Module InterfGraph

Representation of interference graphs for register allocation.

Require Import Coqlib.
Require Import FSets.
Require Import FSetAVL.
Require Import Maps.
Require Import Ordered.
Require Import Registers.
Require Import Locations.

Interference graphs are undirected graphs with two kinds of nodes:

RTL pseudo-registers;
Machine registers.

and four kind of edges:

Conflict edges between two pseudo-registers. (Meaning: these two pseudo-registers must not be assigned the same location.)
Conflict edges between a pseudo-register and a machine register (Meaning: this pseudo-register must not be assigned this machine register.)
Preference edges between two pseudo-registers. (Meaning: the generated code would be more efficient if those two pseudo-registers were assigned the same location, but if this is not possible, the generated code will still be correct.)
Preference edges between a pseudo-register and a machine register (Meaning: the generated code would be more efficient if this pseudo-register was assigned this machine register, but if this is not possible, the generated code will still be correct.)

A graph is represented by four finite sets of edges (one of each kind above). An edge is represented by a pair of two pseudo-registers or a pair (pseudo-register, machine register). In the case of two pseudo-registers (r1, r2), we adopt the convention that r1 <= r2, so as to reflect the undirected nature of the edge.

Module OrderedReg <: OrderedType with Definition t := reg := OrderedPositive.
Module OrderedRegReg := OrderedPair(OrderedReg)(OrderedReg).
Module OrderedMreg := OrderedIndexed(IndexedMreg).
Module OrderedRegMreg := OrderedPair(OrderedReg)(OrderedMreg).

Module SetRegReg := FSetAVL.Make(OrderedRegReg).
Module SetRegMreg := FSetAVL.Make(OrderedRegMreg).

Record graph: Type := mkgraph {
  interf_reg_reg: SetRegReg.t;
  interf_reg_mreg: SetRegMreg.t;
  pref_reg_reg: SetRegReg.t;
  pref_reg_mreg: SetRegMreg.t
}.

Definition empty_graph :=
  mkgraph SetRegReg.empty SetRegMreg.empty
          SetRegReg.empty SetRegMreg.empty.

The following functions add a new edge (if not already present) to the given graph.

Definition ordered_pair (x y: reg) :=
  if plt x y then (x, y) else (y, x).

Definition add_interf (x y: reg) (g: graph) :=
  mkgraph (SetRegReg.add (ordered_pair x y) g.(interf_reg_reg))
          g.(interf_reg_mreg)
          g.(pref_reg_reg)
          g.(pref_reg_mreg).

Definition add_interf_mreg (x: reg) (y: mreg) (g: graph) :=
  mkgraph g.(interf_reg_reg)
          (SetRegMreg.add (x, y) g.(interf_reg_mreg))
          g.(pref_reg_reg)
          g.(pref_reg_mreg).

Definition add_pref (x y: reg) (g: graph) :=
  mkgraph g.(interf_reg_reg)
          g.(interf_reg_mreg)
          (SetRegReg.add (ordered_pair x y) g.(pref_reg_reg))
          g.(pref_reg_mreg).

Definition add_pref_mreg (x: reg) (y: mreg) (g: graph) :=
  mkgraph g.(interf_reg_reg)
          g.(interf_reg_mreg)
          g.(pref_reg_reg)
          (SetRegMreg.add (x, y) g.(pref_reg_mreg)).

interfere x y g holds iff there is a conflict edge in g between the two pseudo-registers x and y.

Definition interfere (x y: reg) (g: graph) : Prop :=
SetRegReg.In (ordered_pair x y) g.(interf_reg_reg).

interfere_mreg x y g holds iff there is a conflict edge in g between the pseudo-register x and the machine register y.

Definition interfere_mreg (x: reg) (y: mreg) (g: graph) : Prop :=
  SetRegMreg.In (x, y) g.(interf_reg_mreg).

Lemma ordered_pair_charact:
  forall x y,
  ordered_pair x y = (x, y) \/ ordered_pair x y = (y, x).

Proof.

unfold ordered_pair; intros.
case (plt x y); intro; tauto.
Qed.

Lemma ordered_pair_sym:
forall x y, ordered_pair y x = ordered_pair x y.

Proof.

  unfold ordered_pair; intros.
  case (plt x y); intro.
  case (plt y x); intro.
  unfold Plt in *; omegaContradiction.
  auto.
  case (plt y x); intro.
  auto.
  assert (Zpos x = Zpos y). unfold Plt in *. omega.
  congruence.
Qed.

Lemma interfere_sym:
forall x y g, interfere x y g -> interfere y x g.

Proof.

unfold interfere; intros.
rewrite ordered_pair_sym. auto.
Qed.

graph_incl g1 g2 holds if g2 contains all the conflict edges of g1 and possibly more.

Definition graph_incl (g1 g2: graph) : Prop :=
  (forall x y, interfere x y g1 -> interfere x y g2) /\
  (forall x y, interfere_mreg x y g1 -> interfere_mreg x y g2).

Lemma graph_incl_trans:
  forall g1 g2 g3, graph_incl g1 g2 -> graph_incl g2 g3 -> graph_incl g1 g3.

Proof.

  unfold graph_incl; intros.
  elim H0; elim H; intros.
  split; eauto.
Qed.

We show that the add_ functions correctly record the desired conflicts, and preserve whatever conflict edges were already present.

Lemma add_interf_correct:
forall x y g,
interfere x y (add_interf x y g).

Proof.

intros. unfold interfere, add_interf; simpl.
apply SetRegReg.add_1. auto.
Qed.

