diff options
Diffstat (limited to 'mppa_k1c/abstractbb/Parallelizability.v')
| -rw-r--r-- | mppa_k1c/abstractbb/Parallelizability.v | 178 |
1 files changed, 92 insertions, 86 deletions
diff --git a/mppa_k1c/abstractbb/Parallelizability.v b/mppa_k1c/abstractbb/Parallelizability.v index 065c0922..d1971e57 100644 --- a/mppa_k1c/abstractbb/Parallelizability.v +++ b/mppa_k1c/abstractbb/Parallelizability.v @@ -21,20 +21,20 @@ Local Open Scope list. Section PARALLEL. Variable ge: genv. -(* parallel run of a macro *) -Fixpoint macro_prun (i: macro) (m tmp old: mem): option mem := +(* parallel run of a inst *) +Fixpoint inst_prun (i: inst) (m tmp old: mem): option mem := match i with | nil => Some m | (x, e)::i' => match exp_eval ge e tmp old with - | Some v' => macro_prun i' (assign m x v') (assign tmp x v') old + | Some v' => inst_prun i' (assign m x v') (assign tmp x v') old | None => None end end. -(* [macro_prun] is generalization of [macro_run] *) -Lemma macro_run_prun i: forall m old, - macro_run ge i m old = macro_prun i m m old. +(* [inst_prun] is generalization of [inst_run] *) +Lemma inst_run_prun i: forall m old, + inst_run ge i m old = inst_prun i m m old. Proof. induction i as [|[y e] i']; simpl; auto. intros m old; destruct (exp_eval ge e m old); simpl; auto. @@ -46,7 +46,7 @@ Fixpoint prun_iw (p: bblock) m old: option mem := match p with | nil => Some m | i::p' => - match macro_prun i m old old with + match inst_prun i m old old with | Some m1 => prun_iw p' m1 old | None => None end @@ -58,9 +58,9 @@ Definition prun (p: bblock) m (om: option mem) := exists p', res_eq om (prun_iw (* a few lemma on equality *) -Lemma macro_prun_equiv i old: forall m1 m2 tmp, +Lemma inst_prun_equiv i old: forall m1 m2 tmp, (forall x, m1 x = m2 x) -> - res_eq (macro_prun i m1 tmp old) (macro_prun i m2 tmp old). + res_eq (inst_prun i m1 tmp old) (inst_prun i m2 tmp old). Proof. induction i as [|[x e] i']; simpl; eauto. intros m1 m2 tmp H; destruct (exp_eval ge e tmp old); simpl; auto. @@ -73,37 +73,43 @@ Lemma prun_iw_equiv p: forall m1 m2 old, Proof. induction p as [|i p']; simpl; eauto. - intros m1 m2 old H. - generalize (macro_prun_equiv i old m1 m2 old H); - destruct (macro_prun i m1 old old); simpl. + generalize (inst_prun_equiv i old m1 m2 old H); + destruct (inst_prun i m1 old old); simpl. + intros (m3 & H3 & H4); rewrite H3; simpl; eauto. + intros H1; rewrite H1; simpl; auto. Qed. + +Lemma prun_iw_app p1: forall m1 old p2, + prun_iw (p1++p2) m1 old = + match prun_iw p1 m1 old with + | Some m2 => prun_iw p2 m2 old + | None => None + end. +Proof. + induction p1; simpl; try congruence. + intros; destruct (inst_prun _ _ _); simpl; auto. +Qed. + Lemma prun_iw_app_None p1: forall m1 old p2, prun_iw p1 m1 old = None -> prun_iw (p1++p2) m1 old = None. Proof. - induction p1; simpl; try congruence. - intros; destruct (macro_prun _ _ _); simpl; auto. + intros m1 old p2 H; rewrite prun_iw_app. rewrite H; auto. Qed. Lemma prun_iw_app_Some p1: forall m1 old m2 p2, prun_iw p1 m1 old = Some m2 -> prun_iw (p1++p2) m1 old = prun_iw p2 m2 old. Proof. - induction p1; simpl; try congruence. - intros; destruct (macro_prun _ _ _); simpl; auto. - congruence. + intros m1 old m2 p2 H; rewrite prun_iw_app. rewrite H; auto. Qed. - End PARALLEL. End ParallelSemantics. - - Fixpoint notIn {A} (x: A) (l:list A): Prop := match l with | nil => True @@ -266,15 +272,15 @@ Qed. (** * Writing frames *) -Fixpoint macro_wframe(i:macro): list R.t := +Fixpoint inst_wframe(i:inst): list R.t := match i with | nil => nil - | a::i' => (fst a)::(macro_wframe i') + | a::i' => (fst a)::(inst_wframe i') end. -Lemma macro_wframe_correct i m' old: forall m tmp, - macro_prun ge i m tmp old = Some m' -> - forall x, notIn x (macro_wframe i) -> m' x = m x. +Lemma inst_wframe_correct i m' old: forall m tmp, + inst_prun ge i m tmp old = Some m' -> + forall x, notIn x (inst_wframe i) -> m' x = m x. Proof. induction i as [|[y e] i']; simpl. - intros m tmp H x H0; inversion_clear H; auto. @@ -283,47 +289,47 @@ Proof. rewrite assign_diff; auto. Qed. -Lemma macro_prun_fequiv i old: forall m1 m2 tmp, - frame_eq (fun x => In x (macro_wframe i)) (macro_prun ge i m1 tmp old) (macro_prun ge i m2 tmp old). +Lemma inst_prun_fequiv i old: forall m1 m2 tmp, + frame_eq (fun x => In x (inst_wframe i)) (inst_prun ge i m1 tmp old) (inst_prun ge i m2 tmp old). Proof. induction i as [|[y e] i']; simpl. - intros m1 m2 tmp; eexists; intuition eauto. - intros m1 m2 tmp. destruct (exp_eval ge e tmp old); simpl; auto. eapply frame_eq_list_split; eauto. clear IHi'. intros m1' m2' x H1 H2. - lapply (macro_wframe_correct i' m1' old (assign m1 y v) (assign tmp y v)); eauto. - lapply (macro_wframe_correct i' m2' old (assign m2 y v) (assign tmp y v)); eauto. + lapply (inst_wframe_correct i' m1' old (assign m1 y v) (assign tmp y v)); eauto. + lapply (inst_wframe_correct i' m2' old (assign m2 y v) (assign tmp y v)); eauto. intros Xm2 Xm1 H H0. destruct H. + subst. rewrite Xm1, Xm2; auto. rewrite !assign_eq. auto. + rewrite <- notIn_iff in H0; tauto. Qed. -Lemma macro_prun_None i m1 m2 tmp old: - macro_prun ge i m1 tmp old = None -> - macro_prun ge i m2 tmp old = None. +Lemma inst_prun_None i m1 m2 tmp old: + inst_prun ge i m1 tmp old = None -> + inst_prun ge i m2 tmp old = None. Proof. - intros H; generalize (macro_prun_fequiv i old m1 m2 tmp). + intros H; generalize (inst_prun_fequiv i old m1 m2 tmp). rewrite H; simpl; auto. Qed. -Lemma macro_prun_Some i m1 m2 tmp old m1': - macro_prun ge i m1 tmp old = Some m1' -> - res_eq (Some (frame_assign m2 (macro_wframe i) m1')) (macro_prun ge i m2 tmp old). +Lemma inst_prun_Some i m1 m2 tmp old m1': + inst_prun ge i m1 tmp old = Some m1' -> + res_eq (Some (frame_assign m2 (inst_wframe i) m1')) (inst_prun ge i m2 tmp old). Proof. - intros H; generalize (macro_prun_fequiv i old m1 m2 tmp). + intros H; generalize (inst_prun_fequiv i old m1 m2 tmp). rewrite H; simpl. intros (m2' & H1 & H2). eexists; intuition eauto. rewrite frame_assign_def. - lapply (macro_wframe_correct i m2' old m2 tmp); eauto. - destruct (notIn_dec x (macro_wframe i)); auto. + lapply (inst_wframe_correct i m2' old m2 tmp); eauto. + destruct (notIn_dec x (inst_wframe i)); auto. intros X; rewrite X; auto. Qed. Fixpoint bblock_wframe(p:bblock): list R.t := match p with | nil => nil - | i::p' => (macro_wframe i)++(bblock_wframe