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Diffstat (limited to 'backend/InterfGraphMapImp.v')
| -rwxr-xr-x | backend/InterfGraphMapImp.v | 9401 |
1 files changed, 0 insertions, 9401 deletions
diff --git a/backend/InterfGraphMapImp.v b/backend/InterfGraphMapImp.v deleted file mode 100755 index cdd8327..0000000 --- a/backend/InterfGraphMapImp.v +++ /dev/null @@ -1,9401 +0,0 @@ -Require Import FSets. -Require Import Recdef. -Require Import ZArith. -Require Import Coq.Init.Wf. -Require Import FSetInterface. -Require Import SetsFacts. -Require Import FMaps. -Require Import OrderedOption. -Require Import FMapAVL. -Require Import Edges. -Require Import MyRegisters. - -Module Register := Regs. - -Import Edge. - -Module VertexSet := FSetAVL.Make Vertex. -Module EdgeSet := FSetAVL.Make Edge. -Module VertexMap := FMapAVL.Make Vertex. -Module MapFacts := Facts VertexMap. -Module RegFacts := MyFacts VertexSet. -Module MEdgeFacts := MyFacts EdgeSet. -Module RegRegProps := MEdgeFacts.Props. -Module Props := RegFacts.Props. - -Definition adj_set x m := -match (VertexMap.find x m) with -| None => VertexSet.empty -| Some x => x -end. - -Record tt : Type := Make_Graph { -vertices : VertexSet.t; -imap : VertexMap.t VertexSet.t; -pmap : VertexMap.t VertexSet.t; -extremities_imap : forall x, VertexMap.In x imap <-> VertexSet.In x vertices; -extremities_pmap : forall x, VertexMap.In x pmap <-> VertexSet.In x vertices; -simple_graph : forall x y, VertexSet.In x (adj_set y imap) /\ - VertexSet.In x (adj_set y pmap) -> False; -sym_imap : forall x y, VertexSet.In x (adj_set y imap) -> - VertexSet.In y (adj_set x imap); -sym_pmap : forall x y, VertexSet.In x (adj_set y pmap) -> - VertexSet.In y (adj_set x pmap); -not_eq_extremities : forall x y, VertexSet.In x (adj_set y imap) \/ - VertexSet.In x (adj_set y pmap) -> - ~Vertex.eq x y -}. - -Definition t := tt. - -Definition V := vertices. - -Definition edgemap_to_edgeset map w := -VertexMap.fold - (fun y imapy s => VertexSet.fold - (fun z s' => EdgeSet.add (y,z,w) s') - imapy - s) - map - EdgeSet.empty. - -Definition AE g := edgemap_to_edgeset (pmap g) (Some N0). - -Definition IE g := edgemap_to_edgeset (imap g) None. - -Definition is_precolored v (g : t) := is_mreg v. - -Definition imap_remove m x := -VertexMap.remove x -(VertexSet.fold - (fun y => VertexMap.add y (VertexSet.remove x (adj_set y m))) - (adj_set x m) - m -). - -Lemma change_fst : forall x y z, -fst_ext (x,y,z) = x. - -Proof. -auto. -Qed. - -Lemma change_snd : forall x y z, -snd_ext (x,y,z) = y. - -Proof. -auto. -Qed. - -Lemma change_weight : forall x y z, -get_weight (x,y,z) = z. - -Proof. -auto. -Qed. - -(* rewriting tactic *) - -Ltac change_rewrite := -repeat (try rewrite change_fst in *;try rewrite change_snd in *;try rewrite change_weight in *). - -(* two tactics for proving equality of edges *) - -Ltac Eq_eq := -apply eq_ordered_eq;unfold E.eq;split;[simpl;split;intuition|simpl;apply OptionN_as_OT.eq_refl]. - -Ltac Eq_comm_eq := rewrite edge_comm;Eq_eq. - -Lemma eq_charac : forall x y, -eq x y -> (Vertex.eq (fst_ext x) (fst_ext y) /\ Vertex.eq (snd_ext x) (snd_ext y)) \/ - (Vertex.eq (fst_ext x) (snd_ext y) /\ Vertex.eq (snd_ext x) (fst_ext y)). - -Proof. -intros x y H;unfold eq in H;unfold ordered_edge in H; -unfold get_edge in H. -destruct (OTFacts.lt_dec (snd_ext x) (fst_ext x)); -destruct (OTFacts.lt_dec (snd_ext y) (fst_ext y)); -unfold E.eq in H;simpl in H;intuition. -Qed. - -Section fold_assoc_interf_map. - -Variable A : Type. - -Inductive eq_set_option : option VertexSet.t -> option VertexSet.t -> Prop := -|None_eq : eq_set_option None None -|Some_eq : forall s s', VertexSet.Equal s s' -> eq_set_option (Some s) (Some s'). - -Definition EqualSetMap m1 m2 := forall x, eq_set_option (VertexMap.find x m1) (VertexMap.find x m2). - -Lemma EqualSetMap_refl : forall m, EqualSetMap m m. - -Proof. -unfold EqualSetMap. intro m. intro x. -destruct (VertexMap.find x m). -constructor. intuition. -constructor. -Qed. - -Lemma EqualSetMap_trans : forall m1 m2 m3, -EqualSetMap m1 m2 -> -EqualSetMap m2 m3 -> -EqualSetMap m1 m3. - -Proof. -intros m1 m2 m3 H H0. -unfold EqualSetMap in *. -intro x. -generalize (H x). clear H. intro H. -generalize (H0 x). clear H0. intro H0. -destruct (VertexMap.find x m1). -inversion H. subst. -rewrite <-H2 in H0. -destruct (VertexMap.find x m3). -constructor. inversion H0. subst. -rewrite H3. assumption. -inversion H0. -destruct (VertexMap.find x m3). -inversion H0. inversion H. subst. rewrite <-H4 in H1. inversion H1. -constructor. -Qed. - -Lemma fold_left_compat_map : forall (f : VertexMap.t VertexSet.t -> A -> VertexMap.t VertexSet.t) l e e', -EqualSetMap e e' -> -(forall e1 e2 a, EqualSetMap e1 e2 -> EqualSetMap (f e1 a) (f e2 a)) -> -EqualSetMap (fold_left f l e) (fold_left f l e'). - -Proof. -intro f;induction l;simpl. -auto. -intros e e' H H0 H1. -apply (IHl (f e a) (f e' a)). -apply H0;assumption. -assumption. -Qed. - -Lemma fold_left_assoc_map : forall l (f : VertexMap.t VertexSet.t -> A -> VertexMap.t VertexSet.t) x h, -(forall (y z : A) s, EqualSetMap (f (f s y) z) (f (f s z) y)) -> -(forall e1 e2 a, EqualSetMap e1 e2 -> EqualSetMap (f e1 a) (f e2 a)) -> -EqualSetMap (fold_left f (h :: l) x) (f (fold_left f l x) h). - -Proof. -induction l;simpl;intros f x h H H0. -apply EqualSetMap_refl. -apply EqualSetMap_trans with (m2 := fold_left f (h :: l) (f x a)). -simpl. apply fold_left_compat_map. apply H. -assumption. -apply IHl. assumption. assumption. -Qed. - -End fold_assoc_interf_map. - -Lemma fold_assoc : forall g g' y0 z s, -(forall x y a, EdgeSet.Equal (g x (g' y a)) (g' y (g x a))) -> -(forall (y z0 : VertexSet.elt) (s0 : EdgeSet.t), -EdgeSet.Equal (g z0 (g y s0)) (g y (g z0 s0))) -> -(forall (e1 e2 : EdgeSet.t) (a1 : VertexSet.elt), -EdgeSet.Equal e1 e2 -> EdgeSet.Equal (g a1 e1) (g a1 e2)) -> -(forall (e1 e2 : EdgeSet.t) (a1 : VertexSet.elt), -EdgeSet.Equal e1 e2 -> EdgeSet.Equal (g' a1 e1) (g' a1 e2)) -> -(forall (y z0 : VertexSet.elt) (s0 : EdgeSet.t), -EdgeSet.Equal (g' z0 (g' y s0)) (g' y (g' z0 s0))) -> -EdgeSet.Equal (VertexSet.fold g z (VertexSet.fold g' y0 s)) - (VertexSet.fold g' y0 (VertexSet.fold g z s)). - -Proof. -intros. -repeat rewrite VertexSet.fold_1. -set (f1 := fun (a : EdgeSet.t) (e : VertexSet.elt) => g e a). -set (f2 := fun (a : EdgeSet.t) (e : VertexSet.elt) => g' e a). -induction (VertexSet.elements z). simpl. -apply EdgeSet.eq_refl. - -set (l' := VertexSet.elements y0) in *. -assert (EdgeSet.Equal (fold_left f2 l' (fold_left f1 (a :: l) s)) - (fold_left f2 l' (f1 (fold_left f1 l s) a))). -apply MEdgeFacts.fold_left_compat_set. -apply MEdgeFacts.fold_left_assoc. - -unfold f2. assumption. -unfold f2. assumption. -unfold f1. assumption. - -apply EdgeSet.eq_trans with (s' := (fold_left f2 l' (f1 (fold_left f1 l s) a))). -set (s' := fold_left f1 l s) in *. -cut (EdgeSet.Equal (fold_left f2 l' (f1 s' a)) (f1 (fold_left f2 l' s') a)). -intro. -apply EdgeSet.eq_trans with (s' := f1 (fold_left f2 l' s') a). -rewrite MEdgeFacts.fold_left_assoc. -apply H1. assumption. -assumption. -assumption. -apply EdgeSet.eq_sym. auto. - -clear IHl. clear H4. -induction l'. simpl. apply EdgeSet.eq_refl. -assert (EdgeSet.Equal (f1 (fold_left f2 (a0 :: l') s') a) - (f1 (f2 (fold_left f2 l' s') a0) a)). -apply H1. -apply MEdgeFacts.fold_left_assoc. -assumption. -assumption. -apply EdgeSet.eq_trans with (s':= (f1 (f2 (fold_left f2 l' s') a0) a)). -rewrite MEdgeFacts.fold_left_assoc. -apply EdgeSet.eq_trans with (s' := f2 (f1 (fold_left f2 l' s') a) a0). -apply H2. assumption. -unfold f1. unfold f2. -unfold EdgeSet.eq. apply EdgeSet.eq_sym. -apply H. -assumption. -assumption. -apply H1. apply EdgeSet.eq_sym. apply MEdgeFacts.fold_left_assoc. -assumption. -assumption. -apply EdgeSet.eq_sym. auto. -Qed. - -Lemma edgemap_to_edgeset_charac : forall m x y (w : option N), -(forall a b, VertexSet.In a (adj_set b m) -> - VertexSet.In b (adj_set a m)) -> -(EdgeSet.In (x,y,w) (edgemap_to_edgeset m w) <-> VertexSet.In y (adj_set x m)). - -Proof. -intros m x y w Hsym. split; intros. -unfold edgemap_to_edgeset in H. -rewrite VertexMap.fold_1 in H. -generalize VertexMap.elements_2. intro. -generalize (H0 _ m). clear H0. intro HH. -induction (VertexMap.elements m). -simpl in H. -elim (EdgeSet.empty_1 H). -set (f := (fun (a : EdgeSet.t) (p : VertexMap.key * VertexSet.t) => - VertexSet.fold - (fun (z : VertexSet.elt) (s' : EdgeSet.t) => - EdgeSet.add (fst p, z, w) s') (snd p) a)) in *. -case_eq a; intros; subst. -rewrite MEdgeFacts.fold_left_assoc in H. -set (s := fold_left f l EdgeSet.empty) in *. -unfold f in H. simpl in H. -assert (EdgeSet.In (x,y,w) s \/ (VertexSet.In y t0 /\ Vertex.eq k x) \/ - (VertexSet.In x t0 /\ Vertex.eq k y)). -clear IHl. intros. -rewrite VertexSet.fold_1 in H. -generalize VertexSet.elements_2. -intro H0. generalize (H0 t0). clear H0. intro Helt. -induction (VertexSet.elements t0). -simpl in H. left. assumption. -rewrite MEdgeFacts.fold_left_assoc in H. -destruct (proj1 (RegRegProps.Dec.F.add_iff _ _ _) H). -fold (eq (k,a,w) (x,y,w)) in H0. -right. -destruct (eq_charac _ _ H0); destruct H1; change_rewrite. -left. split. -apply Helt. left. apply Vertex.eq_sym. assumption. -assumption. -right. split. -apply Helt. left. apply Vertex.eq_sym. assumption. -assumption. -apply IHl0. assumption. -intros. apply Helt. right. assumption. - -intros. apply RegRegProps.add_add. -intros. apply RegRegProps.Dec.F.add_m. apply eq_refl. assumption. - -destruct H0. -apply IHl. -assumption. -intros. apply HH. auto. -destruct H0. -assert (VertexMap.MapsTo x t0 m). -apply HH. -left. -constructor; simpl; intuition. -generalize (VertexMap.find_1 H1). clear H1. intro H1. -unfold adj_set. rewrite H1. intuition. -apply Hsym. -assert (VertexMap.MapsTo y t0 m). -apply HH. -left. -constructor; simpl; intuition. -generalize (VertexMap.find_1 H1). clear H1. intro H1. -unfold adj_set. rewrite H1. intuition. - -unfold f. -intros. set (g := (fun (z0 : VertexSet.elt) (s' : EdgeSet.t) => - EdgeSet.add (fst z, z0, w) s')). -set (g' := fun (z0 : VertexSet.elt) (s' : EdgeSet.t) => - EdgeSet.add (fst y0, z0, w) s'). -apply fold_assoc. -unfold g. unfold g'. -intros. apply RegRegProps.add_add. -unfold g. unfold g'. -intros. apply RegRegProps.add_add. -intros. apply RegRegProps.Dec.F.add_m. apply eq_refl. assumption. -intros. apply RegRegProps.Dec.F.add_m. apply eq_refl. assumption. -unfold g. unfold g'. -intros. apply RegRegProps.add_add. -intros. -unfold f. -rewrite VertexSet.fold_1. -rewrite VertexSet.fold_1. -apply MEdgeFacts.fold_left_compat_set. -assumption. -intros. apply RegRegProps.Dec.F.add_m. apply eq_refl. assumption. - -unfold edgemap_to_edgeset. -rewrite VertexMap.fold_1. -generalize VertexMap.elements_1. intro. -case_eq (VertexMap.find x m); intros. -generalize (H0 _ m x t0). clear H0. intro HH. -induction (VertexMap.elements m). -simpl. -assert (VertexMap.MapsTo x t0 m). -apply VertexMap.find_2. assumption. -generalize (HH H0). intro H2. inversion H2. - -set (f := (fun (a : EdgeSet.t) (p : VertexMap.key * VertexSet.t) => - VertexSet.fold - (fun (z : VertexSet.elt) (s' : EdgeSet.t) => - EdgeSet.add (fst p, z, w) s') (snd p) a)) in *. -rewrite MEdgeFacts.fold_left_assoc. -set (s := fold_left f l EdgeSet.empty) in *. -unfold f. -destruct a. simpl. -rewrite VertexSet.fold_1. -generalize VertexSet.elements_1. -intro H2. generalize (H2 t1 y). clear H2. intro HHH. -induction (VertexSet.elements t1). simpl. -assert (VertexMap.MapsTo x t0 m). -apply VertexMap.find_2. assumption. -generalize (HH H0). intro H2. -inversion H2; subst. -inversion H4. simpl in H3. simpl in H5. subst. -unfold adj_set in H. rewrite H1 in H. -generalize (HHH H). intro. inversion H5. -apply IHl. intro. auto. - -rewrite MEdgeFacts.fold_left_assoc. -generalize (VertexMap.find_2 H1). intro. -generalize (HH H0). clear HH H0. intro HH. -inversion HH; subst. -inversion H2; simpl in *; subst. clear H2 HH. -destruct (Vertex.eq_dec y a). -apply EdgeSet.add_1. -Eq_eq. -apply EdgeSet.add_2. -apply IHl0. -intro H2. generalize (HHH H2). clear HHH H2. intro H2. -inversion H2. subst. -elim (n H4). -assumption. -apply EdgeSet.add_2. -assert (forall l', EdgeSet.In (x,y,w) s -> - EdgeSet.In (x, y, w) - (fold_left - (fun (a0 : EdgeSet.t) (e : VertexSet.elt) => EdgeSet.add (k, e, w) a0) - l' s)). -clear H HH H1 H2 IHl IHl0 HHH Hsym. -intros. -induction l'. simpl. assumption. -rewrite MEdgeFacts.fold_left_assoc. -destruct (Edge.eq_dec (k,a0,w) (x,y,w)). -apply EdgeSet.add_1. auto. -apply EdgeSet.add_2. apply IHl'. - -intros. apply RegRegProps.add_add. -intros. apply RegRegProps.Dec.F.add_m. apply eq_refl. assumption. - -apply H0. -apply IHl. -auto. - -intros. apply RegRegProps.add_add. -intros. apply RegRegProps.Dec.F.add_m. apply eq_refl. assumption. - -unfold f. -intros. set (g := (fun (z0 : VertexSet.elt) (s' : EdgeSet.t) => - EdgeSet.add (fst z, z0, w) s')). -set (g' := fun (z0 : VertexSet.elt) (s' : EdgeSet.t) => - EdgeSet.add (fst y0, z0, w) s'). -apply fold_assoc. -unfold g. unfold g'. -intros. apply RegRegProps.add_add. -unfold g. unfold g'. -intros. apply RegRegProps.add_add. -intros. apply RegRegProps.Dec.F.add_m. apply eq_refl. assumption. -intros. apply RegRegProps.Dec.F.add_m. apply eq_refl. assumption. -unfold g. unfold g'. -intros. apply RegRegProps.add_add. -intros. -unfold f. -rewrite VertexSet.fold_1. -rewrite VertexSet.fold_1. -apply MEdgeFacts.fold_left_compat_set. -assumption. -intros. apply RegRegProps.Dec.F.add_m. apply eq_refl. assumption. - -unfold adj_set in H. rewrite H1 in H. elim (VertexSet.empty_1 H). -Qed. - -Require Import FMapFacts. - -Module InterfFacts := FMapFacts.Facts VertexMap. - -Lemma imap_remove_1 : forall x y m r, -~Vertex.eq r x -> -~Vertex.eq r y -> -VertexSet.In x (adj_set y m) -> -VertexSet.In x (adj_set y (imap_remove m r)). - -Proof. -intros. -unfold imap_remove. -unfold adj_set. -cut (VertexSet.In x match - (VertexMap.find (elt:=VertexSet.t) y - (VertexMap.remove (elt:=VertexSet.t) r - (VertexSet.fold - (fun y0 : VertexSet.elt => - VertexMap.add y0 (VertexSet.remove r (adj_set y0 m))) - (adj_set r m) m))) with - | Some x0 => x0 - | None => VertexSet.empty -end); auto. -rewrite MapFacts.remove_neq_o. - -rewrite VertexSet.fold_1. -induction (VertexSet.elements (adj_set r m)); intros. -simpl. assumption. - -set (f:= (fun (a : VertexMap.t VertexSet.t) (e : VertexSet.elt) => - VertexMap.add e (VertexSet.remove r (adj_set e m)) a)) in *. -assert (EqualSetMap (fold_left f (a :: l) m) (f (fold_left f l m) a)). -apply fold_left_assoc_map. -intros. -unfold f. unfold EqualSetMap. intros. -destruct (Vertex.eq_dec x0 z). -rewrite MapFacts.add_eq_o. -destruct (Vertex.eq_dec x0 y0). -rewrite MapFacts.add_eq_o. -constructor. -apply RegFacts.Props.Dec.F.remove_m. apply Vertex.eq_refl. -unfold adj_set. rewrite (MapFacts.find_o _ (Vertex.eq_trans (Vertex.eq_sym e) e0)). -apply VertexSet.eq_refl. apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_eq_o. -constructor. -apply RegFacts.Props.Dec.F.remove_m. apply Vertex.eq_refl. -apply VertexSet.eq_refl. -apply Regs.eq_sym. auto. -auto. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o. -destruct (Vertex.eq_dec x0 y0). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. -apply RegFacts.Props.Dec.F.remove_m. apply Vertex.eq_refl. -apply VertexSet.eq_refl. -apply Regs.eq_sym. auto. -apply Regs.eq_sym. auto. - -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -destruct (VertexMap.find x0 s); constructor; auto. -apply VertexSet.eq_refl. -auto. -auto. -auto. -auto. - -intros. -unfold f. -unfold EqualSetMap. -intros. -destruct (Vertex.eq_dec x0 a0). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. -apply VertexSet.eq_refl. -apply Vertex.eq_sym. assumption. -apply Vertex.eq_sym. assumption. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -apply H2. auto. auto. -unfold EqualSetMap in H2. -generalize (H2 y). intro H3. -case_eq (VertexMap.find y (fold_left f (a :: l) m)); intros. -rewrite H4 in H3. inversion H3. subst. -unfold f in H6. -destruct (Vertex.eq_dec y a). -rewrite MapFacts.add_eq_o in H6. inversion H6. -rewrite H7. rewrite H8. apply VertexSet.remove_2. -auto. unfold adj_set. rewrite (MapFacts.find_o _ (Vertex.eq_sym e)). assumption. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o in H6. -rewrite H7. unfold f in IHl. -case_eq (VertexMap.find (elt:=VertexSet.t) y - (fold_left - (fun (a : VertexMap.t VertexSet.t) (e : VertexSet.elt) => - VertexMap.add e (VertexSet.remove r (adj_set e m)) a) l m)); intros; -rewrite H5 in *. -inversion H6. subst. -assumption. -inversion H6. -auto. -rewrite H4 in H3. inversion H3. subst. -unfold f in H5. -destruct (Vertex.eq_dec y a). -rewrite MapFacts.add_eq_o in H5. -inversion H5. