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(* *********************************************************************)
(* *)
(* 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. *)
(* *)
(* *********************************************************************)
(** Representation of interference graphs for register allocation. *)
Require Import Coqlib.
Require Import FSets.
Require Import FSetAVL.
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.
destruct (plt y x); destruct (plt x y).
elim (Plt_strict x). eapply Plt_trans; eauto.
auto.
auto.
destruct (Pos.lt_total x y) as [A | [A | A]].
elim n0; auto.
congruence.
elim n; auto.
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.
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