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Require Import Coqlib.
Require Import AST.
Require Import Integers.
Require Import Floats.
Require Import Values.
Require Import Memory.
Require Import Globalenvs.
Require Import Op.
Require Import NeedDomain.
Require Import RTL.
(** Neededness analysis for IA32 operators *)
Definition op1 (nv: nval) := nv :: nil.
Definition op2 (nv: nval) := nv :: nv :: nil.
Definition needs_of_condition (cond: condition): list nval :=
match cond with
| Cmaskzero n | Cmasknotzero n => op1 (maskzero n)
| _ => nil
end.
Definition needs_of_addressing (addr: addressing) (nv: nval): list nval :=
match addr with
| Aindexed n => op1 (modarith nv)
| Aindexed2 n => op2 (modarith nv)
| Ascaled sc ofs => op1 (modarith (modarith nv))
| Aindexed2scaled sc ofs => op2 (modarith nv)
| Aglobal s ofs => nil
| Abased s ofs => op1 (modarith nv)
| Abasedscaled sc s ofs => op1 (modarith (modarith nv))
| Ainstack ofs => nil
end.
Definition needs_of_operation (op: operation) (nv: nval): list nval :=
match op with
| Omove => op1 nv
| Ointconst n => nil
| Ofloatconst n => nil
| Oindirectsymbol id => nil
| Ocast8signed => op1 (sign_ext 8 nv)
| Ocast8unsigned => op1 (zero_ext 8 nv)
| Ocast16signed => op1 (sign_ext 16 nv)
| Ocast16unsigned => op1 (zero_ext 16 nv)
| Oneg => op1 (modarith nv)
| Osub => op2 (default nv)
| Omul => op2 (modarith nv)
| Omulimm n => op1 (modarith nv)
| Omulhs | Omulhu | Odiv | Odivu | Omod | Omodu => op2 (default nv)
| Oand => op2 (bitwise nv)
| Oandimm n => op1 (andimm nv n)
| Oor => op2 (bitwise nv)
| Oorimm n => op1 (orimm nv n)
| Oxor => op2 (bitwise nv)
| Oxorimm n => op1 (bitwise nv)
| Onot => op1 (bitwise nv)
| Oshl => op2 (default nv)
| Oshlimm n => op1 (shlimm nv n)
| Oshr => op2 (default nv)
| Oshrimm n => op1 (shrimm nv n)
| Oshrximm n => op1 (default nv)
| Oshru => op2 (default nv)
| Oshruimm n => op1 (shruimm nv n)
| Ororimm n => op1 (ror nv n)
| Oshldimm n => op1 (default nv)
| Olea addr => needs_of_addressing addr nv
| Onegf | Oabsf => op1 (default nv)
| Oaddf | Osubf | Omulf | Odivf => op2 (default nv)
| Osingleoffloat => op1 (singleoffloat nv)
| Ointoffloat | Ofloatofint => op1 (default nv)
| Omakelong => op2 (default nv)
| Olowlong | Ohighlong => op1 (default nv)
| Ocmp c => needs_of_condition c
end.
Definition operation_is_redundant (op: operation) (nv: nval): bool :=
match op with
| Ocast8signed => sign_ext_redundant 8 nv
| Ocast8unsigned => zero_ext_redundant 8 nv
| Ocast16signed => sign_ext_redundant 16 nv
| Ocast16unsigned => zero_ext_redundant 16 nv
| Oandimm n => andimm_redundant nv n
| Oorimm n => orimm_redundant nv n
| Osingleoffloat => singleoffloat_redundant nv
| _ => false
end.
Ltac InvAgree :=
match goal with
| [H: vagree_list nil _ _ |- _ ] => inv H; InvAgree
| [H: vagree_list (_::_) _ _ |- _ ] => inv H; InvAgree
| _ => idtac
end.
Ltac TrivialExists :=
match goal with
| [ |- exists v, Some ?x = Some v /\ _ ] => exists x; split; auto
| _ => idtac
end.