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o in H5. -unfold f in IHl. -case_eq (VertexMap.find (elt:=VertexSet.t) y - (fold_left - (fun (a : VertexMap.t VertexSet.t) (e : VertexSet.elt) => - VertexMap.add e (VertexSet.remove r (adj_set e m)) a) l m)); intros; -rewrite H6 in *. -inversion H5. -assumption. -auto. -auto. -Qed. - -Lemma imap_remove_2 : forall x y m r, -Vertex.eq r x \/ Vertex.eq r y -> -(forall a b, VertexSet.In a (adj_set b m) -> - VertexSet.In b (adj_set a m)) -> -~VertexSet.In x (adj_set y (imap_remove m r)). - -Proof. -intros x y m r H HH. -unfold imap_remove. -unfold adj_set. -cut (~VertexSet.In x match - (VertexMap.find (elt:=VertexSet.t) y - (VertexMap.remove (elt:=VertexSet.t) r - (VertexSet.fold - (fun y0 : VertexSet.elt => - VertexMap.add y0 (VertexSet.remove r (adj_set y0 m))) - (adj_set r m) m))) with - | Some x0 => x0 - | None => VertexSet.empty -end); auto. -destruct (Vertex.eq_dec r y). -rewrite MapFacts.remove_eq_o. -apply VertexSet.empty_1. -assumption. -rewrite MapFacts.remove_neq_o. -destruct H. - -generalize VertexSet.elements_1. intro HHH. -generalize (HHH (adj_set r m) y). clear HHH. intro HHH. -rewrite VertexSet.fold_1. -induction (VertexSet.elements (adj_set r m)); intros. -simpl. intro H0. -generalize (HH _ _ H0). intro H1. -assert (InA Vertex.eq y nil). -apply HHH. -unfold adj_set. rewrite (MapFacts.find_o _ H). assumption. -inversion H2. - -set (f:= (fun (a : VertexMap.t VertexSet.t) (e : VertexSet.elt) => - VertexMap.add e (VertexSet.remove r (adj_set e m)) a)) in *. -assert (EqualSetMap (fold_left f (a :: l) m) (f (fold_left f l m) a)). -apply fold_left_assoc_map. -intros. -unfold f. unfold EqualSetMap. intros. -destruct (Vertex.eq_dec x0 z). -rewrite MapFacts.add_eq_o. -destruct (Vertex.eq_dec x0 y0). -rewrite MapFacts.add_eq_o. -constructor. -apply RegFacts.Props.Dec.F.remove_m. apply Vertex.eq_refl. -unfold adj_set. rewrite (MapFacts.find_o _ (Vertex.eq_trans (Vertex.eq_sym e) e0)). -apply VertexSet.eq_refl. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_eq_o. -constructor. -apply RegFacts.Props.Dec.F.remove_m. apply Vertex.eq_refl. -apply VertexSet.eq_refl. -apply Regs.eq_sym. auto. -auto. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o. -destruct (Vertex.eq_dec x0 y0). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. -apply RegFacts.Props.Dec.F.remove_m. apply Vertex.eq_refl. -apply VertexSet.eq_refl. -apply Regs.eq_sym. auto. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -destruct (VertexMap.find x0 s); constructor; auto. -apply VertexSet.eq_refl. -auto. -auto. -auto. -auto. - -intros. -unfold f. -unfold EqualSetMap. -intros. -destruct (Vertex.eq_dec x0 a0). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. -apply VertexSet.eq_refl. -apply Vertex.eq_sym. assumption. -apply Vertex.eq_sym. assumption. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -apply H0. auto. auto. - -unfold EqualSetMap in H0. -generalize (H0 y). intro H1. -case_eq (VertexMap.find y (fold_left f (a :: l) m)); intros. -rewrite H2 in H1. inversion H1. subst. -unfold f in H4. -destruct (Vertex.eq_dec y a). -rewrite MapFacts.add_eq_o in H4. inversion H4. -rewrite H5. rewrite H6. apply VertexSet.remove_1. auto. apply Regs.eq_sym. auto. - -rewrite MapFacts.add_neq_o in H4. -rewrite H5. unfold f in IHl. -case_eq (VertexMap.find (elt:=VertexSet.t) y - (fold_left - (fun (a : VertexMap.t VertexSet.t) (e : VertexSet.elt) => - VertexMap.add e (VertexSet.remove r (adj_set e m)) a) l m)); intros; -rewrite H3 in *. -inversion H4. subst. -apply IHl. -intros; intuition. -inversion H7; subst. -elim (n0 H9). assumption. -inversion H4. -auto. -apply VertexSet.empty_1. -elim (n H). -auto. -Qed. - -Lemma imap_remove_3 : forall x y m r, -VertexSet.In x (adj_set y (imap_remove m r)) -> -VertexSet.In x (adj_set y m). - -Proof. -intros. -assert (VertexSet.In x match - (VertexMap.find (elt:=VertexSet.t) y - (VertexMap.remove (elt:=VertexSet.t) r - (VertexSet.fold - (fun y0 : VertexSet.elt => - VertexMap.add y0 (VertexSet.remove r (adj_set y0 m))) - (adj_set r m) m))) with - | Some x0 => x0 - | None => VertexSet.empty -end) by auto. generalize H0. clear H H0. intro H. -destruct (Vertex.eq_dec y r). -rewrite MapFacts.remove_eq_o in H. -elim (VertexSet.empty_1 H). apply Regs.eq_sym. auto. -rewrite MapFacts.remove_neq_o in H. -unfold adj_set. -rewrite VertexSet.fold_1 in H. - -generalize VertexSet.elements_2. intro. -generalize (H0 (adj_set y m) x). clear H0. intro HH. - -induction (VertexSet.elements (adj_set r m)). simpl in H. assumption. - -set (f:= (fun (a : VertexMap.t VertexSet.t) (e : VertexSet.elt) => - VertexMap.add e (VertexSet.remove r (adj_set e m)) a)) in *. -assert (EqualSetMap (fold_left f (a :: l) m) (f (fold_left f l m) a)). -apply fold_left_assoc_map. -intros. -unfold f. unfold EqualSetMap. intros. -destruct (Vertex.eq_dec x0 z). -rewrite MapFacts.add_eq_o. -destruct (Vertex.eq_dec x0 y0). -rewrite MapFacts.add_eq_o. -constructor. -apply RegFacts.Props.Dec.F.remove_m. apply Vertex.eq_refl. -unfold adj_set. rewrite (MapFacts.find_o _ (Vertex.eq_trans (Vertex.eq_sym e) e0)). -apply VertexSet.eq_refl. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_eq_o. -constructor. -apply RegFacts.Props.Dec.F.remove_m. apply Vertex.eq_refl. -apply VertexSet.eq_refl. -apply Regs.eq_sym. auto. -auto. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o. -destruct (Vertex.eq_dec x0 y0). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. -apply RegFacts.Props.Dec.F.remove_m. apply Vertex.eq_refl. -apply VertexSet.eq_refl. -apply Regs.eq_sym. auto. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -destruct (VertexMap.find x0 s); constructor; auto. -apply VertexSet.eq_refl. -auto. -auto. -auto. -auto. - -intros. -unfold f. -unfold EqualSetMap. -intros. -destruct (Vertex.eq_dec x0 a0). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. -apply VertexSet.eq_refl. -apply Vertex.eq_sym. assumption. -apply Vertex.eq_sym. assumption. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -apply H0. auto. auto. - -unfold EqualSetMap in H0. -generalize (H0 y). -case_eq (VertexMap.find y (fold_left f (a :: l) m)); intros. -rewrite H1 in *. inversion H2. subst. -unfold f in H4. -destruct (Vertex.eq_dec y a). -rewrite MapFacts.add_eq_o in H4. -apply HH. -apply VertexSet.elements_1. -unfold adj_set. rewrite (MapFacts.find_o _ e). fold (adj_set a m). -apply VertexSet.remove_3 with (x:=r). -inversion H4. subst. rewrite <-H5. assumption. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o in H4. -fold f in H4. rewrite <-H4 in IHl. -apply IHl. -rewrite <-H5. assumption. -auto. - -inversion H2. subst. rewrite <-H3 in *. rewrite H1 in *. -elim (VertexSet.empty_1 H). -auto. -Qed. - -Lemma imap_remove_4 : forall x m r, -VertexMap.In x (imap_remove m r) -> -(forall a b, VertexSet.In a (adj_set b m) -> - VertexSet.In b (adj_set a m)) -> -VertexMap.In x m. - -Proof. -intros x m r H Hsym. -unfold imap_remove in H. -destruct (Vertex.eq_dec x r). -elim (VertexMap.remove_1 (Vertex.eq_sym e) H). -apply (proj2 (MapFacts.in_find_iff _ _)). -generalize (proj1 (MapFacts.in_find_iff _ _) H). clear H. intro H. -rewrite MapFacts.remove_neq_o in H. -rewrite VertexSet.fold_1 in H. intro H0. elim H. clear H. - -generalize VertexSet.elements_2. intro. -generalize (H (adj_set r m) x). clear H. intro HH. - -induction (VertexSet.elements (adj_set r m)). simpl. -assumption. - -set (f:= (fun (a : VertexMap.t VertexSet.t) (e : VertexSet.elt) => - VertexMap.add e (VertexSet.remove r (adj_set e m)) a)) in *. -assert (EqualSetMap (fold_left f (a :: l) m) (f (fold_left f l m) a)). -apply fold_left_assoc_map. -intros. -unfold f. unfold EqualSetMap. intros. -destruct (Vertex.eq_dec x0 z). -rewrite MapFacts.add_eq_o. -destruct (Vertex.eq_dec x0 y). -rewrite MapFacts.add_eq_o. -constructor. -apply RegFacts.Props.Dec.F.remove_m. apply Vertex.eq_refl. -unfold adj_set. rewrite (MapFacts.find_o _ (Vertex.eq_trans (Vertex.eq_sym e) e0)). -apply VertexSet.eq_refl. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_eq_o. -constructor. -apply RegFacts.Props.Dec.F.remove_m. apply Vertex.eq_refl. -apply VertexSet.eq_refl. -apply Regs.eq_sym. auto. -auto. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o. -destruct (Vertex.eq_dec x0 y). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. -apply RegFacts.Props.Dec.F.remove_m. apply Vertex.eq_refl. -apply VertexSet.eq_refl. -apply Regs.eq_sym. auto. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -destruct (VertexMap.find x0 s); constructor; auto. -apply VertexSet.eq_refl. -auto. -auto. -auto. -auto. - -intros. -unfold f. -unfold EqualSetMap. -intros. -destruct (Vertex.eq_dec x0 a0). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. -apply VertexSet.eq_refl. -apply Vertex.eq_sym. assumption. -apply Vertex.eq_sym. assumption. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -apply H. auto. auto. - -generalize (H x). clear H. intro H. -case_eq (VertexMap.find x (fold_left f (a :: l) m)); intros. -rewrite H1 in H. inversion H. subst. -unfold f in H3. -destruct (Vertex.eq_dec x a). -assert (VertexSet.In r (adj_set x m)). -apply Hsym. apply HH. left. auto. -unfold adj_set in H2. rewrite H0 in H2. elim (VertexSet.empty_1 H2). -rewrite MapFacts.add_neq_o in H3. -fold f in H3. -assert (VertexMap.find x (fold_left f l m) = None). -apply IHl. -intro. apply HH. auto. -rewrite H2 in H3. inversion H3. -auto. -reflexivity. -auto. -Qed. - -Lemma imap_remove_5 : forall r x m, -VertexMap.In x m -> -~Vertex.eq x r -> -VertexMap.In x (imap_remove m r). - -Proof. -intros. -unfold imap_remove. -rewrite MapFacts.in_find_iff. -rewrite MapFacts.remove_neq_o. -rewrite VertexSet.fold_1. -set (f := (fun (a : VertexMap.t VertexSet.t) (e : VertexSet.elt) => - VertexMap.add e (VertexSet.remove r (adj_set e m)) a)). -induction (VertexSet.elements (adj_set r m)). simpl. -rewrite MapFacts.in_find_iff in H. assumption. - -cut (VertexMap.find x (f (fold_left f l m) a) <> None). -intro H1. -cut (EqualSetMap (fold_left f (a :: l) m) (f (fold_left f l m) a)). intro H2. -generalize (H2 x). clear H2. intro H2. inversion H2. -simpl. rewrite <-H4 in *. rewrite <-H5 in *. assumption. -simpl. rewrite <-H3 in *. congruence. - -apply fold_left_assoc_map. -unfold EqualSetMap. unfold f. intros. -destruct (Vertex.eq_dec x0 z). -rewrite MapFacts.add_eq_o. -destruct (Vertex.eq_dec x0 y). -rewrite MapFacts.add_eq_o. -constructor. -apply RegFacts.Props.Dec.F.remove_m. apply Vertex.eq_refl. -unfold adj_set. rewrite (MapFacts.find_o _ (Vertex.eq_trans (Vertex.eq_sym e) e0)). -apply VertexSet.eq_refl. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_eq_o. -constructor. -apply RegFacts.Props.Dec.F.remove_m. apply Vertex.eq_refl. -apply VertexSet.eq_refl. -apply Regs.eq_sym. auto. -auto. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o. -destruct (Vertex.eq_dec x0 y). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. -apply RegFacts.Props.Dec.F.remove_m. apply Vertex.eq_refl. -apply VertexSet.eq_refl. -apply Regs.eq_sym. auto. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -destruct (VertexMap.find x0 s); constructor; auto. -apply VertexSet.eq_refl. -auto. -auto. -auto. -auto. - -intros. -unfold f. -unfold EqualSetMap. -intros. -destruct (Vertex.eq_dec x0 a0). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. -apply VertexSet.eq_refl. -apply Vertex.eq_sym. assumption. -apply Vertex.eq_sym. assumption. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -apply H2. auto. auto. - -set (tmp := fold_left f l m) in *. -unfold f. -destruct (Vertex.eq_dec x a). -rewrite MapFacts.add_eq_o. congruence. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o. -assumption. -auto. -auto. -Qed. - -Lemma extremities_in_remove_vertex_imap v g : -forall x, -VertexMap.In x (imap_remove (imap g) v) <-> -VertexSet.In x (VertexSet.remove v (V g)). - -Proof. -split; intros. -apply VertexSet.remove_2. -unfold imap_remove in H. -intro Helim; apply (VertexMap.remove_1 Helim H). -apply (extremities_imap g x); apply imap_remove_4 with (r:=v); try assumption. -exact (sym_imap g). - -generalize (proj2 (extremities_imap g x)). intro. -generalize (H0 (VertexSet.remove_3 H)). clear H0. intro H0. -apply imap_remove_5. assumption. -intro H1. elim (VertexSet.remove_1 (Vertex.eq_sym H1) H). -Qed. - -Lemma extremities_in_remove_vertex_pmap v g : -forall x, -VertexMap.In x (imap_remove (pmap g) v) <-> -VertexSet.In x (VertexSet.remove v (V g)). - -Proof. -split; intros. -apply VertexSet.remove_2. -unfold imap_remove in H. -intro Helim; apply (VertexMap.remove_1 Helim H). -apply (extremities_pmap g x); apply imap_remove_4 with (r:=v); try assumption. -exact (sym_pmap g). - -generalize (proj2 (extremities_pmap g x)). intro. -generalize (H0 (VertexSet.remove_3 H)). clear H0. intro H0. -apply imap_remove_5. assumption. -intro H1. elim (VertexSet.remove_1 (Vertex.eq_sym H1) H). -Qed. - -Lemma simple_graph_remove_vertex_map v g : -forall x y, -VertexSet.In x (adj_set y (imap_remove (imap g) v)) /\ -VertexSet.In x (adj_set y (imap_remove (pmap g) v)) -> -False. - -Proof. -intros. -apply (simple_graph g x y). -destruct H. -generalize (imap_remove_3 _ _ _ _ H). -generalize (imap_remove_3 _ _ _ _ H0). -auto. -Qed. - -Lemma not_eq_extremities_remove_vertex_map v g : forall x y, -VertexSet.In x (adj_set y (imap_remove (imap g) v)) \/ -VertexSet.In x (adj_set y (imap_remove (pmap g) v)) -> -~Vertex.eq x y. - -Proof. -intros. -apply (not_eq_extremities g). -destruct H;[left|right]; apply (imap_remove_3 _ _ _ _ H). -Qed. - -Lemma sym_imap_remove_vertex v g : -forall (x : VertexSet.elt) (y : VertexMap.key), -VertexSet.In x (adj_set y (imap_remove (imap g) v)) -> -VertexSet.In y (adj_set x (imap_remove (imap g) v)). - -Proof. -intros. -generalize (imap_remove_2 x y (imap g) v). intro H0. -apply imap_remove_1. -intro H1. elim (H0 (or_intror _ H1) (sym_imap g) H). -intro H1. elim (H0 (or_introl _ H1) (sym_imap g) H). -generalize (imap_remove_3 _ _ _ _ H). intro H1. -apply (sym_imap g). assumption. -Qed. - -Lemma sym_pmap_remove_vertex v g : -forall (x : VertexSet.elt) (y : VertexMap.key), -VertexSet.In x (adj_set y (imap_remove (pmap g) v)) -> -VertexSet.In y (adj_set x (imap_remove (pmap g) v)). - -Proof. -intros. -generalize (imap_remove_2 x y (pmap g) v). intro H0. -apply imap_remove_1. -intro H1. elim (H0 (or_intror _ H1) (sym_pmap g) H). -intro H1. elim (H0 (or_introl _ H1) (sym_pmap g) H). -generalize (imap_remove_3 _ _ _ _ H). intro H1. -apply (sym_pmap g). assumption. -Qed. - -Definition remove_vertex v g := -Make_Graph (VertexSet.remove v (V g)) - (imap_remove (imap g) v) - (imap_remove (pmap g) v) - (extremities_in_remove_vertex_imap v g) - (extremities_in_remove_vertex_pmap v g) - (simple_graph_remove_vertex_map v g) - (sym_imap_remove_vertex v g) - (sym_pmap_remove_vertex v g) - (not_eq_extremities_remove_vertex_map v g). - -Definition map_merge e map := -let adj_snd := VertexSet.remove (fst_ext e) (adj_set (snd_ext e) map) in -let adj_fst := VertexSet.remove (snd_ext e) (adj_set (fst_ext e) map) in -let new_fst_adj := VertexSet.union adj_fst adj_snd in -(* -let new_fst_adj_ := VertexSet.union (adj_set (fst_ext e) map) (adj_set (snd_ext e) map) in -let new_fst_adj := VertexSet.remove (fst_ext e) (VertexSet.remove (snd_ext e) new_fst_adj_) in -let m := VertexMap.add (fst_ext e) new_fst_adj map in -*) -let redirect_m := - VertexSet.fold - (fun y m' => - VertexMap.add y - (VertexSet.add (fst_ext e) (VertexSet.remove (snd_ext e) (adj_set y map))) m') - adj_snd -(* - (VertexSet.remove (fst_ext e) (adj_set (snd_ext e) m)) -*) - map in -VertexMap.remove (snd_ext e) -(VertexMap.add (fst_ext e) new_fst_adj redirect_m). - -Definition imap_merge e g := map_merge e (imap g). - -Definition resolve_conflicts y pm padj iadj := -let m' := VertexSet.fold - (fun x m => VertexMap.add x (VertexSet.remove y (adj_set x pm)) m) - (VertexSet.inter padj iadj) - pm -in VertexMap.add y (VertexSet.diff padj iadj) m'. - -Definition pmap_merge e g im := -let pm := map_merge e (pmap g) in -resolve_conflicts (fst_ext e) pm (adj_set (fst_ext e) pm) (adj_set (fst_ext e) im). - -Definition In_graph_edge e g := -EdgeSet.In e (AE g) \/ EdgeSet.In e (IE g). - -Lemma In_graph_edge_dec : forall e g, -{In_graph_edge e g}+{~In_graph_edge e g}. - -Proof. -intros e g. -destruct (RegRegProps.In_dec e (AE g)). -left. left. assumption. -destruct (RegRegProps.In_dec e (IE g)). -left. right. assumption. -right. intro H. destruct H;[elim n|elim n0];assumption. -Qed. - -Lemma aff_edge_dec : forall e, -{aff_edge e}+{~aff_edge e}. - -Proof. -intro e. -case_eq (get_weight e). -intros n H. left. unfold aff_edge. exists n. auto. -intro H. right. intro H0. unfold aff_edge in H0. -destruct H0 as [w H0]. rewrite H0 in H. inversion H. -Qed. - -Definition In_graph (v : Vertex.t) g := VertexSet.In v (V g). - -Add Morphism get_weight : get_weight_m. - -Proof. -intros x y H. -rewrite (weight_ordered_weight x);rewrite (weight_ordered_weight y). -unfold get_weight;unfold E.eq in H. -destruct H as [_ H];inversion H;[|rewrite H2];reflexivity. -Qed. - -Lemma E_weights_aux : forall e map w s, -EdgeSet.In e -(VertexMap.fold - (fun y imapy