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Diffstat (limited to 'backend/Simplify_WL.v')
| -rwxr-xr-x | backend/Simplify_WL.v | 160 |
1 files changed, 0 insertions, 160 deletions
diff --git a/backend/Simplify_WL.v b/backend/Simplify_WL.v deleted file mode 100755 index 4da207a..0000000 --- a/backend/Simplify_WL.v +++ /dev/null @@ -1,160 +0,0 @@ -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. |