ d = R IT2 \/ d = R FT2).
Proof.
  intros srcs dsts.
  set (mu := List.combine srcs dsts).
  assert (forall s d, Parmov.wf_move loc temp_for mu s d ->
            (In s srcs \/ s = R IT2 \/ s = R FT2)
         /\ (In d dsts \/ d = R IT2 \/ d = R FT2)).
  unfold mu; induction 1.
  split.
    left. eapply List.in_combine_l; eauto.
    left. eapply List.in_combine_r; eauto.
  split.
    right. apply temp_for_charact.
    tauto.
  split.
    tauto.
    right. apply temp_for_charact.
  intros. apply H.
  apply (Parmov.parmove2_wf_moves loc Loc.eq temp_for srcs dsts s d H0).
Qed.

Lemma loc_type_temp_for:
  forall l, Loc.type (temp_for l) = Loc.type l.
Proof.
  intros; unfold temp_for. destruct (Loc.type l); reflexivity.
Qed.

Lemma loc_type_combine:
  forall srcs dsts,
  List.map Loc.type srcs = List.map Loc.type dsts ->
  forall s d,
  In (s, d) (List.combine srcs dsts) ->
  Loc.type s = Loc.type d.
Proof.
  induction srcs; destruct dsts; simpl; intros; try discriminate.
  elim H0.
  elim H0; intros. inversion H1; subst. congruence.
  apply IHsrcs with dsts. congruence. auto.
Qed.

Lemma parmove_prop_3:
  forall srcs dsts,
  List.map Loc.type srcs = List.map Loc.type dsts ->
  forall s d,
  In (s, d) (parmove srcs dsts) -> Loc.type s = Loc.type d.
Proof.
  intros srcs dsts TYP.
  set (mu := List.combine srcs dsts).
  assert (forall s d, Parmov.wf_move loc temp_for mu s d ->
            Loc.type s = Loc.type d).
  unfold mu; induction 1.
  eapply loc_type_combine; eauto.
  rewrite loc_type_temp_for; auto.
  rewrite loc_type_temp_for; auto.
  intros. apply H.
  apply (Parmov.parmove2_wf_moves loc Loc.eq temp_for srcs dsts s d H0).
Qed.

Accounting for overlap between locations


Section EQUIVALENCE.

We now prove the correctness of the generated sequence of elementary moves, accounting for possible overlap between locations. The proof is conducted under the following hypotheses: there must be no partial overlap between

Variables srcs dsts: list loc.
Hypothesis LENGTH: List.length srcs = List.length dsts.
Hypothesis NOREPET: Loc.norepet dsts.
Hypothesis NO_OVERLAP: Loc.no_overlap srcs dsts.
Hypothesis NO_SRCS_TEMP: Loc.disjoint srcs temporaries.
Hypothesis NO_DSTS_TEMP: Loc.disjoint dsts temporaries.

no_overlap_dests l holds if location l does not partially overlap a destination location: either it is identical to one of the destinations, or it is disjoint from all destinations.

Definition no_overlap_dests (l: loc) : Prop :=
  forall d, In d dsts -> l = d \/ Loc.diff l d.

We show that no_overlap_dests holds for any destination location and for any source location.

Lemma dests_no_overlap_dests:
  forall l, In l dsts -> no_overlap_dests l.
Proof.
  assert (forall d, Loc.norepet d ->
          forall l1 l2, In l1 d -> In l2 d -> l1 = l2 \/ Loc.diff l1 l2).
  induction 1; simpl; intros.
  contradiction.
  elim H1; intro; elim H2; intro.
  left; congruence.
  right. subst l1. eapply Loc.in_notin_diff; eauto.
  right. subst l2. apply Loc.diff_sym. eapply Loc.in_notin_diff; eauto.
  eauto.
  intros; red; intros. eauto.
Qed.

Lemma notin_dests_no_overlap_dests:
  forall l, Loc.notin l dsts -> no_overlap_dests l.
Proof.
  intros; red; intros.
  right. eapply Loc.in_notin_diff; eauto.
Qed.

Lemma source_no_overlap_dests:
  forall s, In s srcs \/ s = R IT2 \/ s = R FT2 -> no_overlap_dests s.
Proof.
  intros. elim H; intro. exact (NO_OVERLAP s H0).
  elim H0; intro; subst s; red; intros;
  right; apply Loc.diff_sym; apply NO_DSTS_TEMP; auto; simpl; tauto.
Qed.

Lemma source_not_temp1:
  forall s, In s srcs \/ s = R IT2 \/ s = R FT2 -> Loc.diff s (R IT1) /\ Loc.diff s (R FT1).
Proof.
  intros. elim H; intro.
  split; apply NO_SRCS_TEMP; auto; simpl; tauto.
  elim H0; intro; subst s; simpl; split; congruence.
Qed.