s => VertexSet.fold - (fun z s' => EdgeSet.add (y,z,w) s') - imapy - s) - map - s) -> -EdgeSet.In e s \/ get_weight e = w. - -Proof. -intros. -rewrite VertexMap.fold_1 in H. -generalize H. clear H. generalize (VertexMap.elements map) s. -induction l. -simpl. auto. -intros. simpl in H. -set (s' := VertexSet.fold - (fun (z : VertexSet.elt) (s' : EdgeSet.t) => - EdgeSet.add (fst a, z, w) s') (snd a) s0) in H. -generalize (IHl s' H). intro H0. -destruct H0. -unfold s' in H0. - -assert (EdgeSet.In e (VertexSet.fold (fun (z : VertexSet.elt) (s' : EdgeSet.t) => - EdgeSet.add (fst a, z, w) s') (snd a) s0) -> EdgeSet.In e s0 \/ get_weight e = w). -clear H IHl. -rewrite VertexSet.fold_1. -induction (VertexSet.elements (snd a)). -simpl. auto. -rewrite MEdgeFacts.fold_left_assoc. -intro H1. destruct (proj1 (RegRegProps.Dec.F.add_iff _ _ _) H1). -fold (eq (fst a, a0, w) e) in H. -right. rewrite <-H. simpl. reflexivity. -apply IHl0. assumption. - -intros. apply RegRegProps.add_add. -intros. apply RegRegProps.Dec.F.add_m. -apply eq_refl. -assumption. -apply H1. assumption. -right. assumption. -Qed. - -Lemma E_weights : forall e m w, -EdgeSet.In e (edgemap_to_edgeset m w) -> get_weight e = w. - -Proof. -intros. -generalize (E_weights_aux e m w EdgeSet.empty). intro H0. -destruct H0. -assumption. -elim (EdgeSet.empty_1 H0). -assumption. -Qed. - -Lemma IE_weights : forall g e, -EdgeSet.In e (IE g) -> get_weight e = None. - -Proof. -unfold IE. intros. eapply E_weights. eassumption. -Qed. - -Lemma AE_weights : forall g e, -EdgeSet.In e (AE g) -> get_weight e = Some N0. - -Proof. -unfold AE. intros. eapply E_weights. eassumption. -Qed. - -(* extremities of edges are in the graph *) -Lemma In_graph_edge_in_ext : forall e g, -In_graph_edge e g -> In_graph (fst_ext e) g /\ In_graph (snd_ext e) g. - -Proof. -intros. -split. destruct H. -apply (proj1 (extremities_pmap g (fst_ext e))). - -generalize (AE_weights _ _ H). intro Hw. -unfold AE in *. -rewrite (edge_eq e) in H. -simpl in Hw. rewrite Hw in H. -generalize (proj1 (edgemap_to_edgeset_charac _ _ _ _(sym_pmap g)) H). intro H0. -apply (proj2 (InterfFacts.in_find_iff _ _)). -unfold adj_set in H0. -case_eq (VertexMap.find (fst_ext e) (pmap g)); intros; rewrite H1 in H0. -intro Helim. inversion Helim. -elim (VertexSet.empty_1 H0). - -apply (proj1 (extremities_imap g (fst_ext e))). -generalize (IE_weights _ _ H). intro Hw. -unfold IE in *. -rewrite (edge_eq e) in H. -simpl in Hw. rewrite Hw in H. -generalize (proj1 (edgemap_to_edgeset_charac _ _ _ _ (sym_imap g)) H). intro H0. -apply (proj2 (InterfFacts.in_find_iff _ _)). -unfold adj_set in H0. -case_eq (VertexMap.find (fst_ext e) (imap g)); intros; rewrite H1 in H0. -intro Helim. inversion Helim. -elim (VertexSet.empty_1 H0). - -destruct H. -apply (proj1 (extremities_pmap g (snd_ext e))). -generalize (AE_weights _ _ H). intro Hw. -unfold AE in *. -rewrite (edge_eq e) in H. rewrite edge_comm in H. -simpl in Hw. rewrite Hw in H. -generalize (proj1 (edgemap_to_edgeset_charac _ _ _ _ (sym_pmap g)) H). intro H0. -apply (proj2 (InterfFacts.in_find_iff _ _)). -unfold adj_set in H0. -case_eq (VertexMap.find (snd_ext e) (pmap g)); intros; rewrite H1 in H0. -intro Helim. inversion Helim. -elim (VertexSet.empty_1 H0). - -apply (proj1 (extremities_imap g (snd_ext e))). -generalize (IE_weights _ _ H). intro Hw. -unfold IE in *. -rewrite (edge_eq e) in H. rewrite edge_comm in H. -simpl in Hw. rewrite Hw in H. -generalize (proj1 (edgemap_to_edgeset_charac _ _ _ _ (sym_imap g)) H). intro H0. -apply (proj2 (InterfFacts.in_find_iff _ _)). -unfold adj_set in H0. -case_eq (VertexMap.find (snd_ext e) (imap g)); intros; rewrite H1 in H0. -intro Helim. inversion Helim. -elim (VertexSet.empty_1 H0). -Qed. - -Lemma not_eq_extremities_map_merge : forall x y e m, -(forall a b, VertexSet.In a (adj_set b m) -> ~Vertex.eq a b) -> -VertexSet.In x (adj_set y (map_merge e m)) -> -~Vertex.eq x y. - -Proof. -intros x y e m Hsimp H. -unfold map_merge in H. -set (f := (fun (y : VertexSet.elt) (m' : VertexMap.t VertexSet.t) => - VertexMap.add y - (VertexSet.add (fst_ext e) - (VertexSet.remove (snd_ext e) (adj_set y m))) m')) in *. -set (s := (VertexSet.union - (VertexSet.remove (snd_ext e) (adj_set (fst_ext e) m)) - (VertexSet.remove (fst_ext e) (adj_set (snd_ext e) m)))) in *. -set (s' := (VertexSet.remove (fst_ext e) (adj_set (snd_ext e) m))) in *. -intro. -unfold adj_set in H. rewrite MapFacts.remove_neq_o in H. -destruct (Vertex.eq_dec y (fst_ext e)). -rewrite MapFacts.add_eq_o in H. -unfold s in H. -destruct (VertexSet.union_1 H). -generalize (VertexSet.remove_3 H1). clear H1. intro H1. -rewrite H0 in H1. rewrite e0 in H1. -elim (Hsimp _ _ H1). auto. -unfold s' in H1. -rewrite H0 in H1. rewrite e0 in H1. elim (VertexSet.remove_1 (Vertex.eq_refl _) H1). -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o in H. - -rewrite VertexSet.fold_1 in H. -set (f' := fun a e => f e a) in *. -induction (VertexSet.elements s'). simpl in H. -unfold adj_set in H. -fold (adj_set y m) in H. -elim (Hsimp _ _ H). assumption. - -cut (EqualSetMap (fold_left f' (a :: l) m) (f' (fold_left f' l m) a)). intro. -generalize (H1 y). clear H1. intro H1. simpl in H. inversion H1; clear H1. -unfold adj_set in H. rewrite <-H3 in H. elim (VertexSet.empty_1 H). - -rewrite <-H2 in H. -set (tmp := fold_left f' l m) in *. -unfold f' in H3. unfold f in H3. -destruct (Vertex.eq_dec y a). -rewrite MapFacts.add_eq_o in H3. -inversion H3. subst. clear H3. -rewrite H4 in H. clear H4. -rewrite H0 in H. -destruct (proj1 (Props.Dec.F.add_iff _ _ _) H). -elim (n (Vertex.eq_sym H1)). -generalize (VertexSet.remove_3 H1). intro. -elim (Hsimp _ _ H3). auto. apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o in H3. -rewrite <-H3 in IHl. -apply IHl. rewrite <-H4. assumption. -auto. - -apply fold_left_assoc_map. -unfold EqualSetMap. intros. -unfold f'. unfold f. -destruct (Vertex.eq_dec x0 z). -rewrite MapFacts.add_eq_o. -destruct (Vertex.eq_dec x0 y0). -rewrite MapFacts.add_eq_o. -constructor. -unfold adj_set. -rewrite (MapFacts.find_o _ (Vertex.eq_sym e0)). -rewrite (MapFacts.find_o _ (Vertex.eq_sym e1)). -apply VertexSet.eq_refl. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_eq_o. -constructor. -apply VertexSet.eq_refl. -apply Regs.eq_sym. auto. -auto. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o. -destruct (Vertex.eq_dec x0 y0). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. -apply VertexSet.eq_refl. -apply Regs.eq_sym. auto. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -destruct (VertexMap.find x0 s0); constructor; apply VertexSet.eq_refl. -auto. -auto. -auto. -auto. - -unfold EqualSetMap. intros. -unfold f'. unfold f. -destruct (Vertex.eq_dec x0 a0). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. -apply VertexSet.eq_refl. -apply Regs.eq_sym. auto. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -apply H1. auto. -auto. - -auto. - -intro. rewrite MapFacts.remove_eq_o in H. elim (VertexSet.empty_1 H). auto. -Qed. - -Lemma resolve_conflicts_map_0 : forall x y e g, -VertexSet.In x (adj_set y (resolve_conflicts (fst_ext e) (map_merge e (pmap g)) - (adj_set (fst_ext e) (map_merge e (pmap g))) - (adj_set (fst_ext e) (imap_merge e g)))) -> -VertexSet.In x (adj_set y (map_merge e (pmap g))). - -Proof. -intros. -unfold resolve_conflicts in H. -set (f := (fun (x : VertexSet.elt) (m : VertexMap.t VertexSet.t) => - VertexMap.add x (VertexSet.remove (fst_ext e) (adj_set x (map_merge e (pmap g)))) m)) in *. -set (s := (VertexSet.diff (adj_set (fst_ext e) (map_merge e (pmap g))) - (adj_set (fst_ext e) (imap_merge e g)))) in *. -set (inter := (VertexSet.inter (adj_set (fst_ext e) (map_merge e (pmap g))) - (adj_set (fst_ext e) (imap_merge e g)))) in *. -unfold adj_set in H. -destruct (Vertex.eq_dec y (fst_ext e)). rewrite MapFacts.add_eq_o in H. -unfold s in H. -generalize (VertexSet.diff_1 H). intro H0. -unfold adj_set. rewrite (MapFacts.find_o _ e0). assumption. -apply Regs.eq_sym. auto. - -rewrite MapFacts.add_neq_o in H. -rewrite VertexSet.fold_1 in H. -set (f' := fun a e => f e a) in *. -set (m := map_merge e (pmap g)) in *. -induction (VertexSet.elements inter). simpl in H. assumption. -cut (EqualSetMap (fold_left f' (a :: l) m) (f' (fold_left f' l m) a)). intro. -generalize (H0 y). clear H0. simpl in H. intro H0. inversion H0; clear H0. -rewrite <-H2 in H. elim (VertexSet.empty_1 H). -set (tmp := fold_left f' l m) in *. -unfold f' in H2. unfold f in H2. -destruct (Vertex.eq_dec y a). rewrite MapFacts.add_eq_o in H2. -rewrite <-H1 in H. rewrite H3 in H. clear H3. -inversion H2; subst; clear H2. -unfold adj_set. rewrite (MapFacts.find_o _ e0). -apply (VertexSet.remove_3 H). -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o in H2. -apply IHl. rewrite <-H2. rewrite <-H1 in H. rewrite <-H3. assumption. -auto. - -apply fold_left_assoc_map. -unfold EqualSetMap. intros. -unfold f'. unfold f. -destruct (Vertex.eq_dec x0 z). -rewrite MapFacts.add_eq_o. -destruct (Vertex.eq_dec x0 y0). -rewrite MapFacts.add_eq_o. -constructor. -unfold adj_set. -rewrite (MapFacts.find_o _ (Vertex.eq_sym e0)). -rewrite (MapFacts.find_o _ (Vertex.eq_sym e1)). -apply VertexSet.eq_refl. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_eq_o. -constructor. -apply VertexSet.eq_refl. -apply Regs.eq_sym. auto. -auto. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o. -destruct (Vertex.eq_dec x0 y0). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. -apply VertexSet.eq_refl. -apply Regs.eq_sym. auto. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -destruct (VertexMap.find x0 s0); constructor; apply VertexSet.eq_refl. -auto. -auto. -auto. -auto. - -unfold EqualSetMap. intros. -unfold f'. unfold f. -destruct (Vertex.eq_dec x0 a0). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. -apply VertexSet.eq_refl. -apply Regs.eq_sym. auto. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -apply H0. auto. -auto. -auto. -Qed. - -Lemma pmap_merge_sub : forall x y e g, -VertexSet.In x (adj_set y (pmap_merge e g (imap_merge e g))) -> -VertexSet.In x (adj_set y (map_merge e (pmap g))). - -Proof. -exact (fun x y e g H => resolve_conflicts_map_0 _ _ _ _ H). -Qed. - -Lemma pmap_merge_domain_1 : forall x e g, -In_graph_edge e g -> -VertexMap.In x (map_merge e (pmap g)) -> -VertexSet.In x (VertexSet.remove (snd_ext e) (V g)). - -Proof. -intros x e g HH H. -unfold map_merge in H. -set (m := pmap g) in *. -set (s := VertexSet.remove (fst_ext e) (adj_set (snd_ext e) m)) in *. -set (adj := (VertexSet.union - (VertexSet.remove (snd_ext e) (adj_set (fst_ext e) m)) s)) in *. -set (f := (fun (y : VertexSet.elt) (m'0 : VertexMap.t VertexSet.t) => - VertexMap.add y - (VertexSet.add (fst_ext e) - (VertexSet.remove (snd_ext e) (adj_set y m))) m'0)) in *. -rewrite VertexSet.fold_1 in H. -set (f' := fun a e => f e a) in *. -rewrite MapFacts.in_find_iff in H. -assert (~Vertex.eq x (snd_ext e)). -intro. rewrite MapFacts.remove_eq_o in H. congruence. apply Regs.eq_sym. auto. -apply VertexSet.remove_2. auto. -rewrite MapFacts.remove_neq_o in H. -destruct (Vertex.eq_dec x (fst_ext e)). -rewrite e0. apply (proj1 (In_graph_edge_in_ext _ _ HH)). - -cut (forall z, VertexSet.In z s -> VertexSet.In z (V g)). intro HHH. -generalize VertexSet.elements_2. intro H1. -generalize (H1 s x). clear H1. intro Helt. - -induction (VertexSet.elements s). simpl in H. -apply (proj1 (extremities_pmap g x)). -rewrite MapFacts.in_find_iff. -rewrite MapFacts.add_neq_o in H. -assumption. -auto. - -cut (EqualSetMap (fold_left f' (a :: l) m) (f' (fold_left f' l m) a)). intro. -generalize (H1 x). clear H1. intro H1. simpl in H. inversion H1. -rewrite MapFacts.add_neq_o in H. -rewrite <-H3 in H. congruence. auto. -set (tmp := fold_left f' l m) in *. -unfold f' in H3. unfold f in H3. -destruct (Vertex.eq_dec x a). -apply HHH. apply Helt. left. auto. -rewrite MapFacts.add_neq_o in H3. -apply IHl. -rewrite MapFacts.add_neq_o. -congruence. -auto. -intro. apply Helt. right. auto. auto. - -apply fold_left_assoc_map. -unfold EqualSetMap. unfold f'. unfold f. intros. -destruct (Vertex.eq_dec x0 z). -rewrite MapFacts.add_eq_o. -destruct (Vertex.eq_dec x0 y). -rewrite MapFacts.add_eq_o. -constructor. -unfold adj_set. -rewrite (MapFacts.find_o _ (Vertex.eq_sym e0)). -rewrite (MapFacts.find_o _ (Vertex.eq_sym e1)). -apply VertexSet.eq_refl. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_eq_o. -constructor. -apply VertexSet.eq_refl. -apply Regs.eq_sym. auto. -auto. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o. -destruct (Vertex.eq_dec x0 y). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. -apply VertexSet.eq_refl. -apply Regs.eq_sym. auto. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -destruct (VertexMap.find x0 s0); constructor; apply VertexSet.eq_refl. -auto. -auto. -auto. -auto. - -unfold EqualSetMap. unfold f'. unfold f. intros. -destruct (Vertex.eq_dec x0 a0). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. -apply VertexSet.eq_refl. -apply Regs.eq_sym. auto. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -apply H1. -auto. -auto. - -intros. -unfold s in H1. -generalize (VertexSet.remove_3 H1). clear H1. intro H1. -apply (proj1 (extremities_pmap g z)). -rewrite MapFacts.in_find_iff. -generalize (sym_pmap g _ _ H1). clear H1. intro H1. -unfold adj_set in H1. -destruct (VertexMap.find z (pmap g)). -congruence. -elim (VertexSet.empty_1 H1). -auto. -Qed. - -Lemma pmap_merge_domain_2 : forall x e g, -In_graph_edge e g -> -VertexSet.In x (VertexSet.remove (snd_ext e) (V g)) -> -VertexMap.In x (map_merge e (pmap g)). - -Proof. -intros x e g HH H. -unfold map_merge. -set (m := pmap g) in *. -set (s := VertexSet.remove (fst_ext e) (adj_set (snd_ext e) m)) in *. -set (adj := (VertexSet.union - (VertexSet.remove (snd_ext e) (adj_set (fst_ext e) m)) s)) in *. -set (f := (fun (y : VertexSet.elt) (m'0 : VertexMap.t VertexSet.t) => - VertexMap.add y - (VertexSet.add (fst_ext e) - (VertexSet.remove (snd_ext e) (adj_set y m))) m'0)) in *. -rewrite VertexSet.fold_1. -set (f' := fun a e => f e a) in *. -rewrite MapFacts.in_find_iff. -assert (~Vertex.eq x (snd_ext e)). -intro. elim (VertexSet.remove_1 (Vertex.eq_sym H0) H). -generalize (VertexSet.remove_3 H). clear H. intro H. -rewrite MapFacts.remove_neq_o. -destruct (Vertex.eq_dec x (fst_ext e)). -rewrite MapFacts.add_eq_o. congruence. apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o. - -induction (VertexSet.elements s). simpl. -rewrite <-MapFacts.in_find_iff. -apply (proj2 (extremities_pmap g x)). assumption. - -cut (EqualSetMap (fold_left f' (a :: l) m) (f' (fold_left f' l m) a)). intro. -generalize (H1 x). clear H1. intro H1. simpl. inversion H1. -set (tmp := fold_left f' l m) in *. -unfold f' in H4. unfold f in H4. -destruct (Vertex.eq_dec x a). -rewrite MapFacts.add_eq_o in H4. congruence. apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o in H4. congruence. auto. -congruence. - -apply fold_left_assoc_map. -unfold EqualSetMap. unfold f'. unfold f. intros. -destruct (Vertex.eq_dec x0 z). -rewrite MapFacts.add_eq_o. -destruct (Vertex.eq_dec x0 y). -rewrite MapFacts.add_eq_o. -constructor. -unfold adj_set. -rewrite (MapFacts.find_o _ (Vertex.eq_sym e0)). -rewrite (MapFacts.find_o _ (Vertex.eq_sym e1)). -apply VertexSet.eq_refl. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_eq_o. -constructor. -apply VertexSet.eq_refl. -apply Regs.eq_sym. auto. -auto. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o. -destruct (Vertex.eq_dec x0 y). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. -apply VertexSet.eq_refl. -apply Regs.eq_sym. auto. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -destruct (VertexMap.find x0 s0); constructor; apply VertexSet.eq_refl. -auto. -auto. -auto. -auto. - -unfold EqualSetMap. unfold f'. unfold f. intros. -destruct (Vertex.eq_dec x0 a0). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. -apply VertexSet.eq_refl. -apply Regs.eq_sym. auto. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -apply H1. -auto. -auto. - -auto. -auto. -Qed. - -(* there is no loop in the graph *) -Lemma In_graph_edge_diff_ext : forall e g, -In_graph_edge e g -> ~Vertex.eq (snd_ext e) (fst_ext e). - -Proof. -intros. -apply (not_eq_extremities g). -destruct H;[right|left]. - -generalize (AE_weights _ _ H). intro Hw. -unfold AE in *. -rewrite (edge_eq e) in H. -simpl in Hw. rewrite Hw in H. -apply (proj1 (edgemap_to_edgeset_charac _ _ _ _(sym_pmap g)) H). - -generalize (IE_weights _ _ H). intro Hw. -unfold IE in *. -rewrite (edge_eq e) in H. -simpl in Hw. rewrite Hw in H. -apply (proj1 (edgemap_to_edgeset_charac _ _ _ _(sym_imap g)) H). -Qed. - -Lemma extremities_in_merge_map e g : -In_graph_edge e g -> -(forall x, -VertexMap.In x (map_merge e (pmap g)) <-> -VertexSet.In x (VertexSet.remove (snd_ext e) (V g))). - -Proof. -intros e g HH x; split; intro H0. -generalize H0. intro H. -unfold map_merge in H0. -apply VertexSet.remove_2. -intro H1. -elim (VertexMap.remove_1 H1 H0). -generalize (proj1 (MapFacts.remove_in_iff _ _ _) H0). clear H0. -intro H0. destruct H0. -generalize (proj1 (MapFacts.add_in_iff _ _ _ _) H1). clear H1. -intro H1. -destruct H1. -rewrite <-H1. -apply (proj1 (In_graph_edge_in_ext _ _ HH)). -apply (proj1 (extremities_pmap g x)). -rewrite VertexSet.fold_1 in H1. -set (f := (fun (a : VertexMap.t VertexSet.t) (e0 : VertexSet.elt) => - VertexMap.add e0 - (VertexSet.add (fst_ext e) - (VertexSet.remove (snd_ext e) (adj_set e0 (pmap g)))) a)) in *. -set (s := VertexSet.remove (fst_ext e) (adj_set (snd_ext e) (pmap g))) in *. -cut (forall z, In z (VertexSet.elements s) -> - VertexMap.In z (pmap g)). -intros HHH. -induction (VertexSet.elements s). -simpl in H1. assumption. -set (m := pmap g) in *. -cut (VertexMap.In x (f (fold_left f l m) a)). -clear H1. intro H1. -set (tmp := fold_left f l m) in *. -unfold f in H1. -destruct (Vertex.eq_dec x a). -rewrite e0. apply (HHH a). left. auto. -rewrite MapFacts.in_find_iff in H1. -rewrite MapFacts.add_neq_o in H1. -apply IHl. -rewrite MapFacts.in_find_iff. -assumption. -intros. apply HHH. right. assumption. -auto. - -assert (EqualSetMap (fold_left f (a :: l) m) (f (fold_left f l m) a)). -apply fold_left_assoc_map. - -unfold EqualSetMap. intros. -unfold f. -destruct (Vertex.eq_dec x0 z). -rewrite MapFacts.add_eq_o. -destruct (Vertex.eq_dec x0 y). -rewrite MapFacts.add_eq_o. -constructor. -apply RegFacts.Props.Dec.F.add_m. -apply Regs.eq_refl. -apply RegFacts.Props.Dec.F.remove_m. -apply Regs.eq_refl. -unfold adj_set. -rewrite (MapFacts.find_o _ (Vertex.eq_sym e0)). -rewrite (MapFacts.find_o _ e1). -apply VertexSet.eq_refl. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_eq_o. -constructor. -apply RegFacts.Props.Dec.F.add_m. -apply Regs.eq_refl. -apply RegFacts.Props.Dec.F.remove_m. -apply Regs.eq_refl. -unfold adj_set. -rewrite (MapFacts.find_o _ (Vertex.eq_sym e0)). -apply VertexSet.eq_refl. -apply Regs.eq_sym. auto. -auto. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o. -destruct (Vertex.eq_dec x0 y). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. -apply VertexSet.eq_refl. -apply Regs.eq_sym. auto. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -destruct (VertexMap.find x0 s0); constructor. apply VertexSet.eq_refl. -auto. -auto. -auto. -auto. - -unfold EqualSetMap. intros. -unfold f. -destruct (Vertex.eq_dec x0 a0). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. -apply VertexSet.eq_refl. -apply Regs.eq_sym. auto. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -apply H2. auto. -auto. -rewrite MapFacts.in_find_iff. -generalize (H2 x). intro. -inversion H3. -rewrite MapFacts.in_find_iff in H1. -simpl in H1. rewrite <-H5 in H1. assumption. -congruence. - -intros. -exists (adj_set z (pmap g)). apply VertexMap.elements_2. -apply VertexMap.elements_1. -unfold adj_set. -case_eq (VertexMap.find z (pmap g)); intros. -apply VertexMap.find_2. assumption. - -assert (VertexSet.In z s). -apply VertexSet.elements_2. -apply In_InA. apply Regs.eq_refl. assumption. -unfold s in H4. -generalize (VertexSet.remove_3 H4). clear H4. intro H4. -generalize (sym_pmap g _ _ H4). clear H4. intro H4. -unfold adj_set in H4. -rewrite H3 in H4. elim (VertexSet.empty_1 H4). - -unfold map_merge. -rewrite MapFacts.in_find_iff. -rewrite MapFacts.remove_neq_o. -destruct (Vertex.eq_dec x (fst_ext e)). -rewrite MapFacts.add_eq_o. -congruence. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o. -rewrite VertexSet.fold_1. -set (f := (fun (a : VertexMap.t VertexSet.t) (e0 : VertexSet.elt) => - VertexMap.add e0 - (VertexSet.add (fst_ext e) - (VertexSet.remove (snd_ext e) (adj_set e0 (pmap g)))) a)) in *. -set (s := VertexSet.remove (fst_ext e) (adj_set (snd_ext e) (pmap g))) in *. -induction (VertexSet.elements s). simpl. -generalize (proj2 (extremities_pmap g x) (VertexSet.remove_3 H0)). -rewrite MapFacts.in_find_iff. auto. - -set (m := pmap g) in *. -assert (EqualSetMap (fold_left f (a :: l) m) (f (fold_left f l m) a)). -apply fold_left_assoc_map. - -unfold EqualSetMap. intros. -unfold f. -destruct (Vertex.eq_dec x0 z). -rewrite MapFacts.add_eq_o. -destruct (Vertex.eq_dec x0 y). -rewrite MapFacts.add_eq_o. -constructor. -apply RegFacts.Props.Dec.F.add_m. -apply Regs.eq_sym. auto. -apply RegFacts.Props.Dec.F.remove_m. -apply Regs.eq_sym. auto. -unfold adj_set. -rewrite (MapFacts.find_o _ (Vertex.eq_sym e0)). -rewrite (MapFacts.find_o _ e1). -apply VertexSet.eq_refl. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_eq_o. -constructor. -apply RegFacts.Props.Dec.F.add_m. -apply Regs.eq_sym. auto. -apply RegFacts.Props.Dec.F.remove_m. -apply Regs.eq_refl. -apply VertexSet.eq_refl. -apply Regs.eq_sym. auto. -auto. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o. -destruct (Vertex.eq_dec x0 y). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. -apply VertexSet.eq_refl. -apply Regs.eq_sym. auto. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -destruct (VertexMap.find x0 s0); constructor. apply VertexSet.eq_refl. -auto. -auto. -auto. -auto. - -unfold EqualSetMap. intros. -unfold f. -destruct (Vertex.eq_dec x0 a0). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. -apply VertexSet.eq_refl. -apply Regs.eq_sym. auto. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -apply H. auto. -auto. - -generalize (H x). clear H. intro H. -inversion H. -simpl. -set (tmp := fold_left f l m) in *. -unfold f in H3. -destruct (Vertex.eq_dec x a). -rewrite MapFacts.add_eq_o in H3. -congruence. -intuition. -rewrite MapFacts.add_neq_o in H3. -rewrite H3 in IHl. congruence. -auto. -simpl. rewrite <-H1. congruence. -auto. -intro. elim (VertexSet.remove_1 H H0). -Qed. - -Lemma extremities_in_merge_imap e g : -In_graph_edge e g -> -(forall x, -VertexMap.In x (imap_merge e g) <-> -VertexSet.In x (VertexSet.remove (snd_ext e) (V g))). - -Proof. -intros e g HH x; split; intro H0. -generalize H0. intro H. -unfold imap_merge in H0. -unfold map_merge in H0. -apply VertexSet.remove_2. -intro H1. -elim (VertexMap.remove_1 H1 H0). -generalize (proj1 (MapFacts.remove_in_iff _ _ _) H0). clear H0. -intro H0. destruct H0. -generalize (proj1 (MapFacts.add_in_iff _ _ _ _) H1). clear H1. -intro H1. -destruct H1. -rewrite <-H1. -apply (proj1 (In_graph_edge_in_ext _ _ HH)). -apply (proj1 (extremities_imap g x)). -rewrite VertexSet.fold_1 in H1. -set (f := (fun (a : VertexMap.t VertexSet.t) (e0 : VertexSet.elt) => - VertexMap.add e0 - (VertexSet.add (fst_ext e) - (VertexSet.remove (snd_ext e) (adj_set e0 (imap g)))) a)) in *. -set (s := VertexSet.remove (fst_ext e) (adj_set (snd_ext e) (imap g))) in *. -cut (forall z, In z (VertexSet.elements s) -> - VertexMap.In z (imap g)). -intros HHH. -induction (VertexSet.elements s). -simpl in H1. assumption. -set (m := imap g) in *. -cut (VertexMap.In x (f (fold_left f l m) a)). -clear H1. intro H1. -set (tmp := fold_left f l m) in *. -unfold f in H1. -destruct (Vertex.eq_dec x a). -rewrite e0. apply (HHH a). left. auto. -rewrite MapFacts.in_find_iff in H1. -rewrite MapFacts.add_neq_o in H1. -apply IHl. -rewrite MapFacts.in_find_iff. -assumption. -intros. apply HHH. right. assumption. -auto. - -assert (EqualSetMap (fold_left f (a :: l) m) (f (fold_left f l m) a)). -apply fold_left_assoc_map. - -unfold EqualSetMap. intros. -unfold f. -destruct (Vertex.eq_dec x0 z). -rewrite MapFacts.add_eq_o. -destruct (Vertex.eq_dec x0 y). -rewrite MapFacts.add_eq_o. -constructor. -apply RegFacts.Props.Dec.F.add_m. -intuition. -apply RegFacts.Props.Dec.F.remove_m. -intuition. -unfold adj_set. -rewrite (MapFacts.find_o _ (Vertex.eq_sym e0)). -rewrite (MapFacts.find_o _ e1). -apply VertexSet.eq_refl. -intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_eq_o. -constructor. -apply RegFacts.Props.Dec.F.add_m. -intuition. -apply RegFacts.Props.Dec.F.remove_m. -intuition. -unfold adj_set. -rewrite (MapFacts.find_o _ (Vertex.eq_sym e0)). -apply VertexSet.eq_refl. -intuition. -auto. -intuition. -rewrite MapFacts.add_neq_o. -destruct (Vertex.eq_dec x0 y). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. -apply VertexSet.eq_refl. -intuition. -intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -destruct (VertexMap.find x0 s0); constructor. apply VertexSet.eq_refl. -auto. -auto. -auto. -auto. - -unfold EqualSetMap. intros. -unfold f. -destruct (Vertex.eq_dec x0 a0). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. -apply VertexSet.eq_refl. -intuition. -intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -apply H2. auto. -auto. -rewrite MapFacts.in_find_iff. -generalize (H2 x). intro. -inversion H3. -rewrite MapFacts.in_find_iff in H1. -simpl in H1. rewrite <-H5 in H1. assumption. -congruence. - -intros. -exists (adj_set z (imap g)). apply VertexMap.elements_2. -apply VertexMap.elements_1. -unfold adj_set. -case_eq (VertexMap.find z (imap g)); intros. -apply VertexMap.find_2. assumption. - -assert (VertexSet.In z s). -apply VertexSet.elements_2. -apply In_InA. intuition. auto. -unfold s in H4. -generalize (VertexSet.remove_3 H4). clear H4. intro H4. -generalize (sym_imap g _ _ H4). clear H4. intro H4. -unfold adj_set in H4. -rewrite H3 in H4. elim (VertexSet.empty_1 H4). - -unfold imap_merge. -unfold map_merge. -rewrite MapFacts.in_find_iff. -rewrite MapFacts.remove_neq_o. -destruct (Vertex.eq_dec x (fst_ext e)). -rewrite MapFacts.add_eq_o. -congruence. -intuition. -rewrite MapFacts.add_neq_o. -rewrite VertexSet.fold_1. -set (f := (fun (a : VertexMap.t VertexSet.t) (e0 : VertexSet.elt) => - VertexMap.add e0 - (VertexSet.add (fst_ext e) - (VertexSet.remove (snd_ext e) (adj_set e0 (imap g)))) a)) in *. -set (s := VertexSet.remove (fst_ext e) (adj_set (snd_ext e) (imap g))) in *. -induction (VertexSet.elements s). simpl. -generalize (proj2 (extremities_imap g x) (VertexSet.remove_3 H0)). -rewrite MapFacts.in_find_iff. auto. - -set (m := imap g) in *. -assert (EqualSetMap (fold_left f (a :: l) m) (f (fold_left f l m) a)). -apply fold_left_assoc_map. - -unfold EqualSetMap. intros. -unfold f. -destruct (Vertex.eq_dec x0 z). -rewrite MapFacts.add_eq_o. -destruct (Vertex.eq_dec x0 y). -rewrite MapFacts.add_eq_o. -constructor. -apply RegFacts.Props.Dec.F.add_m. -intuition. -apply RegFacts.Props.Dec.F.remove_m. -intuition. -unfold adj_set. -rewrite (MapFacts.find_o _ (Vertex.eq_sym e0)). -rewrite (MapFacts.find_o _ e1). -apply VertexSet.eq_refl. -intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_eq_o. -constructor. -apply RegFacts.Props.Dec.F.add_m. -intuition. -apply RegFacts.Props.Dec.F.remove_m. -intuition. -unfold adj_set. -rewrite (MapFacts.find_o _ (Vertex.eq_sym e0)). -apply VertexSet.eq_refl. -intuition. -auto. -intuition. -rewrite MapFacts.add_neq_o. -destruct (Vertex.eq_dec x0 y). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. -apply VertexSet.eq_refl. -intuition. -intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -destruct (VertexMap.find x0 s0); constructor. apply VertexSet.eq_refl. -auto. -auto. -auto. -auto. - -unfold EqualSetMap. intros. -unfold f. -destruct (Vertex.eq_dec x0 a0). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. -apply VertexSet.eq_refl. -intuition. -intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -apply H. auto. -auto. - -generalize (H x). clear H. intro H. -inversion H. -simpl. -set (tmp := fold_left f l m) in *. -unfold f in H3. -destruct (Vertex.eq_dec x a). -rewrite MapFacts.add_eq_o in H3. -congruence. -intuition. -rewrite MapFacts.add_neq_o in H3. -rewrite H3 in IHl. congruence. -intuition. -simpl. rewrite <-H1. congruence. -auto. -intro. elim (VertexSet.remove_1 H H0). -Qed. - -Lemma sym_map_merge_map : forall e g m x y, -aff_edge e -> -In_graph_edge e g -> -(forall a b, VertexSet.In a (adj_set b m) -> - VertexSet.In b (adj_set a m)) -> -(forall a b, VertexSet.In a (adj_set b (map_merge e m)) -> - ~Vertex.eq a b) -> -~Vertex.eq x (snd_ext e) -> -~Vertex.eq y (snd_ext e) -> -VertexSet.In x (adj_set y (map_merge e m)) -> -VertexSet.In y (adj_set x (map_merge e m)). - -Proof. -intros e g m x y Haff Hin Hsym Hnoteq Hsndx Hsndy H. -assert (~Vertex.eq x y) as Hdiff. -apply Hnoteq. assumption. -unfold map_merge in *. -set (f := (fun (y : VertexSet.elt) (m' : VertexMap.t VertexSet.t) => - VertexMap.add y - (VertexSet.add (fst_ext e) - (VertexSet.remove (snd_ext e) (adj_set y m))) - m')) in *. -set (s := (VertexSet.union - (VertexSet.remove (snd_ext e) - (adj_set (fst_ext e) m)) - (VertexSet.remove (fst_ext e) - (adj_set (snd_ext e) m)))) in *. -set (s' := adj_set (snd_ext e) m) in *. -cut (forall z, VertexSet.In z s' -> ~Vertex.eq z (fst_ext e) -> VertexSet.In z s). intro Himp. -unfold adj_set. rewrite MapFacts.remove_neq_o. -destruct (Vertex.eq_dec x (fst_ext e)). -rewrite MapFacts.add_eq_o. - -unfold adj_set in H. -rewrite MapFacts.remove_neq_o in H. -destruct (Vertex.eq_dec y (fst_ext e)). -rewrite MapFacts.add_eq_o in H. -rewrite e1. rewrite <-e0. assumption. -intuition. - -rewrite MapFacts.add_neq_o in H. -case_eq (VertexMap.find y (VertexSet.fold f (VertexSet.remove (fst_ext e) s') m)); intros. -rewrite H0 in H. - -rewrite VertexSet.fold_1 in H0. -set (f' := fun a e => f e a) in *. - -assert (eq_set_option (VertexMap.find (elt:=VertexSet.t) y - (fold_left f' (VertexSet.elements (VertexSet.remove (fst_ext e) s')) - m)) (Some t0)). rewrite H0. constructor. apply VertexSet.eq_refl. -generalize H1. clear H0 H1. intro H0. -generalize VertexSet.elements_2. intro. -generalize (H1 (VertexSet.remove (fst_ext e) s')). clear H1. intro HH. -induction (VertexSet.elements (VertexSet.remove (fst_ext e) s')). -unfold s. apply VertexSet.union_2. -apply VertexSet.remove_2. -auto. -apply Hsym. rewrite <-e0. -unfold adj_set. inversion H0. subst. rewrite H3. assumption. - -cut (EqualSetMap (fold_left f' (a :: l) m) (f' (fold_left f' l m) a)). intro. -generalize (H1 y). clear H1. intro H1. simpl in H0. inversion H1. -inversion H0. congruence. -set (tmp := fold_left f' l m) in *. -unfold f' in H3. unfold f in H3. -destruct (Vertex.eq_dec y a). -rewrite e1. -apply Himp. -apply VertexSet.remove_3 with (x:= fst_ext e). -apply HH. left. intuition. -assert (VertexSet.In a (VertexSet.remove (fst_ext e) s')). -apply HH. left. intuition. -intro. elim (VertexSet.remove_1 (Vertex.eq_sym H6) H5). -rewrite MapFacts.add_neq_o in H3. clear H1. -apply IHl. -inversion H0. subst. -rewrite <-H1 in H2. clear H1. inversion H2. subst. clear H2. -rewrite <-H3. constructor. -rewrite <-H4. assumption. - -intros. apply HH. right. assumption. -auto. - -apply fold_left_assoc_map. -unfold EqualSetMap. unfold f'. unfold f. -intros. -destruct (Vertex.eq_dec x0 z). -rewrite MapFacts.add_eq_o. -destruct (Vertex.eq_dec x0 y0). -rewrite MapFacts.add_eq_o. -constructor. -unfold adj_set. -rewrite (MapFacts.find_o _ (Vertex.eq_sym e1)). -rewrite (MapFacts.find_o _ (Vertex.eq_sym e2)). -apply VertexSet.eq_refl. -intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_eq_o. -constructor. apply VertexSet.eq_refl. -intuition. -auto. -intuition. -rewrite MapFacts.add_neq_o. -destruct (Vertex.eq_dec x0 y0). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. apply VertexSet.eq_refl. -intuition. -intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -destruct (VertexMap.find x0 s0); constructor; apply VertexSet.eq_refl. -auto. -auto. -auto. -auto. - -unfold EqualSetMap. intros. -unfold f'. unfold f. -destruct (Vertex.eq_dec x0 a0). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. -apply VertexSet.eq_refl. -intuition. -intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -apply H1. auto. auto. - -rewrite H0 in H. -elim (VertexSet.empty_1 H). -auto. -auto. -intuition. - -rewrite MapFacts.add_neq_o. -unfold adj_set in H. -rewrite MapFacts.remove_neq_o in H. -destruct (Vertex.eq_dec y (fst_ext e)). -rewrite MapFacts.add_eq_o in H. - -rewrite VertexSet.fold_1. -generalize VertexSet.elements_1. intro H0. -generalize (H0 (VertexSet.remove (fst_ext e) s') x). clear H0. intro H0. -induction (VertexSet.elements (VertexSet.remove (fst_ext e) s')). simpl. -unfold s in H. -destruct (VertexSet.union_1 H). -generalize (VertexSet.remove_3 H1). clear H1. intro H1. -generalize (Hsym _ _ H1). clear H1. intro H1. -rewrite <-e0 in H1. assumption. -generalize (VertexSet.remove_3 H1). clear H1. intro H1. -assert (VertexSet.In x (VertexSet.remove (fst_ext e) s')) by - (apply VertexSet.remove_2; auto). -generalize (H0 H2). intro H3. inversion H3. -set (f' := fun a e => f e a) in *. - -cut (EqualSetMap (fold_left f' (a :: l) m) (f' (fold_left f' l m) a)). intro. -generalize (H1 x). clear H1. intro H1. simpl. inversion H1. -set (tmp := fold_left f' l m) in *. -unfold f' in H4. unfold f in H4. -destruct (Vertex.eq_dec x a). -rewrite MapFacts.add_eq_o in H4. congruence. -intuition. -rewrite MapFacts.add_neq_o in H4. rewrite <-H4 in IHl. -apply IHl. intro. -generalize (H0 H2). clear H2. intro H2. -inversion H2; subst. -elim (n0 H6). -assumption. -auto. - -set (tmp := fold_left f' l m) in *. -unfold f' in H3. unfold f in H3. -destruct (Vertex.eq_dec x a). -rewrite MapFacts.add_eq_o in H3. -inversion H3. subst. clear H3. clear H1. -rewrite H4. apply VertexSet.add_1. intuition. intuition. -rewrite MapFacts.add_neq_o in H3. rewrite <-H3 in IHl. -rewrite H4. apply IHl. intro. -generalize (H0 H5). clear H5. intro H5. -inversion H5; subst. -elim (n0 H7). -assumption. -auto. - -apply fold_left_assoc_map. -unfold EqualSetMap. unfold f'. unfold f. -intros. -destruct (Vertex.eq_dec x0 z). -rewrite MapFacts.add_eq_o. -destruct (Vertex.eq_dec x0 y0). -rewrite MapFacts.add_eq_o. -constructor. -unfold adj_set. -rewrite (MapFacts.find_o _ (Vertex.eq_sym e1)). -rewrite (MapFacts.find_o _ (Vertex.eq_sym e2)). -apply VertexSet.eq_refl. -intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_eq_o. -constructor. apply VertexSet.eq_refl. -intuition. -auto. -intuition. -rewrite MapFacts.add_neq_o. -destruct (Vertex.eq_dec x0 y0). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. apply VertexSet.eq_refl. -intuition. -intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -destruct (VertexMap.find x0 s0); constructor; apply VertexSet.eq_refl. -auto. -auto. -auto. -auto. - -unfold EqualSetMap. intros. -unfold f'. unfold f. -destruct (Vertex.eq_dec x0 a0). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. -apply VertexSet.eq_refl. -intuition. -intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -apply H1. auto. auto. - -intuition. - -rewrite MapFacts.add_neq_o in H. -rewrite VertexSet.fold_1 in H. rewrite VertexSet.fold_1. -set (f' := fun a e => f e a) in *. -induction (VertexSet.elements (VertexSet.remove (fst_ext e) s')). simpl in H. simpl. -apply (Hsym _ _ H). - -cut (EqualSetMap (fold_left f' (a :: l) m) (f' (fold_left f' l m) a)). intro. -generalize (H0 x). generalize (H0 y). clear H0. intros H0 HH0. simpl. simpl in H. inversion H0. -rewrite <-H2 in H. elim (VertexSet.empty_1 H). -set (tmp := fold_left f' l m) in *. -unfold f' in H2. unfold f in H2. -destruct (Vertex.eq_dec y a). -rewrite MapFacts.add_eq_o in H2. -inversion HH0. -unfold f' in H6. unfold f in H6. -destruct (Vertex.eq_dec x a). -elim Hdiff. apply (Vertex.eq_trans e1 (Vertex.eq_sym e0)). -rewrite MapFacts.add_neq_o in H6. - -cut (VertexMap.In x tmp). -intro H7. rewrite MapFacts.in_find_iff in H7. congruence. -cut (VertexMap.In x m -> VertexMap.In x tmp). -intro H7. apply H7. clear H7. -rewrite MapFacts.in_find_iff. intro. - -rewrite <-H1 in *. rewrite H3 in H. clear H0 HH0 H1 H5 H6 IHl. -inversion H2. subst. clear H2. clear H3. -destruct (proj1 (Props.Dec.F.add_iff _ _ _) H). elim n. auto. -generalize (VertexSet.remove_3 H0). clear H H0. intro H. -generalize (Hsym _ _ H). clear H. intro H. -unfold adj_set in H. rewrite H4 in H. elim (VertexSet.empty_1 H). - -clear IHl H H0 HH0 H1 H2 H3 H5 H6. intro. -unfold tmp. -induction l. simpl. assumption. - -cut (EqualSetMap (fold_left f' (a0 :: l) m) (f' (fold_left f' l m) a0)). intro. -generalize (H0 x). clear H0. simpl. intro H0. inversion H0. -set (tmp' := fold_left f' l m) in *. -unfold f' in H3. unfold f in H3. -destruct (Vertex.eq_dec x a0). -rewrite MapFacts.add_eq_o in H3. congruence. intuition. -rewrite MapFacts.add_neq_o in H3. -rewrite MapFacts.in_find_iff in IHl. congruence. intuition. -rewrite MapFacts.in_find_iff. congruence. - -apply fold_left_assoc_map. -unfold EqualSetMap. unfold f'. unfold f. -intros. -destruct (Vertex.eq_dec x0 z). -rewrite MapFacts.add_eq_o. -destruct (Vertex.eq_dec x0 y0). -rewrite MapFacts.add_eq_o. -constructor. -unfold adj_set. -rewrite (MapFacts.find_o _ (Vertex.eq_sym e1)). -rewrite (MapFacts.find_o _ (Vertex.eq_sym e2)). -apply VertexSet.eq_refl. -intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_eq_o. -constructor. apply VertexSet.eq_refl. -intuition. -auto. -intuition. -rewrite MapFacts.add_neq_o. -destruct (Vertex.eq_dec x0 y0). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. apply VertexSet.eq_refl. -intuition. -intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -destruct (VertexMap.find x0 s1); constructor; apply VertexSet.eq_refl. -auto. -auto. -auto. -auto. - -unfold EqualSetMap. intros. -unfold f'. unfold f. -destruct (Vertex.eq_dec x0 a1). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. -apply VertexSet.eq_refl. -intuition. -intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -apply H0. auto. auto. - -auto. -clear H0 HH0. -rewrite <-H1 in H. -unfold f' in H5. unfold f in H5. -destruct (Vertex.eq_dec x a). -elim Hdiff. apply Vertex.eq_trans with (y:=a); auto. -rewrite MapFacts.add_neq_o in H5. -rewrite <-H5 in *. -inversion H2. subst. clear H2. -rewrite H3 in H. -destruct (proj1 (Props.Dec.F.add_iff _ _ _) H). elim n. auto. -generalize (VertexSet.remove_3 H0). clear H H0. intro H. - -case_eq (VertexMap.find y tmp); intros. -rewrite H0 in IHl. -rewrite H6. apply IHl. - -clear IHl H1 H6 H4 H3 H5. -unfold tmp in H0. -unfold adj_set in H. rewrite (MapFacts.find_o _ (Vertex.eq_sym e0)) in H. -assert (eq_set_option (VertexMap.find y (fold_left f' l m)) (Some t0)). -rewrite H0. constructor. apply VertexSet.eq_refl. -generalize H1. clear H0 H1. intro H0. -induction l. simpl in H0. inversion H0. subst. clear H0. rewrite <-H1 in H. rewrite <-H3. assumption. -cut (EqualSetMap (fold_left f' (a0 :: l) m) (f' (fold_left f' l m) a0)). intro. -generalize (H1 y). clear H1. simpl in H0. intro H1. inversion H1. clear H1. -set (tmp' := fold_left f' l m) in *. -unfold f' in H4. unfold f in H4. -destruct (Vertex.eq_dec y a0). -rewrite MapFacts.add_eq_o in H4. congruence. intuition. -rewrite MapFacts.add_neq_o in H4. - -case_eq (VertexMap.find y m); intros. -clear H. - -assert (VertexMap.In y tmp'). -clear H3 H4 IHl H0. -unfold tmp'. induction l. simpl. -rewrite MapFacts.in_find_iff. congruence. - -cut (EqualSetMap (fold_left f' (a1 :: l) m) (f' (fold_left f' l m) a1)). intro. -generalize (H y). clear H. intro H. simpl. inversion H. clear H. -set (tmp'' := fold_left f' l m) in *. -unfold f' in H3. unfold f in H3. -destruct (Vertex.eq_dec y a1). -rewrite MapFacts.add_eq_o in H3. congruence. intuition. -rewrite MapFacts.add_neq_o in H3. -rewrite MapFacts.in_find_iff in IHl. rewrite <-H3 in IHl. congruence. -auto. -rewrite MapFacts.in_find_iff. rewrite <-H0. congruence. - -apply fold_left_assoc_map. -unfold EqualSetMap. unfold f'. unfold f. -intros. -destruct (Vertex.eq_dec x0 z). -rewrite MapFacts.add_eq_o. -destruct (Vertex.eq_dec x0 y0). -rewrite MapFacts.add_eq_o. -constructor. -unfold adj_set. -rewrite (MapFacts.find_o _ (Vertex.eq_sym e1)). -rewrite (MapFacts.find_o _ (Vertex.eq_sym e2)). -apply VertexSet.eq_refl. -intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_eq_o. -constructor. apply VertexSet.eq_refl. -intuition. -auto. -intuition. -rewrite MapFacts.add_neq_o. -destruct (Vertex.eq_dec x0 y0). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. apply VertexSet.eq_refl. -intuition. -intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -destruct (VertexMap.find x0 s2); constructor; apply VertexSet.eq_refl. -auto. -auto. -auto. -auto. - -unfold EqualSetMap. intros. -unfold f'. unfold f. -destruct (Vertex.eq_dec x0 a2). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. -apply VertexSet.eq_refl. -intuition. -intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -apply H. auto. auto. - -rewrite MapFacts.in_find_iff in H. congruence. -rewrite H1 in H. elim (VertexSet.empty_1 H). -auto. - -rewrite <-H2 in *. inversion H0. subst. clear H0 H1. - -set (tmp' := fold_left f' l m) in *. -unfold f' in H3. unfold f in H3. -destruct (Vertex.eq_dec y a0). rewrite MapFacts.add_eq_o in H3. -inversion H3. subst. clear H3. -case_eq (VertexMap.find y m); intros. -rewrite H0 in H. -rewrite <-H7. rewrite H4. -apply VertexSet.add_2. -apply VertexSet.remove_2. -auto. -unfold adj_set. rewrite (MapFacts.find_o _ (Vertex.eq_sym e1)). -rewrite H0. assumption. -rewrite H0 in H. elim (VertexSet.empty_1 H). -intuition. -rewrite MapFacts.add_neq_o in H3. - -apply IHl. -rewrite <-H3. constructor. -rewrite <-H4. rewrite <-H7. apply VertexSet.eq_refl. -auto. - -apply fold_left_assoc_map. -unfold EqualSetMap. unfold f'. unfold f. -intros. -destruct (Vertex.eq_dec x0 z). -rewrite MapFacts.add_eq_o. -destruct (Vertex.eq_dec x0 y0). -rewrite MapFacts.add_eq_o. -constructor. -unfold adj_set. -rewrite (MapFacts.find_o _ (Vertex.eq_sym e1)). -rewrite (MapFacts.find_o _ (Vertex.eq_sym e2)). -apply VertexSet.eq_refl. -intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_eq_o. -constructor. apply VertexSet.eq_refl. -intuition. -auto. -intuition. -rewrite MapFacts.add_neq_o. -destruct (Vertex.eq_dec x0 y0). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. apply VertexSet.eq_refl. -intuition. -intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -destruct (VertexMap.find x0 s2); constructor; apply VertexSet.eq_refl. -auto. -auto. -auto. -auto. - -unfold EqualSetMap. intros. -unfold f'. unfold f. -destruct (Vertex.eq_dec x0 a1). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. -apply VertexSet.eq_refl. -intuition. -intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -apply H1. auto. auto. - -unfold adj_set in H. rewrite (MapFacts.find_o _ (Vertex.eq_sym e0)) in H. -clear IHl H1 H6 H4 H5 H3. -unfold tmp in H0. -induction l. simpl in H0. -rewrite H0 in H. elim (VertexSet.empty_1 H). - -cut (EqualSetMap (fold_left f' (a0 :: l) m) (f' (fold_left f' l m) a0)). intro. -generalize (H1 y). clear H1. intro H1. simpl in H0. inversion H1. -set (tmp' := fold_left f' l m) in *. -unfold f' in H4. unfold f in H4. -destruct (Vertex.eq_dec y a0). -rewrite MapFacts.add_eq_o in H4. congruence. intuition. -rewrite MapFacts.add_neq_o in H4. -apply IHl. auto. auto. - -congruence. - -apply fold_left_assoc_map. -unfold EqualSetMap. unfold f'. unfold f. -intros. -destruct (Vertex.eq_dec x0 z). -rewrite MapFacts.add_eq_o. -destruct (Vertex.eq_dec x0 y0). -rewrite MapFacts.add_eq_o. -constructor. -unfold adj_set. -rewrite (MapFacts.find_o _ (Vertex.eq_sym e1)). -rewrite (MapFacts.find_o _ (Vertex.eq_sym e2)). -apply VertexSet.eq_refl. -intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_eq_o. -constructor. apply VertexSet.eq_refl. -intuition. -auto. -intuition. -rewrite MapFacts.add_neq_o. -destruct (Vertex.eq_dec x0 y0). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. apply VertexSet.eq_refl. -intuition. -intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -destruct (VertexMap.find x0 s2); constructor; apply VertexSet.eq_refl. -auto. -auto. -auto. -auto. - -unfold EqualSetMap. intros. -unfold f'. unfold f. -destruct (Vertex.eq_dec x0 a1). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. -apply VertexSet.eq_refl. -intuition. -intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -apply H1. auto. auto. - -auto. -intuition. -rewrite MapFacts.add_neq_o in H2. -rewrite <-H2 in *. rewrite <-H1 in *. - -inversion HH0. -unfold f' in H6. unfold f in H6. -destruct (Vertex.eq_dec x a). -rewrite MapFacts.add_eq_o in H6. congruence. intuition. -rewrite MapFacts.add_neq_o in H6. -rewrite <-H6 in IHl. -apply IHl. rewrite <-H3. assumption. -auto. - -unfold f' in H5. unfold f in H5. -destruct (Vertex.eq_dec x a). -rewrite MapFacts.add_eq_o in H5. -clear H0 HH0. - -inversion H5. subst. clear H5. -rewrite H6. -apply VertexSet.add_2. -apply VertexSet.remove_2. -auto. -apply Hsym. -rewrite <-e0. -rewrite H3 in H. clear IHl H1 H4 H6. -unfold tmp in H2. -assert (eq_set_option (Some s'0) (VertexMap.find y (fold_left f' l m))). -rewrite <-H2. constructor. apply VertexSet.eq_refl. -generalize H0. clear H0 H2. intro H2. -induction l. simpl in H2. -unfold adj_set. inversion H2. subst. rewrite <-H4. assumption. - -cut (EqualSetMap (fold_left f' (a0 :: l) m) (f' (fold_left f' l m) a0)). intro. -generalize (H0 y). clear H0. simpl in H2. intro H0. inversion H0. clear H0. -rewrite <-H4 in H2. inversion H2. -set (tmp' := fold_left f' l m) in *. -unfold f' in H4. unfold f in H4. -destruct (Vertex.eq_dec y a0). -rewrite MapFacts.add_eq_o in H4. inversion H4. subst. clear H4. -inversion H2. subst. rewrite <-H6 in H1. inversion H1. subst. -apply VertexSet.remove_3 with (x:=snd_ext e). -apply VertexSet.add_3 with (x:= fst_ext e). -auto. -unfold adj_set. rewrite (MapFacts.find_o _ e1). rewrite <-H5. rewrite <-H7. assumption. -intuition. -rewrite MapFacts.add_neq_o in H4. -apply IHl. -rewrite <-H4. constructor. rewrite <-H1 in H2. inversion H2. subst. -rewrite H8. assumption. -auto. - -apply fold_left_assoc_map. -unfold EqualSetMap. unfold f'. unfold f. -intros. -destruct (Vertex.eq_dec x0 z). -rewrite MapFacts.add_eq_o. -destruct (Vertex.eq_dec x0 y0). -rewrite MapFacts.add_eq_o. -constructor. -unfold adj_set. -rewrite (MapFacts.find_o _ (Vertex.eq_sym e1)). -rewrite (MapFacts.find_o _ (Vertex.eq_sym e2)). -apply VertexSet.eq_refl. intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_eq_o. -constructor. apply VertexSet.eq_refl. -intuition. -auto. -intuition. -rewrite MapFacts.add_neq_o. -destruct (Vertex.eq_dec x0 y0). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. apply VertexSet.eq_refl. -intuition. -intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -destruct (VertexMap.find x0 s2); constructor; apply VertexSet.eq_refl. -auto. -auto. -auto. -auto. - -unfold EqualSetMap. intros. -unfold f'. unfold f. -destruct (Vertex.eq_dec x0 a1). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. -apply VertexSet.eq_refl. -intuition. -intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -apply H0. auto. auto. -intuition. -rewrite MapFacts.add_neq_o in H5. -rewrite <-H5 in *. -rewrite H6. apply IHl. -rewrite <-H3. assumption. -auto. -auto. - -apply fold_left_assoc_map. -unfold EqualSetMap. unfold f'. unfold f. -intros. -destruct (Vertex.eq_dec x0 z). -rewrite MapFacts.add_eq_o. -destruct (Vertex.eq_dec x0 y0). -rewrite MapFacts.add_eq_o. -constructor. -unfold adj_set. -rewrite (MapFacts.find_o _ (Vertex.eq_sym e0)). -rewrite (MapFacts.find_o _ (Vertex.eq_sym e1)). -apply VertexSet.eq_refl. -intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_eq_o. -constructor. apply VertexSet.eq_refl. -intuition. -auto. -intuition. -rewrite MapFacts.add_neq_o. -destruct (Vertex.eq_dec x0 y0). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. apply VertexSet.eq_refl. -intuition. -intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -destruct (VertexMap.find x0 s0); constructor; apply VertexSet.eq_refl. -auto. -auto. -auto. -auto. - -unfold EqualSetMap. intros. -unfold f'. unfold f. -destruct (Vertex.eq_dec x0 a0). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. -apply VertexSet.eq_refl. -intuition. -intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -apply H0. auto. auto. - -auto. -auto. -auto. -auto. - -clear H. -unfold s. unfold s'. intros z H Hne. -apply VertexSet.union_3. apply VertexSet.remove_2; auto. -Qed. - -Lemma not_eq_extremities_merge_map : forall e g, -aff_edge e -> -forall x y, -VertexSet.In x (adj_set y (imap_merge e g)) \/ -VertexSet.In x (adj_set y (pmap_merge e g (imap_merge e g))) -> -~Vertex.eq x y. - -Proof. -intros. destruct H0. -apply (not_eq_extremities_map_merge x y e (imap g)). -intros. apply (not_eq_extremities g). left. assumption. assumption. -apply (not_eq_extremities_map_merge x y e (pmap g)). -intros. apply (not_eq_extremities g). right. assumption. -apply pmap_merge_sub. assumption. -Qed. - -Lemma sym_merge : forall e g, -aff_edge e -> -In_graph_edge e g -> -forall x y, -VertexSet.In x (adj_set y (map_merge e (pmap g))) -> -VertexSet.In y (adj_set x (map_merge e (pmap g))). - -Proof. -intros e g Haff Hin x y H. -apply sym_map_merge_map with (g:=g); auto. -apply (sym_pmap g). -intros. apply (not_eq_extremities_map_merge _ _ e (pmap g)). -intros. apply (not_eq_extremities g). right. auto. -assumption. - -assert (~Vertex.eq y (snd_ext e)) as Hy. -intro. -generalize (extremities_in_merge_map e g Hin y). -rewrite MapFacts.in_find_iff. intro. -unfold adj_set in H. -case_eq (VertexMap.find y (map_merge e (pmap g))); intros. -rewrite H2 in H1. -assert (VertexSet.In y (VertexSet.remove (snd_ext e) (V g))). -rewrite <-H1. congruence. -elim (VertexSet.remove_1 (Vertex.eq_sym H0) H3). -rewrite H2 in H. elim (VertexSet.empty_1 H). - -intro. -unfold map_merge in H. - -set (f := (fun (y : VertexSet.elt) (m' : VertexMap.t VertexSet.t) => - VertexMap.add y - (VertexSet.add (fst_ext e) - (VertexSet.remove (snd_ext e) (adj_set y (pmap g)))) - m')) in *. -set (s := (VertexSet.union - (VertexSet.remove (snd_ext e) - (adj_set (fst_ext e) (pmap g))) - (VertexSet.remove (fst_ext e) - (adj_set (snd_ext e) (pmap g))))) in *. -set (s' := adj_set (snd_ext e) (pmap g)) in *. - -unfold adj_set in H. -rewrite MapFacts.remove_neq_o in H. -destruct (Vertex.eq_dec y (fst_ext e)). -rewrite MapFacts.add_eq_o in H. -unfold s in H. -destruct (VertexSet.union_1 H). -elim (VertexSet.remove_1 (Vertex.eq_sym H0) H1). -generalize (VertexSet.remove_3 H1). clear H1. intro H1. -unfold s' in H1. - -elim (not_eq_extremities g x (snd_ext e)). -right. assumption. -assumption. -intuition. -rewrite MapFacts.add_neq_o in H. - -rewrite VertexSet.fold_1 in H. -set (f' := fun a e => f e a) in *. -generalize VertexSet.elements_1. intro. -generalize (H1 (VertexSet.remove (fst_ext e) s') y). clear H1. intro H1. -induction (VertexSet.elements (VertexSet.remove (fst_ext e) s')). -simpl in H. -unfold s' in H1. -assert (InA Vertex.eq y nil). -apply H1. -apply VertexSet.remove_2. auto. -apply (sym_pmap g). rewrite <-H0. assumption. inversion H2. - -cut (EqualSetMap (fold_left f' (a :: l) (pmap g)) (f' (fold_left f' l (pmap g)) a)). intro. -generalize (H2 y). clear H2. intro H2. simpl in H. inversion H2; clear H2. -rewrite <-H4 in *. -elim (VertexSet.empty_1 H). -set (tmp := fold_left f' l (pmap g)) in *. -unfold f' in H4. unfold f in H4. -destruct (Vertex.eq_dec y a). -rewrite MapFacts.add_eq_o in H4. -rewrite <-H3 in *. clear H3. inversion H4. subst. clear H4. -rewrite H5 in H. clear H5. -destruct (proj1 (Props.Dec.F.add_iff _ _ _) H). -elim (In_graph_edge_diff_ext _ _ Hin). -apply Vertex.eq_trans with (y := x); auto. -elim (VertexSet.remove_1 (Vertex.eq_sym H0) H2). -intuition. -rewrite MapFacts.add_neq_o in H4. rewrite <-H4 in IHl. -apply IHl. -rewrite <-H5. rewrite <-H3 in H. assumption. -intro. generalize (H1 H2). clear H2. intro H2. -inversion H2; subst. -elim (n0 H7). -assumption. -auto. - -apply fold_left_assoc_map. -unfold EqualSetMap. unfold f'. unfold f. -intros. -destruct (Vertex.eq_dec x0 z). -rewrite MapFacts.add_eq_o. -destruct (Vertex.eq_dec x0 y0). -rewrite MapFacts.add_eq_o. -constructor. -unfold adj_set. -rewrite (MapFacts.find_o _ (Vertex.eq_sym e0)). -rewrite (MapFacts.find_o _ (Vertex.eq_sym e1)). -apply VertexSet.eq_refl. -intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_eq_o. -constructor. apply VertexSet.eq_refl. -intuition. -auto. -intuition. -rewrite MapFacts.add_neq_o. -destruct (Vertex.eq_dec x0 y0). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. apply VertexSet.eq_refl. -intuition. -intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -destruct (VertexMap.find x0 s0); constructor; apply VertexSet.eq_refl. -auto. -auto. -auto. -auto. - -unfold EqualSetMap. intros. -unfold f'. unfold f. -destruct (Vertex.eq_dec x0 a0). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. -apply VertexSet.eq_refl. -intuition. -intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -apply H2. auto. auto. - -auto. -auto. - -intro. -generalize (extremities_in_merge_map e g Hin y). -rewrite MapFacts.in_find_iff. intro. -unfold adj_set in H. -case_eq (VertexMap.find y (map_merge e (pmap g))); intros. -rewrite H2 in H1. -assert (VertexSet.In y (VertexSet.remove (snd_ext e) (V g))). -rewrite <-H1. congruence. -elim (VertexSet.remove_1 (Vertex.eq_sym H0) H3). -rewrite H2 in H. elim (VertexSet.empty_1 H). -Qed. - -Lemma sym_imap_merge_map : forall e g, -aff_edge e -> -In_graph_edge e g -> -forall x y, -VertexSet.In x (adj_set y (imap_merge e g)) -> -VertexSet.In y (adj_set x (imap_merge e g)). - -Proof. -intros e g Haff Hin x y H. -apply sym_map_merge_map with (g:=g); auto. -apply (sym_imap g). -intros. apply (not_eq_extremities_merge_map e g Haff). left. assumption. - -assert (~Vertex.eq y (snd_ext e)) as Hy. -intro. -generalize (extremities_in_merge_imap e g Hin y). -rewrite MapFacts.in_find_iff. intro. -unfold adj_set in H. -case_eq (VertexMap.find y (imap_merge e g)); intros. -rewrite H2 in H1. -assert (VertexSet.In y (VertexSet.remove (snd_ext e) (V g))). -rewrite <-H1. congruence. -elim (VertexSet.remove_1 (Vertex.eq_sym H0) H3). -rewrite H2 in H. elim (VertexSet.empty_1 H). - -intro. -unfold imap_merge in H. -unfold map_merge in H. - -set (f := (fun (y : VertexSet.elt) (m' : VertexMap.t VertexSet.t) => - VertexMap.add y - (VertexSet.add (fst_ext e) - (VertexSet.remove (snd_ext e) (adj_set y (imap g)))) - m')) in *. -set (s := (VertexSet.union - (VertexSet.remove (snd_ext e) - (adj_set (fst_ext e) (imap g))) - (VertexSet.remove (fst_ext e) - (adj_set (snd_ext e) (imap g))))) in *. -set (s' := adj_set (snd_ext e) (imap g)) in *. - -unfold adj_set in H. -rewrite MapFacts.remove_neq_o in H. -destruct (Vertex.eq_dec y (fst_ext e)). -rewrite MapFacts.add_eq_o in H. -unfold s in H. -destruct (VertexSet.union_1 H). -elim (VertexSet.remove_1 (Vertex.eq_sym H0) H1). -generalize (VertexSet.remove_3 H1). clear H1. intro H1. -unfold s' in H1. - -elim (not_eq_extremities g x (snd_ext e)). -left. assumption. -assumption. -intuition. -rewrite MapFacts.add_neq_o in H. - -rewrite VertexSet.fold_1 in H. -set (f' := fun a e => f e a) in *. -generalize VertexSet.elements_1. intro. -generalize (H1 (VertexSet.remove (fst_ext e) s') y). clear H1. intro H1. -induction (VertexSet.elements (VertexSet.remove (fst_ext e) s')). -simpl in H. -unfold s' in H1. -assert (InA Vertex.eq y nil). -apply H1. -apply VertexSet.remove_2. auto. -apply (sym_imap g). rewrite <-H0. assumption. inversion H2. - -cut (EqualSetMap (fold_left f' (a :: l) (imap g)) (f' (fold_left f' l (imap g)) a)). intro. -generalize (H2 y). clear H2. intro H2. simpl in H. inversion H2; clear H2. -rewrite <-H4 in *. -elim (VertexSet.empty_1 H). -set (tmp := fold_left f' l (imap g)) in *. -unfold f' in H4. unfold f in H4. -destruct (Vertex.eq_dec y a). -rewrite MapFacts.add_eq_o in H4. -rewrite <-H3 in *. clear H3. inversion H4. subst. clear H4. -rewrite H5 in H. clear H5. -destruct (proj1 (Props.Dec.F.add_iff _ _ _) H). -elim (In_graph_edge_diff_ext _ _ Hin). -apply Vertex.eq_trans with (y := x); auto. -elim (VertexSet.remove_1 (Vertex.eq_sym H0) H2). -intuition. -rewrite MapFacts.add_neq_o in H4. rewrite <-H4 in IHl. -apply IHl. -rewrite <-H5. rewrite <-H3 in H. assumption. -intro. generalize (H1 H2). clear H2. intro H2. -inversion H2; subst. -elim (n0 H7). -assumption. -auto. - -apply fold_left_assoc_map. -unfold EqualSetMap. unfold f'. unfold f. -intros. -destruct (Vertex.eq_dec x0 z). -rewrite MapFacts.add_eq_o. -destruct (Vertex.eq_dec x0 y0). -rewrite MapFacts.add_eq_o. -constructor. -unfold adj_set. -rewrite (MapFacts.find_o _ (Vertex.eq_sym e0)). -rewrite (MapFacts.find_o _ (Vertex.eq_sym e1)). -apply VertexSet.eq_refl. -intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_eq_o. -constructor. apply VertexSet.eq_refl. -intuition. -auto. -intuition. -rewrite MapFacts.add_neq_o. -destruct (Vertex.eq_dec x0 y0). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. apply VertexSet.eq_refl. -intuition. -intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -destruct (VertexMap.find x0 s0); constructor; apply VertexSet.eq_refl. -auto. -auto. -auto. -auto. - -unfold EqualSetMap. intros. -unfold f'. unfold f. -destruct (Vertex.eq_dec x0 a0). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. -apply VertexSet.eq_refl. -intuition. -intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -apply H2. auto. auto. - -auto. -auto. - -intro. -generalize (extremities_in_merge_imap e g Hin y). -rewrite MapFacts.in_find_iff. intro. -unfold adj_set in H. -case_eq (VertexMap.find y (imap_merge e g)); intros. -rewrite H2 in H1. -assert (VertexSet.In y (VertexSet.remove (snd_ext e) (V g))). -rewrite <-H1. congruence. -elim (VertexSet.remove_1 (Vertex.eq_sym H0) H3). -rewrite H2 in H. elim (VertexSet.empty_1 H). -Qed. - -Lemma pmap_merge_in_merge : forall x e g, -aff_edge e -> -In_graph_edge e g -> -VertexMap.In x (pmap_merge e g (imap_merge e g)) -> -VertexMap.In x (map_merge e (pmap g)). - -Proof. -intros x e g Haff HH H. -unfold pmap_merge in H. -unfold resolve_conflicts in H. -set (f := (fun (x : VertexSet.elt) (m : VertexMap.t VertexSet.t) => - VertexMap.add x - (VertexSet.remove (fst_ext e) - (adj_set x (map_merge e (pmap g)))) m)) in *. -rewrite MapFacts.in_find_iff in *. intro H0. -destruct (Vertex.eq_dec x (fst_ext e)). -rewrite (MapFacts.find_o _ e0) in H0. -assert (VertexMap.In (fst_ext e) (map_merge e (pmap g))). -apply pmap_merge_domain_2. assumption. -apply VertexSet.remove_2. apply (In_graph_edge_diff_ext _ _ HH). -apply (proj1 (In_graph_edge_in_ext _ _ HH)). -rewrite MapFacts.in_find_iff in H1. congruence. -rewrite MapFacts.add_neq_o in H. -set (s' := (VertexSet.inter (adj_set (fst_ext e) (map_merge e (pmap g))) - (adj_set (fst_ext e) (imap_merge e g)))) in *. -set (m := map_merge e (pmap g)) in *. -rewrite VertexSet.fold_1 in H. -set (f' := fun a e => f e a) in *. -assert (forall z, InA Vertex.eq z (VertexSet.elements s') -> - VertexMap.In z m) as Hin. -intros. -apply pmap_merge_domain_2. assumption. -generalize (VertexSet.elements_2 H1). clear H1. intro H1. -unfold s' in H1. generalize (VertexSet.inter_1 H1). -generalize (VertexSet.inter_2 H1). clear H1. intros. -rewrite <-(extremities_in_merge_imap e g HH). -generalize (sym_imap_merge_map e g Haff HH _ _ H1). intro. -rewrite MapFacts.in_find_iff. intro. -unfold adj_set in H3. rewrite H4 in H3. elim (VertexSet.empty_1 H3). -induction (VertexSet.elements s'). simpl in H. congruence. -cut (EqualSetMap (fold_left f' (a :: l) m) (f' (fold_left f' l m) a)). intro. -generalize (H1 x). clear H1. simpl in H. intro H1. inversion H1; clear H1. -congruence. -set (tmp := fold_left f' l m) in *. -unfold f' in H3. unfold f in H3. -destruct (Vertex.eq_dec x a). -rewrite MapFacts.add_eq_o in H3. inversion H3; subst; clear H3. -rewrite <-H2 in H. -rewrite (MapFacts.find_o _ e0) in H0. -assert (VertexMap.In a m). -apply Hin. left. auto. -rewrite MapFacts.in_find_iff in H1. congruence. -intuition. -rewrite MapFacts.add_neq_o in H3. -apply IHl. congruence. -intros. apply Hin. right. auto. -auto. - - -apply fold_left_assoc_map. -unfold EqualSetMap. unfold f'. unfold f. -intros. -destruct (Vertex.eq_dec x0 z). -rewrite MapFacts.add_eq_o. -destruct (Vertex.eq_dec x0 y). -rewrite MapFacts.add_eq_o. -constructor. -unfold adj_set. -rewrite (MapFacts.find_o _ (Vertex.eq_sym e0)). -rewrite (MapFacts.find_o _ (Vertex.eq_sym e1)). -apply VertexSet.eq_refl. -intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_eq_o. -constructor. apply VertexSet.eq_refl. -intuition. -auto. -intuition. -rewrite MapFacts.add_neq_o. -destruct (Vertex.eq_dec x0 y). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. apply VertexSet.eq_refl. -intuition. -intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -destruct (VertexMap.find x0 s); constructor; apply VertexSet.eq_refl. -auto. -auto. -auto. -auto. - -unfold EqualSetMap. intros. -unfold f'. unfold f. -destruct (Vertex.eq_dec x0 a0). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. -apply VertexSet.eq_refl. -intuition. -intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -apply H1. auto. auto. -auto. -Qed. - -Lemma pmap_merge_merge_in : forall x e g, -In_graph_edge e g -> -VertexMap.In x (map_merge e (pmap g)) -> -VertexMap.In x (pmap_merge e g (imap_merge e g)). - -Proof. -intros x e g HH H. -unfold pmap_merge. -unfold resolve_conflicts. -set (f := (fun (x0 : VertexSet.elt) (m : VertexMap.t VertexSet.t) => - VertexMap.add x0 - (VertexSet.remove (fst_ext e) (adj_set x0 (map_merge e (pmap g)))) - m)) in *. -set (s := (VertexSet.inter (adj_set (fst_ext e) (map_merge e (pmap g))) - (adj_set (fst_ext e) (imap_merge e g)))) in *. -rewrite MapFacts.in_find_iff. -destruct (Vertex.eq_dec x (fst_ext e)). -rewrite MapFacts.add_eq_o. congruence. -intuition. -rewrite MapFacts.add_neq_o. -intro H0. -rewrite VertexSet.fold_1 in H0. -set (f' := fun a e => f e a) in *. -assert (VertexMap.find x (map_merge e (pmap g)) = None). -induction (VertexSet.elements s). simpl in H0. assumption. -set (m := map_merge e (pmap g)) in *. -cut (EqualSetMap (fold_left f' (a :: l) m) (f' (fold_left f' l m) a)). intro. -generalize (H1 x). clear H1. intro H1. simpl in H0. inversion H1; clear H1. -set (tmp := fold_left f' l m) in *. -unfold f' in H4. unfold f in H4. -destruct (Vertex.eq_dec x a). rewrite MapFacts.add_eq_o in H4. congruence. -intuition. -rewrite MapFacts.add_neq_o in H4. -apply IHl. auto. -auto. - -congruence. -apply fold_left_assoc_map. -unfold EqualSetMap. unfold f'. unfold f. -intros. -destruct (Vertex.eq_dec x0 z). -rewrite MapFacts.add_eq_o. -destruct (Vertex.eq_dec x0 y). -rewrite MapFacts.add_eq_o. -constructor. -unfold adj_set. -rewrite (MapFacts.find_o _ (Vertex.eq_sym e0)). -rewrite (MapFacts.find_o _ (Vertex.eq_sym e1)). -apply VertexSet.eq_refl. -intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_eq_o. -constructor. apply VertexSet.eq_refl. -intuition. -auto. -intuition. -rewrite MapFacts.add_neq_o. -destruct (Vertex.eq_dec x0 y). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. apply VertexSet.eq_refl. -intuition. -intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -destruct (VertexMap.find x0 s0); constructor; apply VertexSet.eq_refl. -auto. -auto. -auto. -auto. - -unfold EqualSetMap. intros. -unfold f'. unfold f. -destruct (Vertex.eq_dec x0 a0). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. -apply VertexSet.eq_refl. -intuition. -intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -apply H1. auto. auto. -auto. - -rewrite MapFacts.in_find_iff in H. congruence. -auto. -Qed. - -Lemma extremities_in_merge_pmap e g : -aff_edge e -> -In_graph_edge e g -> -(forall x, -VertexMap.In x (pmap_merge e g (imap_merge e g)) <-> -VertexSet.In x (VertexSet.remove (snd_ext e) (V g))). - -Proof. -intros e g Haff HH x;split; intro H0. -apply pmap_merge_domain_1. assumption. -apply pmap_merge_in_merge. assumption. -assumption. assumption. -apply pmap_merge_merge_in. assumption. -apply pmap_merge_domain_2. assumption. -assumption. -Qed. - -Lemma merge_conflicts_aux_1 : forall e g, -aff_edge e -> -In_graph_edge e g -> -forall x y, -VertexSet.In x (adj_set y (imap_merge e g)) -> -~Vertex.eq y (fst_ext e) -> -~Vertex.eq x (fst_ext e) -> -VertexSet.In x (adj_set y (imap g)). - -Proof. -intros. -unfold imap_merge in H1. -unfold map_merge in H1. -set (m := imap g) in *. -set (f := fun (y : VertexSet.elt) (m' : VertexMap.t VertexSet.t) => - VertexMap.add y - (VertexSet.add (fst_ext e) - (VertexSet.remove (snd_ext e) (adj_set y m))) - m') in *. -set (s := (VertexSet.union - (VertexSet.remove (snd_ext e) - (adj_set (fst_ext e) m)) - (VertexSet.remove (fst_ext e) - (adj_set (snd_ext e) m)))) in *. -set (s' := (adj_set (snd_ext e) m)) in *. -unfold adj_set in H1. -destruct (Vertex.eq_dec y (snd_ext e)). -rewrite MapFacts.remove_eq_o in H1. elim (VertexSet.empty_1 H1). -intuition. -rewrite MapFacts.remove_neq_o in H1. rewrite MapFacts.add_neq_o in H1. -rewrite VertexSet.fold_1 in H1. set (f' := fun a e => f e a) in *. -induction (VertexSet.elements (VertexSet.remove (fst_ext e) s')). simpl in H1. -assumption. -cut (EqualSetMap (fold_left f' (a :: l) m) (f' (fold_left f' l m) a)). intro. -generalize (H4 y). clear H4. intro H4. simpl in H1. inversion H4; subst; clear H4. -rewrite <-H6 in H1. elim (VertexSet.empty_1 H1). -set (tmp := fold_left f' l m) in *. -unfold f' in H6. unfold f in H6. -destruct (Vertex.eq_dec y a). rewrite MapFacts.add_eq_o in H6. -rewrite <-H5 in H1. rewrite H7 in H1. -inversion H6; subst; clear H6. -destruct (proj1 (Props.Dec.F.add_iff _ _ _) H1). elim H3; auto. -generalize (VertexSet.remove_3 H4). intro. -unfold adj_set. rewrite (MapFacts.find_o _ e0). assumption. -intuition. -rewrite MapFacts.add_neq_o in H6. -rewrite <-H6 in IHl. -apply IHl. rewrite <-H7. rewrite <-H5 in H1. assumption. -auto. - -apply fold_left_assoc_map. -unfold EqualSetMap. unfold f'. unfold f. -intros. -destruct (Vertex.eq_dec x0 z). -rewrite MapFacts.add_eq_o. -destruct (Vertex.eq_dec x0 y0). -rewrite MapFacts.add_eq_o. -constructor. -unfold adj_set. -rewrite (MapFacts.find_o _ (Vertex.eq_sym e0)). -rewrite (MapFacts.find_o _ (Vertex.eq_sym e1)). -apply VertexSet.eq_refl. -intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_eq_o. -constructor. apply VertexSet.eq_refl. -intuition. -auto. -intuition. -rewrite MapFacts.add_neq_o. -destruct (Vertex.eq_dec x0 y0). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. apply VertexSet.eq_refl. -intuition. -intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -destruct (VertexMap.find x0 s0); constructor; apply VertexSet.eq_refl. -auto. -auto. -auto. -auto. - -unfold EqualSetMap. intros. -unfold f'. unfold f. -destruct (Vertex.eq_dec x0 a0). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. -apply VertexSet.eq_refl. -intuition. -intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -apply H4. auto. auto. - -auto. -auto. -Qed. - -Lemma merge_conflicts_aux_2 : forall e g, -aff_edge e -> -In_graph_edge e g -> -forall x y, -VertexSet.In x (adj_set y (map_merge e (pmap g))) -> -~Vertex.eq y (fst_ext e) -> -~Vertex.eq x (fst_ext e) -> -VertexSet.In x (adj_set y (pmap g)). - -Proof. -intros. -unfold map_merge in H1. -set (m := pmap g) in *. -set (f := fun (y : VertexSet.elt) (m' : VertexMap.t VertexSet.t) => - VertexMap.add y - (VertexSet.add (fst_ext e) - (VertexSet.remove (snd_ext e) (adj_set y m))) - m') in *. -set (s := (VertexSet.union - (VertexSet.remove (snd_ext e) - (adj_set (fst_ext e) m)) - (VertexSet.remove (fst_ext e) - (adj_set (snd_ext e) m)))) in *. -set (s' := (adj_set (snd_ext e) m)) in *. -unfold adj_set in H1. -destruct (Vertex.eq_dec y (snd_ext e)). -rewrite MapFacts.remove_eq_o in H1. elim (VertexSet.empty_1 H1). -intuition. -rewrite MapFacts.remove_neq_o in H1. rewrite MapFacts.add_neq_o in H1. -rewrite VertexSet.fold_1 in H1. set (f' := fun a e => f e a) in *. -induction (VertexSet.elements (VertexSet.remove (fst_ext e) s')). simpl in H1. -assumption. -cut (EqualSetMap (fold_left f' (a :: l) m) (f' (fold_left f' l m) a)). intro. -generalize (H4 y). clear H4. intro H4. simpl in H1. inversion H4; subst; clear H4. -rewrite <-H6 in H1. elim (VertexSet.empty_1 H1). -set (tmp := fold_left f' l m) in *. -unfold f' in H6. unfold f in H6. -destruct (Vertex.eq_dec y a). rewrite MapFacts.add_eq_o in H6. -rewrite <-H5 in H1. rewrite H7 in H1. -inversion H6; subst; clear H6. -destruct (proj1 (Props.Dec.F.add_iff _ _ _) H1). elim H3; auto. -generalize (VertexSet.remove_3 H4). intro. -unfold adj_set. rewrite (MapFacts.find_o _ e0). assumption. -intuition. -rewrite MapFacts.add_neq_o in H6. -rewrite <-H6 in IHl. -apply IHl. rewrite <-H7. rewrite <-H5 in H1. assumption. -auto. - -apply fold_left_assoc_map. -unfold EqualSetMap. unfold f'. unfold f. -intros. -destruct (Vertex.eq_dec x0 z). -rewrite MapFacts.add_eq_o. -destruct (Vertex.eq_dec x0 y0). -rewrite MapFacts.add_eq_o. -constructor. -unfold adj_set. -rewrite (MapFacts.find_o _ (Vertex.eq_sym e0)). -rewrite (MapFacts.find_o _ (Vertex.eq_sym e1)). -apply VertexSet.eq_refl. -intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_eq_o. -constructor. apply VertexSet.eq_refl. -intuition. -auto. -intuition. -rewrite MapFacts.add_neq_o. -destruct (Vertex.eq_dec x0 y0). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. apply VertexSet.eq_refl. -intuition. -intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -destruct (VertexMap.find x0 s0); constructor; apply VertexSet.eq_refl. -auto. -auto. -auto. -auto. - -unfold EqualSetMap. intros. -unfold f'. unfold f. -destruct (Vertex.eq_dec x0 a0). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. -apply VertexSet.eq_refl. -intuition. -intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -apply H4. auto. auto. - -auto. -auto. -Qed. - -Lemma merge_conflicts : forall e g, -aff_edge e -> -In_graph_edge e g -> -forall x y, -VertexSet.In x (adj_set y (imap_merge e g)) -> -VertexSet.In x (adj_set y (map_merge e (pmap g))) -> -~Vertex.eq y (fst_ext e) -> -~Vertex.eq x (fst_ext e) -> -False. - -Proof. -intros. -apply (simple_graph g x y). -split. -apply (merge_conflicts_aux_1 e g H H0 x y H1 H3 H4). -apply (merge_conflicts_aux_2 e g H H0 x y H2 H3 H4). -Qed. - -Lemma resolve_conflicts_map_3 : forall x y e g, -aff_edge e -> -In_graph_edge e g -> -VertexSet.In x (adj_set y (imap_merge e g)) /\ -VertexSet.In x (adj_set y (resolve_conflicts (fst_ext e) (map_merge e (pmap g)) - (adj_set (fst_ext e) (map_merge e (pmap g))) - (adj_set (fst_ext e) (imap_merge e g)))) - --> False. - -Proof. -intros x y e g Haff Hin H. destruct H. -unfold resolve_conflicts in H0. -set (f := (fun (x : VertexSet.elt) (m : VertexMap.t VertexSet.t) => - VertexMap.add x - (VertexSet.remove (fst_ext e) - (adj_set x (map_merge e (pmap g)))) m)) in *. -set (s := (VertexSet.diff (adj_set (fst_ext e) (map_merge e (pmap g))) - (adj_set (fst_ext e) (imap_merge e g)))) in *. -set (s' := (VertexSet.inter (adj_set (fst_ext e) (map_merge e (pmap g))) - (adj_set (fst_ext e) (imap_merge e g)))) in *. -rewrite VertexSet.fold_1 in H0. -set (f' := fun a e => f e a) in *. -generalize VertexSet.elements_1. intro HH. -generalize (HH s' y). clear HH. intro HH. -induction (VertexSet.elements s'). simpl in H0. -destruct (Vertex.eq_dec y (fst_ext e)). -unfold adj_set in H0. rewrite MapFacts.add_eq_o in H0. -elim (VertexSet.diff_2 H0). -unfold adj_set. rewrite (MapFacts.find_o _ (Vertex.eq_sym e0)). assumption. -intuition. -unfold adj_set in H0. rewrite MapFacts.add_neq_o in H0. - -assert (InA Vertex.eq y nil). -apply HH. -unfold s'. apply VertexSet.inter_3. -cut (Vertex.eq x (fst_ext e)). intro. -unfold adj_set. rewrite (MapFacts.find_o _ (Vertex.eq_sym H1)). -apply sym_merge; auto. -destruct (Vertex.eq_dec x (fst_ext e)). -intuition. -apply False_ind. apply (merge_conflicts e g Haff Hin x y); auto. - -destruct (Vertex.eq_dec x (fst_ext e)). -unfold adj_set. rewrite (MapFacts.find_o _ (Vertex.eq_sym e0)). -apply (sym_imap_merge_map e g Haff Hin). assumption. -apply False_ind. -apply (merge_conflicts e g Haff Hin x y); auto. inversion H1. -auto. - -set (m := map_merge e (pmap g)) in *. -cut (EqualSetMap (fold_left f' (a :: l) m) (f' (fold_left f' l m) a)). intro. -generalize (H1 y). clear H1. simpl in H0. intro H1. inversion H1; clear H1. -set (tmp := fold_left f' l m) in *. -unfold f' in H4. unfold f in H4. -destruct (Vertex.eq_dec y a). rewrite MapFacts.add_eq_o in H4. congruence. -intuition. -rewrite MapFacts.add_neq_o in H4. -destruct (Vertex.eq_dec y (fst_ext e)). -unfold adj_set in H0. rewrite MapFacts.add_eq_o in H0. -unfold adj_set in IHl. rewrite MapFacts.add_eq_o in IHl. -apply IHl. assumption. intros. -generalize (HH H1). intro H2. inversion H2; subst. -elim (n H6). auto. -intuition. -intuition. - -unfold adj_set in H0. rewrite MapFacts.add_neq_o in H0. -rewrite <-H3 in H0. elim (VertexSet.empty_1 H0). -auto. -auto. - -set (tmp := fold_left f' l m) in *. -unfold f' in H3. unfold f in H3. -destruct (Vertex.eq_dec y a). rewrite MapFacts.add_eq_o in H3. -destruct (Vertex.eq_dec y (fst_ext e)). -unfold adj_set in H0. -rewrite MapFacts.add_eq_o in H0. -elim (VertexSet.diff_2 H0). -unfold adj_set. rewrite (MapFacts.find_o _ (Vertex.eq_sym e1)). assumption. -intuition. -unfold adj_set in H0. rewrite MapFacts.add_neq_o in H0. -rewrite <-H2 in H0. -inversion H3; subst; clear H3. -rewrite H4 in H0. clear H4. -destruct (Vertex.eq_dec x (fst_ext e)). -elim (VertexSet.remove_1 (Vertex.eq_sym e1) H0). -apply (merge_conflicts e g Haff Hin x y); auto. -unfold adj_set. rewrite (MapFacts.find_o _ e0). apply (VertexSet.remove_3 H0). -auto. -intuition. - -rewrite MapFacts.add_neq_o in H3. -unfold adj_set in IHl. -destruct (Vertex.eq_dec y (fst_ext e)). -rewrite MapFacts.add_eq_o in IHl. -unfold adj_set in H0. -rewrite MapFacts.add_eq_o in H0. -apply IHl. assumption. intro. -generalize (HH H1). intro. -inversion H5; subst. -elim (n H7). -assumption. -intuition. -intuition. -rewrite MapFacts.add_neq_o in IHl. rewrite <-H3 in IHl. -unfold adj_set in H0. rewrite MapFacts.add_neq_o in H0. -rewrite <-H2 in H0. -apply IHl. rewrite <-H4. assumption. -intro. generalize (HH H1). intro. -inversion H5; subst. -elim (n H7). -assumption. -auto. -auto. -auto. - -apply fold_left_assoc_map. -unfold EqualSetMap. unfold f'. unfold f. -intros. -destruct (Vertex.eq_dec x0 z). -rewrite MapFacts.add_eq_o. -destruct (Vertex.eq_dec x0 y0). -rewrite MapFacts.add_eq_o. -constructor. -unfold adj_set. -rewrite (MapFacts.find_o _ (Vertex.eq_sym e0)). -rewrite (MapFacts.find_o _ (Vertex.eq_sym e1)). -apply VertexSet.eq_refl. -intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_eq_o. -constructor. apply VertexSet.eq_refl. -intuition. -auto. -intuition. -rewrite MapFacts.add_neq_o. -destruct (Vertex.eq_dec x0 y0). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. apply VertexSet.eq_refl. -intuition. -intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -destruct (VertexMap.find x0 s0); constructor; apply VertexSet.eq_refl. -auto. -auto. -auto. -auto. - -unfold EqualSetMap. intros. -unfold f'. unfold f. -destruct (Vertex.eq_dec x0 a0). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. -apply VertexSet.eq_refl. -intuition. -intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -apply H1. auto. auto. -Qed. - -Lemma simple_graph_merge_map : forall e g, -aff_edge e -> -In_graph_edge e g -> -forall x y, -VertexSet.In x (adj_set y (imap_merge e g)) /\ -VertexSet.In x (adj_set y (pmap_merge e g (imap_merge e g))) -> -False. - -Proof. -exact (fun e g H H0 x y H1 => - (resolve_conflicts_map_3 x y e g H H0 H1)). -Qed. - -Lemma sym_map_merge_pmap : forall e g x y, -aff_edge e -> -In_graph_edge e g -> -VertexSet.In x (adj_set y (map_merge e (pmap g))) -> -VertexSet.In y (adj_set x (map_merge e (pmap g))). - -Proof. -intros e g x y Haff Hin H. -apply sym_map_merge_map with (g:=g); auto. -apply (sym_pmap g). -intros. -apply (not_eq_extremities_map_merge a b e (pmap g)). -intros. apply (not_eq_extremities g). right. assumption. assumption. - -assert (~Vertex.eq y (snd_ext e)) as Hy. -intro. -assert (VertexSet.In y (VertexSet.remove (snd_ext e) (V g))). -apply (pmap_merge_domain_1 y _ _ Hin). -rewrite MapFacts.in_find_iff. intro. -unfold adj_set in H. rewrite H1 in H. elim (VertexSet.empty_1 H). -elim (VertexSet.remove_1 (Vertex.eq_sym H0) H1). - -intro. -unfold map_merge in H. - -set (f := (fun (y : VertexSet.elt) (m' : VertexMap.t VertexSet.t) => - VertexMap.add y - (VertexSet.add (fst_ext e) - (VertexSet.remove (snd_ext e) (adj_set y (pmap g)))) - m')) in *. -set (s := (VertexSet.union - (VertexSet.remove (snd_ext e) - (adj_set (fst_ext e) (pmap g))) - (VertexSet.remove (fst_ext e) - (adj_set (snd_ext e) (pmap g))))) in *. -set (s' := adj_set (snd_ext e) (pmap g)) in *. - -unfold adj_set in H. -rewrite MapFacts.remove_neq_o in H. -destruct (Vertex.eq_dec y (fst_ext e)). -rewrite MapFacts.add_eq_o in H. -unfold s in H. -destruct (VertexSet.union_1 H). -elim (VertexSet.remove_1 (Vertex.eq_sym H0) H1). -generalize (VertexSet.remove_3 H1). clear H1. intro H1. -unfold s' in H1. - -elim (not_eq_extremities g x (snd_ext e)). -right. assumption. -assumption. -intuition. -rewrite MapFacts.add_neq_o in H. - -rewrite VertexSet.fold_1 in H. -set (f' := fun a e => f e a) in *. -generalize VertexSet.elements_1. intro. -generalize (H1 (VertexSet.remove (fst_ext e) s') y). clear H1. intro H1. -induction (VertexSet.elements (VertexSet.remove (fst_ext e) s')). -simpl in H. -unfold s' in H1. -assert (InA Vertex.eq y nil). -apply H1. -apply VertexSet.remove_2. auto. -apply (sym_pmap g). rewrite <-H0. assumption. inversion H2. - -cut (EqualSetMap (fold_left f' (a :: l) (pmap g)) (f' (fold_left f' l (pmap g)) a)). intro. -generalize (H2 y). clear H2. intro H2. simpl in H. inversion H2; clear H2. -rewrite <-H4 in *. -elim (VertexSet.empty_1 H). -set (tmp := fold_left f' l (pmap g)) in *. -unfold f' in H4. unfold f in H4. -destruct (Vertex.eq_dec y a). -rewrite MapFacts.add_eq_o in H4. -rewrite <-H3 in *. clear H3. inversion H4. subst. clear H4. -rewrite H5 in H. clear H5. -destruct (proj1 (Props.Dec.F.add_iff _ _ _) H). -elim (In_graph_edge_diff_ext _ _ Hin). -apply Vertex.eq_trans with (y := x); auto. -elim (VertexSet.remove_1 (Vertex.eq_sym H0) H2). -intuition. -rewrite MapFacts.add_neq_o in H4. rewrite <-H4 in IHl. -apply IHl. -rewrite <-H5. rewrite <-H3 in H. assumption. -intro. generalize (H1 H2). clear H2. intro H2. -inversion H2; subst. -elim (n0 H7). -assumption. -auto. - -apply fold_left_assoc_map. -unfold EqualSetMap. unfold f'. unfold f. -intros. -destruct (Vertex.eq_dec x0 z). -rewrite MapFacts.add_eq_o. -destruct (Vertex.eq_dec x0 y0). -rewrite MapFacts.add_eq_o. -constructor. -unfold adj_set. -rewrite (MapFacts.find_o _ (Vertex.eq_sym e0)). -rewrite (MapFacts.find_o _ (Vertex.eq_sym e1)). -apply VertexSet.eq_refl. -intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_eq_o. -constructor. apply VertexSet.eq_refl. -intuition. -auto. -intuition. -rewrite MapFacts.add_neq_o. -destruct (Vertex.eq_dec x0 y0). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. apply VertexSet.eq_refl. -intuition. -intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -destruct (VertexMap.find x0 s0); constructor; apply VertexSet.eq_refl. -auto. -auto. -auto. -auto. - -unfold EqualSetMap. intros. -unfold f'. unfold f. -destruct (Vertex.eq_dec x0 a0). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. -apply VertexSet.eq_refl. -intuition. -intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -apply H2. auto. auto. - -auto. -auto. - -intro. -assert (VertexSet.In y (VertexSet.remove (snd_ext e) (V g))). -apply (pmap_merge_domain_1 y _ _ Hin). -rewrite MapFacts.in_find_iff. intro. -unfold adj_set in H. rewrite H1 in H. elim (VertexSet.empty_1 H). -elim (VertexSet.remove_1 (Vertex.eq_sym H0) H1). -Qed. - -Lemma sym_resolve : forall x y e g, -aff_edge e -> -In_graph_edge e g -> -VertexSet.In x (adj_set y (resolve_conflicts (fst_ext e) (map_merge e (pmap g)) - (adj_set (fst_ext e) (map_merge e (pmap g))) - (adj_set (fst_ext e) (imap_merge e g)))) -> -VertexSet.In y (adj_set x (resolve_conflicts (fst_ext e) (map_merge e (pmap g)) - (adj_set (fst_ext e) (map_merge e (pmap g))) - (adj_set (fst_ext e) (imap_merge e g)))). - -Proof. -unfold resolve_conflicts; intros x y e g p q H. -set (f := (fun (x0 : VertexSet.elt) (m : VertexMap.t VertexSet.t) => - VertexMap.add x0 - (VertexSet.remove (fst_ext e) - (adj_set x0 (map_merge e (pmap g)))) m)) in *. -set (s := (VertexSet.diff (adj_set (fst_ext e) (map_merge e (pmap g))) - (adj_set (fst_ext e) (imap_merge e g)))) in *. -set (s' := (VertexSet.inter (adj_set (fst_ext e) (map_merge e (pmap g))) - (adj_set (fst_ext e) (imap_merge e g)))) in *. -set (m := map_merge e (pmap g)) in *. -rewrite VertexSet.fold_1 in *. -set (f' := fun a e => f e a) in *. -generalize VertexSet.elements_1. intro. -generalize (H0 s' y). clear H0. intro HH. -generalize VertexSet.elements_2. intro. -generalize (H0 s'). clear H0. intro HHH. -induction (VertexSet.elements s'). simpl in *. -destruct (Vertex.eq_dec x (fst_ext e)). -unfold adj_set. rewrite MapFacts.add_eq_o. -destruct (Vertex.eq_dec y (fst_ext e)). -unfold adj_set in H. rewrite MapFacts.add_eq_o in H. -generalize (VertexSet.diff_1 H). intro H0. -rewrite e0 in H0. -unfold m in H0. unfold map_merge in H0. -unfold adj_set in H0. -rewrite MapFacts.remove_neq_o in H0. -rewrite MapFacts.add_eq_o in H0. -fold (adj_set (fst_ext e) (pmap g)) in H0. -fold (adj_set (snd_ext e) (pmap g)) in H0. -destruct (VertexSet.union_1 H0). -generalize (VertexSet.remove_3 H1). intro. -elim (not_eq_extremities g (fst_ext e) (fst_ext e)). right. auto. -intuition. -elim (VertexSet.remove_1 (Vertex.eq_refl _) H1). -intuition. -apply (In_graph_edge_diff_ext _ _ q). intuition. -unfold adj_set in H. rewrite MapFacts.add_neq_o in H. -fold (adj_set y m) in H. -destruct (Props.In_dec y (adj_set (fst_ext e) (imap_merge e g))). -assert (InA Vertex.eq y nil). -apply HH. apply VertexSet.inter_3. -unfold adj_set. rewrite (MapFacts.find_o _ (Vertex.eq_sym e0)). -apply (sym_map_merge_pmap e g _ _ p q H). assumption. inversion H0. -apply VertexSet.diff_3. -unfold adj_set. rewrite (MapFacts.find_o _ (Vertex.eq_sym e0)). -apply (sym_map_merge_pmap e g _ _ p q H). assumption. -auto. -intuition. - -unfold adj_set. rewrite MapFacts.add_neq_o. -fold (adj_set x m). -destruct (Vertex.eq_dec y (fst_ext e)). -unfold adj_set in H. rewrite MapFacts.add_eq_o in H. -apply (sym_map_merge_pmap e g _ _ p q). -unfold adj_set. rewrite (MapFacts.find_o _ e0). -apply (VertexSet.diff_1 H). -intuition. -unfold adj_set in H. rewrite MapFacts.add_neq_o in H. -apply (sym_map_merge_pmap e g _ _ p q H). -auto. -auto. - -cut (EqualSetMap (fold_left f' (a :: l) m) (f' (fold_left f' l m) a)). intro. -destruct (Vertex.eq_dec x (fst_ext e)). -unfold adj_set. rewrite MapFacts.add_eq_o. -destruct (Vertex.eq_dec y (fst_ext e)). -unfold adj_set in H. rewrite MapFacts.add_eq_o in H. -rewrite e1. rewrite <-e0. assumption. -intuition. -unfold adj_set in H. rewrite MapFacts.add_neq_o in H. -generalize (H0 y). simpl in H. intro H1. inversion H1; clear H1. -rewrite <-H3 in H. -elim (VertexSet.empty_1 H). -rewrite <-H2 in H. clear H2. -set (tmp := fold_left f' l m) in *. -unfold f' in H3. unfold f in H3. -destruct (Vertex.eq_dec y a). rewrite MapFacts.add_eq_o in H3. -inversion H3; subst; clear H3. -rewrite H4 in H. elim (VertexSet.remove_1 (Vertex.eq_sym e0) H). -intuition. -rewrite MapFacts.add_neq_o in H3. -unfold adj_set in IHl. rewrite MapFacts.add_neq_o in IHl. - rewrite MapFacts.add_eq_o in IHl. -rewrite <-H3 in IHl. -apply IHl. rewrite <-H4. assumption. -intros. generalize (HH H1). intro. -inversion H2; subst. -elim (n0 H6). -assumption. -auto. -intuition. -auto. -auto. -auto. -intuition. - -unfold adj_set. rewrite MapFacts.add_neq_o. -destruct (Vertex.eq_dec y (fst_ext e)). -unfold adj_set in H. rewrite MapFacts.add_eq_o in H. -generalize (H0 x). intro H1. inversion H1; subst; clear H1. -set (tmp := fold_left f' l m) in *. -unfold f' in H4. unfold f in H4. -destruct (Vertex.eq_dec x a). -rewrite MapFacts.add_eq_o in H4. congruence. intuition. -destruct (Vertex.eq_dec y a). -assert (VertexSet.In y s'). apply HHH. -left. auto. -rewrite e0 in H1. -unfold s' in H1. -elim (not_eq_extremities_merge_map e g p (fst_ext e) (fst_ext e)). -left. apply (VertexSet.inter_2 