p') + | i::p' => (inst_wframe i)++(bblock_wframe p') end. Local Hint Resolve Permutation_app_head Permutation_app_tail Permutation_app_comm. @@ -344,11 +350,11 @@ Proof. induction p as [|i p']; simpl. - intros m H; inversion_clear H; auto. - intros m H x; rewrite notIn_app; intros (H1 & H2). - remember (macro_prun i m old old) as om. + remember (inst_prun i m old old) as om. destruct om as [m1|]; simpl. + eapply eq_trans. eapply IHp'; eauto. - eapply macro_wframe_correct; eauto. + eapply inst_wframe_correct; eauto. + inversion H. Qed. @@ -357,13 +363,13 @@ Lemma prun_iw_fequiv p old: forall m1 m2, Proof. induction p as [|i p']; simpl. - intros m1 m2; eexists; intuition eauto. - - intros m1 m2; generalize (macro_prun_fequiv i old m1 m2 old). - remember (macro_prun i m1 old old) as om. + - intros m1 m2; generalize (inst_prun_fequiv i old m1 m2 old). + remember (inst_prun i m1 old old) as om. destruct om as [m1'|]; simpl. + intros (m2' & H1 & H2). rewrite H1; simpl. eapply frame_eq_list_split; eauto. clear IHp'. intros m1'' m2'' x H3 H4. rewrite in_app_iff. - intros X X2. assert (X1: In x (macro_wframe i)). { destruct X; auto. rewrite <- notIn_iff in X2; tauto. } + intros X X2. assert (X1: In x (inst_wframe i)). { destruct X; auto. rewrite <- notIn_iff in X2; tauto. } clear X. lapply (bblock_wframe_correct p' m1'' old m1'); eauto. lapply (bblock_wframe_correct p' m2'' old m2'); eauto. @@ -391,7 +397,7 @@ Fixpoint is_det (p: bblock): Prop := match p with | nil => True | i::p' => - disjoint (macro_wframe i) (bblock_wframe p') (* no WRITE-AFTER-WRITE *) + disjoint (inst_wframe i) (bblock_wframe p') (* no WRITE-AFTER-WRITE *) /\ is_det p' end. @@ -413,32 +419,32 @@ Theorem is_det_correct p p': Proof. induction 1 as [ | i p p' | i1 i2 p | p1 p2 p3 ]; simpl; eauto. - intros [H0 H1] m old. - remember (macro_prun ge i m old old) as om0. + remember (inst_prun ge i m old old) as om0. destruct om0 as [ m0 | ]; simpl; auto. - rewrite disjoint_app_r. intros ([Z1 Z2] & Z3 & Z4) m old. - remember (macro_prun ge i2 m old old) as om2. + remember (inst_prun ge i2 m old old) as om2. destruct om2 as [ m2 | ]; simpl; auto. - + remember (macro_prun ge i1 m old old) as om1. + + remember (inst_prun ge i1 m old old) as om1. destruct om1 as [ m1 | ]; simpl; auto. - * lapply (macro_prun_Some i2 m m1 old old m2); simpl; auto. - lapply (macro_prun_Some i1 m m2 old old m1); simpl; auto. + * lapply (inst_prun_Some i2 m m1 old old m2); simpl; auto. + lapply (inst_prun_Some i1 m m2 old old m1); simpl; auto. intros (m1' & Hm1' & Xm1') (m2' & Hm2' & Xm2'). rewrite Hm1', Hm2'; simpl. eapply prun_iw_equiv. intros x; rewrite <- Xm1', <- Xm2'. clear Xm2' Xm1' Hm1' Hm2' m1' m2'. rewrite frame_assign_def. rewrite disjoint_sym in Z1; unfold disjoint in Z1. - destruct (notIn_dec x (macro_wframe i1)) as [ X1 | X1 ]. - { rewrite frame_assign_def; destruct (notIn_dec x (macro_wframe i2)) as [ X2 | X2 ]; auto. - erewrite (macro_wframe_correct i2 m2 old m old); eauto. - erewrite (macro_wframe_correct i1 m1 old m old); eauto. + destruct (notIn_dec x (inst_wframe i1)) as [ X1 | X1 ]. + { rewrite frame_assign_def; destruct (notIn_dec x (inst_wframe i2)) as [ X2 | X2 ]; auto. + erewrite (inst_wframe_correct i2 m2 old m old); eauto. + erewrite (inst_wframe_correct i1 m1 old m old); eauto. } rewrite frame_assign_notIn; auto. - * erewrite macro_prun_None; eauto. simpl; auto. - + remember (macro_prun ge i1 m old old) as om1. + * erewrite inst_prun_None; eauto. simpl; auto. + + remember (inst_prun ge i1 m old old) as om1. destruct om1 as [ m1 | ]; simpl; auto. - erewrite macro_prun_None; eauto. + erewrite inst_prun_None; eauto. - intros; eapply res_eq_trans. eapply IHPermutation1; eauto. eapply IHPermutation2; eauto. @@ -449,7 +455,7 @@ Qed. Fixpoint exp_frame (e: exp): list R.t := match e with - | Name x => x::nil + | PReg x => x::nil | Op o le => list_exp_frame le | Old e => exp_frame e end @@ -473,23 +479,23 @@ Proof. intros; (eapply H1 || eapply H2); rewrite in_app_iff; auto. Qed. -Fixpoint macro_frame (i: macro): list R.t := +Fixpoint inst_frame (i: inst): list R.t := match i with | nil => nil - | a::i' => (fst a)::(exp_frame (snd a) ++ macro_frame i') + | a::i' => (fst a)::(exp_frame (snd a) ++ inst_frame i') end. -Lemma macro_wframe_frame i x: In x (macro_wframe i) -> In x (macro_frame i). +Lemma inst_wframe_frame i x: In x (inst_wframe i) -> In x (inst_frame i). Proof. induction i as [ | [y e] i']; simpl; intuition. Qed. -Lemma macro_frame_correct i wframe old1 old2: forall m tmp1 tmp2, - (disjoint (macro_frame i) wframe) -> +Lemma inst_frame_correct i wframe old1 old2: forall m tmp1 tmp2, + (disjoint (inst_frame i) wframe) -> (forall x, notIn x wframe -> old1 x = old2 x) -> (forall x, notIn x wframe -> tmp1 x = tmp2 x) -> - macro_prun ge i m tmp1 old1 = macro_prun ge i m tmp2 old2. + inst_prun ge i m tmp1 old1 = inst_prun ge i m tmp2 old2. Proof. induction i as [|[x e] i']; simpl; auto. intros m tmp1 tmp2; rewrite disjoint_cons_l, disjoint_app_l. @@ -509,8 +515,8 @@ Fixpoint pararec (p: bblock) (wframe: list R.t): Prop := match p with | nil => True | i::p' => - disjoint (macro_frame i) wframe (* no USE-AFTER-WRITE *) - /\ pararec p' ((macro_wframe i) ++ wframe) + disjoint (inst_frame i) wframe (* no USE-AFTER-WRITE *) + /\ pararec p' ((inst_wframe i) ++ wframe) end. Lemma pararec_disjoint (p: bblock): forall wframe, pararec p wframe -> disjoint (bblock_wframe p) wframe. @@ -521,7 +527,7 @@ Proof. generalize (IHp' _ H1). rewrite disjoint_app_r. intuition. eapply disjoint_incl_l. 2: eapply H0. - unfold incl. eapply macro_wframe_frame; eauto. + unfold incl. eapply inst_wframe_frame; eauto. Qed. Lemma pararec_det p: forall wframe, pararec p wframe -> is_det p. @@ -540,13 +546,13 @@ Lemma pararec_correct p old: forall wframe m, Proof. elim p; clear p; simpl; auto. intros i p' X wframe m [H H0] H1. - erewrite macro_run_prun, macro_frame_correct; eauto. - remember (macro_prun ge i m old old) as om0. + erewrite inst_run_prun, inst_frame_correct; eauto. + remember (inst_prun ge i m old old) as om0. destruct om0 as [m0 | ]; try congruence. eapply X; eauto. intro x; rewrite notIn_app. intros [H3 H4]. rewrite <- H1; auto. - eapply macro_wframe_correct; eauto. + eapply inst_wframe_correct; eauto. Qed. Definition parallelizable (p: bblock) := pararec p nil. @@ -570,9 +576,9 @@ End