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Diffstat (limited to 'backend/Simplify_WL.v')
| -rwxr-xr-x | backend/Simplify_WL.v | 160 |
1 files changed, 160 insertions, 0 deletions
diff --git a/backend/Simplify_WL.v b/backend/Simplify_WL.v new file mode 100755 index 00000000..4da207a9 --- /dev/null +++ b/backend/Simplify_WL.v @@ -0,0 +1,160 @@ +Require Import FSets. +Require Import InterfGraphMapImp. +Require Import Remove_Vertex_WL. +Require Import WS. +Require Import Affinity_relation. +Require Import Interference_adjacency. +Require Import Edges. +Require Import IRC_graph. + +Import Edge RegFacts Props OTFacts. + +Definition simplify_wl r ircg K := + let g := irc_g ircg in + let wl := irc_wl ircg in + let simplify := get_simplifyWL wl in + let freeze := get_freezeWL wl in + let spillWL := get_spillWL wl in + let movesWL := get_movesWL wl in + let pre := precolored g in + let int_adj := interference_adj r g in + let not_pre_int_adj := VertexSet.diff int_adj pre in + let newlow := VertexSet.filter (fun x => eq_K K (VertexSet.cardinal (interference_adj x g))) not_pre_int_adj in + let (free, simp) := VertexSet.partition (move_related g) newlow in + let simplifyWL_ := VertexSet.union simplify simp in + let simplifyWL' := VertexSet.remove r simplifyWL_ in + let freezeWL' := VertexSet.union freeze free in + let spillWL' := VertexSet.diff spillWL newlow in + let movesWL' := movesWL in (spillWL', freezeWL', simplifyWL', movesWL'). + +Lemma WS_simplify_aux : forall r ircg, +VertexSet.In r (get_simplifyWL (irc_wl ircg)) -> +WS_properties (remove_vertex r (irc_g ircg)) (VertexSet.cardinal (pal ircg)) + (simplify_wl r ircg (VertexSet.cardinal (pal ircg))). + +Proof. +intros. +generalize (WS_props_equal (remove_vertex r (irc_g ircg)) (VertexSet.cardinal (pal ircg)) + (remove_wl_2 r ircg (VertexSet.cardinal (pal ircg))) + (simplify_wl r ircg (VertexSet.cardinal (pal ircg)))). +generalize (WS_remove_wl_2 r ircg). +unfold remove_wl_2, simplify_wl, get_simplifyWL, + get_freezeWL, get_spillWL, get_movesWL. +set (g' := remove_vertex r (irc_g ircg)) in *. +set (k := VertexSet.cardinal (pal ircg)) in *. +set (g := irc_g ircg) in *. +set (wl := irc_wl ircg) in *. +set ( simplify := get_simplifyWL wl ) in *. +set ( freeze := get_freezeWL wl ) in *. +set ( spillWL := get_spillWL wl ) in *. +set ( int_adj := interference_adj r g ) in *. +set ( not_pre_int_adj := VertexSet.diff int_adj (precolored g) ) in *. +set ( pre_adj := preference_adj r g ) in *. +set ( not_pre_pre_adj := VertexSet.diff pre_adj (precolored g) ) in *. +set ( low := VertexSet.filter (fun x => eq_K k (VertexSet.cardinal (interference_adj x g))) not_pre_int_adj ) in *. +set ( simpfree := VertexSet.partition (move_related g) low ) in *. +case_eq simpfree. intros free simp Hsf. +unfold simpfree in Hsf. +set ( nmr := VertexSet.filter (fun x => eq_K 1 (VertexSet.cardinal (preference_adj x g)) && has_low_degree g k x) not_pre_pre_adj) in *. +set ( simplifyWL__ := VertexSet.union simplify simp ) in *. +set ( simplifyWL_ := VertexSet.union simplifyWL__ nmr) in *. +set ( simplifyWL' := VertexSet.remove r simplifyWL_ ) in *. +set ( freezeWL__ := VertexSet.diff freeze nmr ) in *. +set ( freezeWL_ := VertexSet.union freezeWL__ free ) in *. +set ( freezeWL' := VertexSet.remove r freezeWL_ ) in *. +set ( spillWL_ := VertexSet.diff spillWL low ) in *. +set ( spillWL' := VertexSet.remove r spillWL_ ) in *. +set ( movesWL' := not_incident_edges r (get_movesWL wl) g) in *. simpl. +generalize (In_simplify_props _ _ _ _ _ _ _ _ H (refl_equal _) (HWS_irc ircg)). intro Hr. + +assert (VertexSet.Equal (preference_adj r