Lemma dest_noteq_diff:
  forall d l,
  In d dsts \/ d = R IT2 \/ d = R FT2 ->
  l <> d ->
  no_overlap_dests l ->
  Loc.diff l d.
Proof.
  intros. elim H; intro.
  elim (H1 d H2); intro. congruence. auto.
  assert (forall r, l <> R r -> Loc.diff l (R r)).
    intros. destruct l; simpl. congruence. destruct s; auto.
  elim H2; intro; subst d; auto.
Qed.

locmap_equiv e1 e2 holds if the location maps e1 and e2 assign the same values to all locations except temporaries IT1, FT1 and except locations that partially overlap a destination.

Definition locmap_equiv (e1 e2: Locmap.t): Prop :=
  forall l,
  no_overlap_dests l -> Loc.diff l (R IT1) -> Loc.diff l (R FT1) -> e2 l = e1 l.

The following predicates characterize the effect of one move move (effect_move) and of a sequence of elementary moves (effect_seqmove). We allow the code generated for one move to use the temporaries IT1 and FT1 in any way it needs.

Definition effect_move (src dst: loc) (e e': Locmap.t): Prop :=
  e' dst = e src /\
  forall l, Loc.diff l dst -> Loc.diff l (R IT1) -> Loc.diff l (R FT1) -> e' l = e l.

Inductive effect_seqmove: list (loc * loc) -> Locmap.t -> Locmap.t -> Prop :=
  | effect_seqmove_nil: forall e,
      effect_seqmove nil e e
  | effect_seqmove_cons: forall s d m e1 e2 e3,
      effect_move s d e1 e2 ->
      effect_seqmove m e2 e3 ->
      effect_seqmove ((s, d) :: m) e1 e3.

The following crucial lemma shows that locmap_equiv is preserved by executing one move d <- s, once using the effect_move predicate that accounts for partial overlap and the use of temporaries IT1, FT1, or via the Parmov.update function that does not account for any of these.

Lemma effect_move_equiv:
  forall s d e1 e2 e1',
  (In s srcs \/ s = R IT2 \/ s = R FT2) ->
  (In d dsts \/ d = R IT2 \/ d = R FT2) ->
  locmap_equiv e1 e2 -> effect_move s d e1 e1' ->
  locmap_equiv e1' (Parmov.update loc Loc.eq val d (e2 s) e2).
Proof.
  intros. destruct H2. red; intros.
  unfold Parmov.update. destruct (Loc.eq l d).
  subst l. elim (source_not_temp1 _ H); intros.
  rewrite H2. apply H1; auto. apply source_no_overlap_dests; auto.
  rewrite H3; auto. apply dest_noteq_diff; auto.
Qed.

We then extend the previous lemma to a sequence mu of elementary moves.

Lemma effect_seqmove_equiv:
  forall mu e1 e1',
  effect_seqmove mu e1 e1' ->
  forall e2,
  (forall s d, In (s, d) mu ->
     (In s srcs \/ s = R IT2 \/ s = R FT2) /\
     (In d dsts \/ d = R IT2 \/ d = R FT2)) ->
  locmap_equiv e1 e2 ->
  locmap_equiv e1' (exec_seq mu e2).
Proof.
  induction 1; intros.
  simpl. auto.
  simpl. apply IHeffect_seqmove.
  intros. apply H1. apply in_cons; auto.
  destruct (H1 s d (in_eq _ _)).
  eapply effect_move_equiv; eauto.
Qed.

Here is the main result in this file: executing the sequence of moves returned by the parmove function results in the desired state for locations: the final values of destination locations are the initial values of source locations, and all locations that are disjoint from the temporaries and the destinations keep their initial values.

Lemma effect_parmove:
  forall e e',
  effect_seqmove (parmove srcs dsts) e e' ->
  List.map e' dsts = List.map e srcs /\
  forall l, Loc.notin l dsts -> Loc.notin l temporaries -> e' l = e l.
Proof.
  set (mu := parmove srcs dsts). intros.
  assert (locmap_equiv e e) by (red; auto).
  generalize (effect_seqmove_equiv mu e e' H e (parmove_prop_2 srcs dsts) H0).
  intro.
  generalize (parmove_prop_1 srcs dsts LENGTH NOREPET NO_SRCS_TEMP NO_DSTS_TEMP e).
  fold mu. intros [A B].
  split. rewrite <- A. apply list_map_exten; intros.
  apply H1. apply dests_no_overlap_dests; auto.
  apply NO_DSTS_TEMP; auto; simpl; tauto.
  apply NO_DSTS_TEMP; auto; simpl; tauto.
  intros. transitivity (exec_seq mu e l).
  symmetry. apply H1. apply notin_dests_no_overlap_dests; auto.
  eapply Loc.in_notin_diff; eauto. simpl; tauto.
  eapply Loc.in_notin_diff; eauto. simpl; tauto.
  apply B. apply Loc.notin_not_in; auto.
  apply Loc.diff_not_eq. eapply Loc.in_notin_diff; eauto. simpl; tauto.
  apply Loc.diff_not_eq. eapply Loc.in_notin_diff; eauto. simpl; tauto.
Qed.

End EQUIVALENCE.

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