PARALLELI. End ParallelizablityChecking. -Module Type ResourceSet. +Module Type PseudoRegSet. -Declare Module R: ResourceNames. +Declare Module R: PseudoRegisters. (** We assume a datatype [t] refining (list R.t) @@ -596,7 +602,7 @@ Parameter union_match_frame: forall s1 s2 l1 l2, match_frame s1 l1 -> match_fram Parameter is_disjoint: t -> t -> bool. Parameter is_disjoint_match_frame: forall s1 s2 l1 l2, match_frame s1 l1 -> match_frame s2 l2 -> (is_disjoint s1 s2)=true -> disjoint l1 l2. -End ResourceSet. +End PseudoRegSet. Lemma lazy_andb_bool_true (b1 b2: bool): b1 &&& b2 = true <-> b1 = true /\ b2 = true. @@ -607,7 +613,7 @@ Qed. -Module ParallelChecks (L: SeqLanguage) (S:ResourceSet with Module R:=L.LP.R). +Module ParallelChecks (L: SeqLanguage) (S:PseudoRegSet with Module R:=L.LP.R). Include ParallelizablityChecking L. @@ -618,20 +624,20 @@ Local Hint Resolve S.empty_match_frame S.add_match_frame S.union_match_frame S.i (** Now, refinement of each operation toward parallelizable *) -Fixpoint macro_wsframe(i:macro): S.t := +Fixpoint inst_wsframe(i:inst): S.t := match i with | nil => S.empty - | a::i' => S.add (fst a) (macro_wsframe i') + | a::i' => S.add (fst a) (inst_wsframe i') end. -Lemma macro_wsframe_correct i: S.match_frame (macro_wsframe i) (macro_wframe i). +Lemma inst_wsframe_correct i: S.match_frame (inst_wsframe i) (inst_wframe i). Proof. induction i; simpl; auto. Qed. Fixpoint exp_sframe (e: exp): S.t := match e with - | Name x => S.add x S.empty + | PReg x => S.add x S.empty | Op o le => list_exp_sframe le | Old e => exp_sframe e end @@ -647,27 +653,27 @@ Proof. induction e using exp_mut with (P0:=fun l => S.match_frame (list_exp_sframe l) (list_exp_frame l)); simpl; auto. Qed. -Fixpoint macro_sframe (i: macro): S.t := +Fixpoint inst_sframe (i: inst): S.t := match i with | nil => S.empty - | a::i' => S.add (fst a) (S.union (exp_sframe (snd a)) (macro_sframe i')) + | a::i' => S.add (fst a) (S.union (exp_sframe (snd a)) (inst_sframe i')) end. Local Hint Resolve exp_sframe_correct. -Lemma macro_sframe_correct i: S.match_frame (macro_sframe i) (macro_frame i). +Lemma inst_sframe_correct i: S.match_frame (inst_sframe i) (inst_frame i). Proof. induction i as [|[y e] i']; simpl; auto. Qed. -Local Hint Resolve macro_wsframe_correct macro_sframe_correct. +Local Hint Resolve inst_wsframe_correct inst_sframe_correct. Fixpoint is_pararec (p: bblock) (wsframe: S.t): bool := match p with | nil => true | i::p' => - S.is_disjoint (macro_sframe i) wsframe (* no USE-AFTER-WRITE *) - &&& is_pararec p' (S.union (macro_wsframe i) wsframe) + S.is_disjoint (inst_sframe i) wsframe (* no USE-AFTER-WRITE *) + &&& is_pararec p' (S.union (inst_wsframe i) wsframe) end. Lemma is_pararec_correct (p: bblock): forall s l, S.match_frame s l -> (is_pararec p s)=true -> (pararec p l). @@ -700,7 +706,7 @@ End ParallelChecks. Require Import PArith. Require Import MSets.MSetPositive. -Module PosResourceSet <: ResourceSet with Module R:=Pos. +Module PosPseudoRegSet <: PseudoRegSet with Module R:=Pos. Module R:=Pos. @@ -770,4 +776,4 @@ Proof. intros H4 H5; eapply is_disjoint_spec_true; eauto. Qed. -End PosResourceSet. +End PosPseudoRegSet. |