Section SOUNDNESS.
Variable ge: genv.
Variable sp: block.
Variables m m': mem.
Hypothesis PERM: forall b ofs k p, Mem.perm m b ofs k p -> Mem.perm m' b ofs k p.
Lemma needs_of_condition_sound:
forall cond args b args',
eval_condition cond args m = Some b ->
vagree_list args args' (needs_of_condition cond) ->
eval_condition cond args' m' = Some b.
Proof.
intros. destruct cond; simpl in H;
try (eapply default_needs_of_condition_sound; eauto; fail);
simpl in *; FuncInv; InvAgree.
- eapply maskzero_sound; eauto.
- destruct (Val.maskzero_bool v i) as [b'|] eqn:MZ; try discriminate.
erewrite maskzero_sound; eauto.
Qed.
Lemma needs_of_addressing_sound:
forall (ge: genv) sp addr args v nv args',
eval_addressing ge (Vptr sp Int.zero) addr args = Some v ->
vagree_list args args' (needs_of_addressing addr nv) ->
exists v',
eval_addressing ge (Vptr sp Int.zero) addr args' = Some v'
/\ vagree v v' nv.
Proof.
unfold needs_of_addressing; intros.
destruct addr; simpl in *; FuncInv; InvAgree; TrivialExists;
auto using add_sound, mul_sound with na.
apply add_sound; auto with na. apply add_sound; rewrite modarith_idem; auto.
apply add_sound; auto. apply add_sound; rewrite modarith_idem; auto with na.
apply mul_sound; rewrite modarith_idem; auto with na.
Qed.
Lemma needs_of_operation_sound:
forall op args v nv args',
eval_operation ge (Vptr sp Int.zero) op args m = Some v ->
vagree_list args args' (needs_of_operation op nv) ->
nv <> Nothing ->
exists v',
eval_operation ge (Vptr sp Int.zero) op args' m' = Some v'
/\ vagree v v' nv.
Proof.
unfold needs_of_operation; intros; destruct op; try (eapply default_needs_of_operation_sound; eauto; fail);
simpl in *; FuncInv; InvAgree; TrivialExists.
- apply sign_ext_sound; auto. compute; auto.
- apply zero_ext_sound; auto. omega.
- apply sign_ext_sound; auto. compute; auto.
- apply zero_ext_sound; auto. omega.
- apply neg_sound; auto.
- apply mul_sound; auto.
- apply mul_sound; auto with na.
- apply and_sound; auto.
- apply andimm_sound; auto.
- apply or_sound; auto.
- apply orimm_sound; auto.
- apply xor_sound; auto.
- apply xor_sound; auto with na.
- apply notint_sound; auto.
- apply shlimm_sound; auto.
- apply shrimm_sound; auto.
- apply shruimm_sound; auto.
- apply ror_sound; auto.
- eapply needs_of_addressing_sound; eauto.
- apply singleoffloat_sound; auto.
- destruct (eval_condition c args m) as [b|] eqn:EC; simpl in H2.
erewrite needs_of_condition_sound by eauto.
subst v; simpl. auto with na.
subst v; auto with na.
Qed.
Lemma operation_is_redundant_sound:
forall op nv arg1 args v arg1' args',
operation_is_redundant op nv = true ->
eval_operation ge (Vptr sp Int.zero) op (arg1 :: args) m = Some v ->
vagree_list (arg1 :: args) (arg1' :: args') (needs_of_operation op nv) ->
vagree v arg1' nv.
Proof.
intros. destruct op; simpl in *; try discriminate; inv H1; FuncInv; subst.
- apply sign_ext_redundant_sound; auto. omega.
- apply zero_ext_redundant_sound; auto. omega.
- apply sign_ext_redundant_sound; auto. omega.
- apply zero_ext_redundant_sound; auto. omega.
- apply andimm_redundant_sound; auto.
- apply orimm_redundant_sound; auto.
- apply singleoffloat_redundant_sound; auto.
Qed.
End SOUNDNESS.
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