Lemma add_interf_incl:
forall a b g, graph_incl g (add_interf a b g).

Proof.

  intros. split; intros.
  unfold add_interf, interfere; simpl.
  apply SetRegReg.add_2. exact H.
  exact H.
Qed.

Lemma add_interf_mreg_correct:
forall x y g,
interfere_mreg x y (add_interf_mreg x y g).

Proof.

intros. unfold interfere_mreg, add_interf_mreg; simpl.
apply SetRegMreg.add_1. auto.
Qed.

Lemma add_interf_mreg_incl:
forall a b g, graph_incl g (add_interf_mreg a b g).

Proof.

  intros. split; intros.
  exact H.
  unfold add_interf_mreg, interfere_mreg; simpl.
  apply SetRegMreg.add_2. exact H.
Qed.

Lemma add_pref_incl:
forall a b g, graph_incl g (add_pref a b g).

Proof.

  intros. split; intros.
  exact H.
  exact H.
Qed.

Lemma add_pref_mreg_incl:
forall a b g, graph_incl g (add_pref_mreg a b g).

Proof.

  intros. split; intros.
  exact H.
  exact H.
Qed.

all_interf_regs g returns the set of pseudo-registers that are nodes of g.

Definition add_intf2 (r1r2: reg * reg) (u: Regset.t) : Regset.t :=
  Regset.add (fst r1r2) (Regset.add (snd r1r2) u).
Definition add_intf1 (r1m2: reg * mreg) (u: Regset.t) : Regset.t :=
  Regset.add (fst r1m2) u.

Definition all_interf_regs (g: graph) : Regset.t :=
  let s1 := SetRegMreg.fold add_intf1 g.(interf_reg_mreg) Regset.empty in
  let s2 := SetRegMreg.fold add_intf1 g.(pref_reg_mreg) s1 in
  let s3 := SetRegReg.fold add_intf2 g.(interf_reg_reg) s2 in
  SetRegReg.fold add_intf2 g.(pref_reg_reg) s3.

Lemma in_setregreg_fold:
  forall g r1 r2 u,
  SetRegReg.In (r1, r2) g \/ Regset.In r1 u /\ Regset.In r2 u ->
  Regset.In r1 (SetRegReg.fold add_intf2 g u) /\
  Regset.In r2 (SetRegReg.fold add_intf2 g u).

Proof.

  set (add_intf2' := fun u r1r2 => add_intf2 r1r2 u).
  assert (forall l r1 r2 u,
    InA OrderedRegReg.eq (r1,r2) l \/ Regset.In r1 u /\ Regset.In r2 u ->
    Regset.In r1 (List.fold_left add_intf2' l u) /\
    Regset.In r2 (List.fold_left add_intf2' l u)).
  induction l; intros; simpl.
  elim H; intro. inversion H0. auto.
  apply IHl. destruct a as [a1 a2].
  elim H; intro. inversion H0; subst.
  red in H2. simpl in H2. destruct H2. subst r1 r2.
  right; unfold add_intf2', add_intf2; simpl; split.
  apply Regset.add_1. auto.
  apply Regset.add_2. apply Regset.add_1. auto.
  tauto.
  right; unfold add_intf2', add_intf2; simpl; split;
  apply Regset.add_2; apply Regset.add_2; tauto.

  intros. rewrite SetRegReg.fold_1. apply H.
  intuition.
Qed.

Lemma in_setregreg_fold':
  forall g r u,
  Regset.In r u ->
  Regset.In r (SetRegReg.fold add_intf2 g u).

Proof.

intros. exploit in_setregreg_fold. eauto.
intros [A B]. eauto.
Qed.

Lemma in_setregmreg_fold:
  forall g r1 mr2 u,
  SetRegMreg.In (r1, mr2) g \/ Regset.In r1 u ->
  Regset.In r1 (SetRegMreg.fold add_intf1 g u).

Proof.

  set (add_intf1' := fun u r1mr2 => add_intf1 r1mr2 u).
  assert (forall l r1 mr2 u,
    InA OrderedRegMreg.eq (r1,mr2) l \/ Regset.In r1 u ->
    Regset.In r1 (List.fold_left add_intf1' l u)).
  induction l; intros; simpl.
  elim H; intro. inversion H0. auto.
  apply IHl with mr2. destruct a as [a1 a2].
  elim H; intro. inversion H0; subst.
  red in H2. simpl in H2. destruct H2. subst r1 mr2.
  right; unfold add_intf1', add_intf1; simpl.
  apply Regset.add_1; auto.
  tauto.
  right; unfold add_intf1', add_intf1; simpl.
  apply Regset.add_2; auto.

  intros. rewrite SetRegMreg.fold_1. apply H with mr2.
  intuition.
Qed.

Lemma all_interf_regs_correct_1:
  forall r1 r2 g,
  SetRegReg.In (r1, r2) g.(interf_reg_reg) ->
  Regset.In r1 (all_interf_regs g) /\
  Regset.In r2 (all_interf_regs g).

Proof.

  intros. unfold all_interf_regs.
  apply in_setregreg_fold. right.
  apply in_setregreg_fold. tauto.
Qed.

Lemma all_interf_regs_correct_2:
  forall r1 mr2 g,
  SetRegMreg.In (r1, mr2) g.(interf_reg_mreg) ->
  Regset.In r1 (all_interf_regs g).

Proof.

  intros. unfold all_interf_regs.
  apply in_setregreg_fold'. apply in_setregreg_fold'.
  apply in_setregmreg_fold with mr2. right.
  apply in_setregmreg_fold with mr2. eauto.
Qed.