g) VertexSet.empty) as Hempty_pre. +apply not_move_related_empty_pref. intuition. +assert (VertexSet.Equal nmr VertexSet.empty) as Hempty_nmr. +unfold nmr. +split; intros. +generalize (VertexSet.filter_1 (compat_move_up _ _) H0). intro. +unfold not_pre_pre_adj in H1. generalize (VertexSet.diff_1 H1). intro. +unfold pre_adj in H2. rewrite Hempty_pre in H2. elim (VertexSet.empty_1 H2). +elim (VertexSet.empty_1 H0). + +intros HWS H0. apply H0. + +(* simplify worklist is only decreasing from r and increasing from simp, + since nmr is empty, because preference adj r g is empty *) +split; intros. +apply VertexSet.remove_2. +intro. elim (VertexSet.remove_1 H2 H1). +destruct (VertexSet.union_1 (VertexSet.remove_3 H1)). +assumption. +rewrite Hempty_nmr in H2. elim (VertexSet.empty_1 H2). + +apply VertexSet.remove_2. +intro. elim (VertexSet.remove_1 H2 H1). +apply VertexSet.union_2. apply (VertexSet.remove_3 H1). + +(* r is not deleted from freezewl, since it is not move-related and + no vertex is removed from freeze, since free is empty, because + preference adj r g is empty *) + +set (s := VertexSet.union (VertexSet.diff (snd (fst (fst wl))) nmr) free). +split; intros. +unfold s in H1. destruct (VertexSet.union_1 (VertexSet.remove_3 H1)). +apply VertexSet.union_2. apply (VertexSet.diff_1 H2). +apply VertexSet.union_3. assumption. + +apply VertexSet.remove_2. intro. rewrite <-H2 in H1. clear H2. +unfold s in H1. +destruct (VertexSet.union_1 H1). +generalize (In_freeze_props _ _ _ _ _ _ _ _ H2 (refl_equal _) (HWS_irc ircg)). intro. +destruct H3. destruct H4. destruct Hr. destruct H7. +rewrite H7 in H4. inversion H4. +assert (free = fst (VertexSet.partition (move_related g) low)). +rewrite Hsf. auto. +rewrite H3 in H2. rewrite VertexSet.partition_1 in H2. +generalize (VertexSet.filter_2 (compat_bool_move _ ) H2). intro. +destruct Hr. destruct H6. unfold g in H4. rewrite H6 in H4. inversion H4. +apply compat_bool_move. + +unfold s. destruct (VertexSet.union_1 H1). +apply VertexSet.union_2. apply VertexSet.diff_3. +assumption. +intro. rewrite Hempty_nmr in H3. elim (VertexSet.empty_1 H3). +apply VertexSet.union_3. assumption. + +(* identically, r is not removed from spillwl, since it is not of high-degree, + but degrees remain the same even if r is not move related *) +set (s := VertexSet.diff (fst (fst (fst wl))) low). +split; intros. +apply (VertexSet.remove_3 H1). +apply VertexSet.remove_2. intro. rewrite <-H2 in H1. clear H2. +unfold s in H1. +generalize (VertexSet.diff_1 H1). intro. +generalize (In_spill_props _ _ _ _ _ _ _ _ H2 (refl_equal _) (HWS_irc ircg)). intro. +destruct H3. destruct Hr. rewrite H5 in H3. inversion H3. +assumption. + +(* there is no preference edge incident to r *) +split; intros. +rewrite not_incident_edges_1 in H1. destruct H1. assumption. +intros. apply (In_move_props _ _ _ _ _ _ _ _ H2 (refl_equal _) (HWS_irc ircg)). +rewrite not_incident_edges_1. split. +assumption. +intro. destruct Hr. destruct H4. +cut (move_related g r = true). intro. unfold g in H6. rewrite H4 in H6. inversion H6. +generalize (In_move_props _ _ _ _ _ _ _ _ H1 (refl_equal _) (HWS_irc ircg)). intro. +destruct H6. +apply move_related_charac2 with (e:=a); assumption. +intros. apply (In_move_props _ _ _ _ _ _ _ _ H2 (refl_equal _) (HWS_irc ircg)). + +(* WS_remove_wl respects the invariant *) +assumption. +Qed. + +Lemma WS_simplify : forall r ircg, +VertexSet.In r (get_simplifyWL (irc_wl ircg)) -> +WS_properties (remove_vertex r (irc_g ircg)) (irc_k ircg) + (simplify_wl r ircg (irc_k ircg)). + +Proof. +intros. rewrite <-(Hk ircg). apply WS_simplify_aux. auto. +Qed. |