H1). -intuition. -rewrite MapFacts.add_neq_o in H4. -unfold adj_set in IHl. rewrite MapFacts.add_eq_o in IHl. - rewrite MapFacts.add_neq_o in IHl. -rewrite <-H4 in IHl. -assert (VertexSet.In y VertexSet.empty). -apply IHl. assumption. -intro. generalize (HH H1). intro. -inversion H2; subst. -elim (n1 H6). auto. -intros. apply HHH. right. auto. -elim (VertexSet.empty_1 H1). -auto. -intuition. -auto. - -unfold adj_set in IHl. rewrite MapFacts.add_eq_o in IHl. - rewrite MapFacts.add_neq_o in IHl. -simpl. rewrite <-H2. rewrite H4. -set (tmp := fold_left f' l m) in *. -unfold f' in H3. unfold f in H3. -destruct (Vertex.eq_dec x a). -rewrite MapFacts.add_eq_o in H3. -inversion H3; subst; clear H3. -assert (VertexSet.In x s'). -apply HHH. left. auto. -elim (VertexSet.diff_2 H (VertexSet.inter_2 H1)). intuition. -rewrite MapFacts.add_neq_o in H3. -rewrite <-H3 in IHl. -destruct (Vertex.eq_dec y a). -assert (VertexSet.In y s'). apply HHH. left. intuition. -unfold s' in H1. -elim (not_eq_extremities_merge_map e g p (fst_ext e) (fst_ext e)). -left. rewrite e0 in H1. apply (VertexSet.inter_2 H1). intuition. - -apply IHl. -assumption. -intro. generalize (HH H1). intro. -inversion H5; subst. -elim (n1 H7). -auto. - -intros. apply HHH. right. auto. -auto. -auto. -intuition. -intuition. - -unfold adj_set in H. rewrite MapFacts.add_neq_o in H. -simpl in H. -unfold adj_set in IHl. rewrite MapFacts.add_neq_o in IHl. - rewrite MapFacts.add_neq_o in IHl. -generalize (H0 y). intro H1. inversion H1; clear H1. -rewrite <-H3 in H. elim (VertexSet.empty_1 H). -rewrite <-H2 in H. -set (tmp := fold_left f' l m) in *. -unfold f' in H3. unfold f in H3. -destruct (Vertex.eq_dec y a). rewrite MapFacts.add_eq_o in H3. -inversion H3; subst; clear H3. -generalize H. clear H. intro H. rewrite H4 in H. -generalize (VertexSet.remove_3 H). clear H. intro H. -unfold adj_set in H. rewrite (MapFacts.find_o _ (Vertex.eq_sym e0)) in H. -fold (adj_set y m) in H. -generalize (sym_map_merge_pmap e g _ _ p q H). clear H. intro H. -fold m in H. -clear IHl H0 H2 HH HHH. - -set (l' := a :: l) in *. -induction l'. simpl. assumption. - -(* - unfold f'. unfold f. -destruct (Vertex.eq_dec x a). rewrite MapFacts.add_eq_o. -apply VertexSet.remove_2; auto. -unfold adj_set. rewrite (MapFacts.find_o _ (Vertex.eq_sym e1)). auto. -auto. -rewrite MapFacts.add_neq_o. assumption. -auto. -*) - -cut (EqualSetMap (fold_left f' (a0 :: l') m) (f' (fold_left f' l' m) a0)). intro. -generalize (H0 x). clear H0. intro H0. simpl. inversion H0; clear H0. -set (tmp' := fold_left f' l' m) in *. -unfold f' in H3. unfold f in H3. -destruct (Vertex.eq_dec x a0). -rewrite MapFacts.add_eq_o in H3. congruence. intuition. -rewrite MapFacts.add_neq_o in H3. -rewrite <-H3 in IHl'. elim (VertexSet.empty_1 IHl'). -auto. -set (tmp' := fold_left f' l' m) in *. -unfold f' in H2. unfold f in H2. -destruct (Vertex.eq_dec x a0). -rewrite MapFacts.add_eq_o in H2. inversion H2; subst; clear H2. -rewrite H3. apply VertexSet.remove_2. auto. -unfold adj_set. rewrite (MapFacts.find_o _ (Vertex.eq_sym e1)). assumption. -intuition. -rewrite MapFacts.add_neq_o in H2. -rewrite <-H2 in IHl'. -rewrite H3. auto. -auto. - -apply fold_left_assoc_map. -unfold EqualSetMap. unfold f'. unfold f. -intros. -destruct (Vertex.eq_dec x0 z). -rewrite MapFacts.add_eq_o. -destruct (Vertex.eq_dec x0 y0). -rewrite MapFacts.add_eq_o. -constructor. -unfold adj_set. -rewrite (MapFacts.find_o _ (Vertex.eq_sym e2)). -rewrite (MapFacts.find_o _ (Vertex.eq_sym e1)). -apply VertexSet.eq_refl. intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_eq_o. -constructor. apply VertexSet.eq_refl. -intuition. -auto. -intuition. -rewrite MapFacts.add_neq_o. -destruct (Vertex.eq_dec x0 y0). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. apply VertexSet.eq_refl. -intuition. -intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -destruct (VertexMap.find x0 s1); constructor; apply VertexSet.eq_refl. -auto. -auto. -auto. -auto. - -unfold EqualSetMap. intros. -unfold f'. unfold f. -destruct (Vertex.eq_dec x0 a1). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. -apply VertexSet.eq_refl. -intuition. -intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -apply H0. auto. intuition. - -intuition. -rewrite MapFacts.add_neq_o in H3. -rewrite <-H3 in IHl. - -generalize (H0 x). intro. inversion H1; clear H1. -unfold f' in H7. unfold f in H7. -destruct (Vertex.eq_dec x a). -rewrite MapFacts.add_eq_o in H7. congruence. intuition. -rewrite MapFacts.add_neq_o in H7. -rewrite <-H7 in IHl. simpl. rewrite <-H6. -apply IHl. -rewrite <-H4. auto. - -intro. generalize (HH H1). intro. -inversion H5; subst. -elim (n1 H9). -auto. - -intros. apply HHH. right. auto. -auto. - -unfold f' in H6. unfold f in H6. -destruct (Vertex.eq_dec x a). -rewrite MapFacts.add_eq_o in H6. -inversion H6; subst; clear H6. -simpl. rewrite <-H5. -rewrite H7. -apply VertexSet.remove_2. auto. - -case_eq (VertexMap.find x tmp); intros. -rewrite H1 in IHl. -assert (VertexSet.In y t0). -apply IHl. rewrite <-H4. assumption. - -intro. generalize (HH H6). intro. -inversion H8; subst. -elim (n1 H10). -auto. - -intros. apply HHH. right. auto. -clear H HH HHH IHl H0 H4 H2 H3 H5 H7. -assert (eq_set_option (Some t0) (VertexMap.find x tmp)). -rewrite H1. constructor. apply VertexSet.eq_refl. clear H1. -unfold tmp in H. -unfold adj_set. rewrite (MapFacts.find_o _ (Vertex.eq_sym e0)). -fold (adj_set x m). -induction l. simpl in H. inversion H. -unfold adj_set. rewrite <-H1. rewrite <-H2. auto. - -cut (EqualSetMap (fold_left f' (a0 :: l) m) (f' (fold_left f' l m) a0)). intro. -generalize (H0 x). clear H0. intro H0. simpl in H. inversion H0; subst. -rewrite <-H2 in H. inversion H. -set (tmp' := fold_left f' l m) in *. -unfold f' in H2. unfold f in H2. -destruct (Vertex.eq_dec x a0). -rewrite MapFacts.add_eq_o in H2. -inversion H; subst. rewrite <-H1 in H5. -inversion H5; subst; clear H5. -rewrite H7 in H6. rewrite H3 in H6. inversion H2; subst; clear H2 H3 H7. -unfold adj_set. rewrite (MapFacts.find_o _ e1). apply (VertexSet.remove_3 H6). intuition. -rewrite MapFacts.add_neq_o in H2. rewrite <-H2 in IHl. -apply IHl. constructor. rewrite <-H1 in H. inversion H; subst. -rewrite H7. auto. -auto. - -apply fold_left_assoc_map. -unfold EqualSetMap. unfold f'. unfold f. -intros. -destruct (Vertex.eq_dec x0 z). -rewrite MapFacts.add_eq_o. -destruct (Vertex.eq_dec x0 y0). -rewrite MapFacts.add_eq_o. -constructor. -unfold adj_set. -rewrite (MapFacts.find_o _ (Vertex.eq_sym e2)). -rewrite (MapFacts.find_o _ (Vertex.eq_sym e1)). -apply VertexSet.eq_refl. intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_eq_o. -constructor. apply VertexSet.eq_refl. -intuition. -auto. -intuition. -rewrite MapFacts.add_neq_o. -destruct (Vertex.eq_dec x0 y0). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. apply VertexSet.eq_refl. -intuition. -intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -destruct (VertexMap.find x0 s2); constructor; apply VertexSet.eq_refl. -auto. -auto. -auto. -auto. - -unfold EqualSetMap. intros. -unfold f'. unfold f. -destruct (Vertex.eq_dec x0 a1). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. -apply VertexSet.eq_refl. -intuition. -intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -apply H0. auto. auto. - -rewrite H1 in IHl. -assert (VertexSet.In y VertexSet.empty). -apply IHl. -rewrite <-H4. auto. -intro. generalize (HH H6). intro. -inversion H8; subst. -elim (n1 H10). -auto. - -intros. apply HHH. right. auto. -elim (VertexSet.empty_1 H6). intuition. - -rewrite MapFacts.add_neq_o in H6. -rewrite <-H6 in IHl. simpl. rewrite <-H5. -rewrite H7. apply IHl. -rewrite <-H4. auto. -intro. generalize (HH H1). intro. -inversion H8; subst. -elim (n1 H10). -auto. - -intros. apply HHH. right. auto. -auto. -auto. -auto. -auto. -auto. -auto. - -apply fold_left_assoc_map. -unfold EqualSetMap. unfold f'. unfold f. -intros. -destruct (Vertex.eq_dec x0 z). -rewrite MapFacts.add_eq_o. -destruct (Vertex.eq_dec x0 y0). -rewrite MapFacts.add_eq_o. -constructor. -unfold adj_set. -rewrite (MapFacts.find_o _ (Vertex.eq_sym e0)). -rewrite (MapFacts.find_o _ (Vertex.eq_sym e1)). -apply VertexSet.eq_refl. intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_eq_o. -constructor. apply VertexSet.eq_refl. -intuition. -auto. -intuition. -rewrite MapFacts.add_neq_o. -destruct (Vertex.eq_dec x0 y0). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. apply VertexSet.eq_refl. -intuition. -intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -destruct (VertexMap.find x0 s0); constructor; apply VertexSet.eq_refl. -auto. -auto. -auto. -auto. - -unfold EqualSetMap. intros. -unfold f'. unfold f. -destruct (Vertex.eq_dec x0 a0). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. -apply VertexSet.eq_refl. -intuition. -intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -apply H0. auto. auto. -Qed. - -Lemma sym_pmap_merge_map e g : -aff_edge e -> -In_graph_edge e g -> -forall x y, -VertexSet.In x (adj_set y (pmap_merge e g (imap_merge e g))) -> -VertexSet.In y (adj_set x (pmap_merge e g (imap_merge e g))). - -Proof. -intros. apply sym_resolve; assumption. -Qed. - -Definition merge e g - (q : In_graph_edge e g) - (p : aff_edge e) := - let im := imap_merge e g in - let pm := pmap_merge e g im in - Make_Graph (VertexSet.remove (snd_ext e) (V g)) - im - pm - (extremities_in_merge_imap e g q) - (extremities_in_merge_pmap e g p q) - (simple_graph_merge_map e g p q) - (sym_imap_merge_map e g p q) - (sym_pmap_merge_map e g p q) - (not_eq_extremities_merge_map e g p). - -Lemma In_graph_dec : forall v g, -{In_graph v g} + {~In_graph v g}. - -Proof. -exact (fun v g => Props.In_dec v (V g)). -Qed. - -Definition pmap_delete_preferences v g := -let pm := pmap g in -let m := VertexMap.add v VertexSet.empty pm in -VertexSet.fold - (fun y m' => VertexMap.add y (VertexSet.remove v (adj_set y pm)) m') - (adj_set v pm) - m. - -Lemma delete_preference_sub : forall x v g, -VertexSet.Subset (adj_set x (pmap_delete_preferences v g)) - (adj_set x (pmap g)). - -Proof. -unfold VertexSet.Subset;intros. -unfold pmap_delete_preferences in H. -rewrite VertexSet.fold_1 in H. - -generalize VertexSet.elements_2. -intro H0. generalize (H0 (adj_set v (pmap g))). clear H0. intro H0. -induction (VertexSet.elements (adj_set v (pmap g))); intros. -simpl in H. -unfold adj_set in H. -destruct (Vertex.eq_dec v x). -rewrite InterfFacts.add_eq_o in H. -elim (VertexSet.empty_1 H). -assumption. -rewrite InterfFacts.add_neq_o in H. auto. -assumption. -cut (EqualSetMap - (fold_left - (fun (a : VertexMap.t VertexSet.t) (e : VertexSet.elt) => - VertexMap.add e (VertexSet.remove v (adj_set e (pmap g))) a) (a0 :: l) - (VertexMap.add v VertexSet.empty (pmap g))) - (VertexMap.add a0 - (VertexSet.remove v - (adj_set a0 (pmap g))) - (fold_left - (fun (a : VertexMap.t VertexSet.t) (e : VertexSet.elt) => - VertexMap.add e (VertexSet.remove v (adj_set e (pmap g))) a) l - (VertexMap.add v VertexSet.empty (pmap g))))). - -intro. -assert (VertexSet.In a (adj_set x - (VertexMap.add a0 - (VertexSet.remove v - (adj_set a0 (pmap g))) - (fold_left - (fun (a : VertexMap.t VertexSet.t) (e : VertexSet.elt) => - VertexMap.add e (VertexSet.remove v (adj_set e (pmap g))) a) l - (VertexMap.add v VertexSet.empty (pmap g)))))). -unfold EqualSetMap in H1. generalize (H1 x). intro H2. -inversion H2. -unfold adj_set. unfold adj_set in H5. rewrite <-H5. -unfold adj_set in *. simpl in H. rewrite <-H4 in H. -assumption. -unfold adj_set in *. rewrite <-H4. -unfold adj_set in *. simpl in H. rewrite <-H3 in H. -rewrite <-H5. assumption. -generalize (H1 x). clear H1. intro H1. inversion H1. -simpl in H. unfold adj_set in H. unfold adj_set in H4. rewrite <-H4 in H. -elim (VertexSet.empty_1 H). -clear H1. -destruct (Vertex.eq_dec x a0). -unfold adj_set in H2. rewrite InterfFacts.add_eq_o in H2. -generalize (VertexSet.remove_3 H2). unfold adj_set. -rewrite (InterfFacts.find_o _ e). auto. intuition. -apply IHl. -rewrite InterfFacts.add_neq_o in H4. -unfold adj_set in *. rewrite <-H4. -rewrite <-H5. simpl in H. rewrite <-H3 in *. -assumption. -auto. -intuition. - -apply fold_left_assoc_map. - -intros. -unfold EqualSetMap. intros. -destruct (Vertex.eq_dec x0 z). -rewrite InterfFacts.add_eq_o. -destruct (Vertex.eq_dec x0 y). -rewrite InterfFacts.add_eq_o. -constructor. -generalize (Vertex.eq_trans (Vertex.eq_sym e0) e). intro HH. -unfold adj_set. -rewrite (InterfFacts.find_o _ HH). apply VertexSet.eq_refl. intuition. - -rewrite InterfFacts.add_neq_o. -rewrite InterfFacts.add_eq_o. -constructor. apply VertexSet.eq_refl. -intuition. -auto. -intuition. - -rewrite InterfFacts.add_neq_o. -destruct (Vertex.eq_dec x0 y). -rewrite InterfFacts.add_eq_o. -rewrite InterfFacts.add_eq_o. -constructor. apply VertexSet.eq_refl. -intuition. -intuition. - -intros. -unfold EqualSetMap. intros. -destruct (Vertex.eq_dec x0 y). -rewrite InterfFacts.add_eq_o. -rewrite InterfFacts.add_eq_o. -constructor. apply VertexSet.eq_refl. -intuition. -intuition. -rewrite InterfFacts.add_neq_o. -rewrite InterfFacts.add_neq_o. -rewrite InterfFacts.add_neq_o. -apply EqualSetMap_refl. -auto. -auto. -auto. -auto. - -unfold EqualSetMap. intros. -destruct (Vertex.eq_dec x0 a1). -rewrite InterfFacts.add_eq_o. -rewrite InterfFacts.add_eq_o. -constructor. apply VertexSet.eq_refl. -intuition. -intuition. -rewrite InterfFacts.add_neq_o. -rewrite InterfFacts.add_neq_o. -apply (H1 x0). -auto. -auto. -Qed. - -Lemma in_pmap_delete_in : forall x v g, -In_graph v g -> -VertexMap.In x (pmap_delete_preferences v g) -> -VertexMap.In x (pmap g). - -Proof. -intros x v g H1 H. -unfold pmap_delete_preferences in H. -rewrite VertexSet.fold_1 in H. - -generalize VertexSet.elements_2. intro HH. -generalize (HH (adj_set v (pmap g))). clear HH. intro HH. - -induction (VertexSet.elements (adj_set v (pmap g))); intros. -simpl in H. -destruct (proj1 (InterfFacts.add_in_iff _ _ _ _) H). -apply (proj2 (extremities_pmap g x)). -rewrite <-H0. assumption. -assumption. - -cut (EqualSetMap - (fold_left - (fun (a : VertexMap.t VertexSet.t) (e : VertexSet.elt) => - VertexMap.add e (VertexSet.remove v (adj_set e (pmap g))) a) (a :: l) - (VertexMap.add v VertexSet.empty (pmap g))) - (VertexMap.add a - (VertexSet.remove v - (adj_set a (pmap g))) - (fold_left - (fun (a : VertexMap.t VertexSet.t) (e : VertexSet.elt) => - VertexMap.add e (VertexSet.remove v (adj_set e (pmap g))) a) l - (VertexMap.add v VertexSet.empty (pmap g))))). intro. -unfold EqualSetMap in H0. - -generalize (proj1 (InterfFacts.in_find_iff _ _) H). clear H. intro H. -generalize (H0 x). intro H2. -inversion H2. -unfold adj_set in *. simpl in H. rewrite <-H4 in H. -elim H. auto. -clear H2. -destruct (Vertex.eq_dec x a). -rewrite InterfFacts.add_eq_o in H4. -apply (proj2 (InterfFacts.in_find_iff _ _)). -assert (VertexSet.In v (adj_set x (pmap g))). -apply (sym_pmap g). -apply HH. left. auto. -unfold adj_set in H2. -destruct (VertexMap.find x (pmap g)). -intro Helim. inversion Helim. -elim (VertexSet.empty_1 H2). intuition. -rewrite InterfFacts.add_neq_o in H4. -apply IHl. -apply (proj2 (InterfFacts.in_find_iff _ _)). -rewrite <-H4. intro Helim. inversion Helim. -intuition. -auto. - -apply fold_left_assoc_map. -intros. -unfold EqualSetMap. intros. -destruct (Vertex.eq_dec x0 z). -rewrite InterfFacts.add_eq_o. -destruct (Vertex.eq_dec x0 y). -rewrite InterfFacts.add_eq_o. -constructor. -generalize (Vertex.eq_trans (Vertex.eq_sym e0) e). intro HHH. -unfold adj_set. -rewrite (InterfFacts.find_o _ HHH). apply VertexSet.eq_refl. intuition. - -rewrite InterfFacts.add_neq_o. -rewrite InterfFacts.add_eq_o. -constructor. apply VertexSet.eq_refl. -intuition. -auto. -intuition. - -rewrite InterfFacts.add_neq_o. -destruct (Vertex.eq_dec x0 y). -rewrite InterfFacts.add_eq_o. -rewrite InterfFacts.add_eq_o. -constructor. apply VertexSet.eq_refl. -intuition. -intuition. - -intros. -unfold EqualSetMap. intros. -destruct (Vertex.eq_dec x0 y). -rewrite InterfFacts.add_eq_o. -rewrite InterfFacts.add_eq_o. -constructor. apply VertexSet.eq_refl. -intuition. -intuition. -rewrite InterfFacts.add_neq_o. -rewrite InterfFacts.add_neq_o. -rewrite InterfFacts.add_neq_o. -apply EqualSetMap_refl. -auto. -auto. -auto. -auto. - -unfold EqualSetMap. intros. -destruct (Vertex.eq_dec x0 a0). -rewrite InterfFacts.add_eq_o. -rewrite InterfFacts.add_eq_o. -constructor. apply VertexSet.eq_refl. -intuition. -intuition. -rewrite InterfFacts.add_neq_o. -rewrite InterfFacts.add_neq_o. -apply (H0 x0). -auto. -auto. -Qed. - -Lemma extremities_in_delete_preferences_imap (v : Vertex.t) g : -forall x, -VertexMap.In x (imap g) <-> VertexSet.In x (V g). - -Proof. -exact (fun v g => extremities_imap g). -Qed. - -Lemma extremities_in_delete_preferences_pmap v g : -In_graph v g -> -(forall x, -VertexMap.In x (pmap_delete_preferences v g) <-> -VertexSet.In x (V g)). - -Proof. -split; intros. -apply (proj1 (extremities_pmap g x)). -apply in_pmap_delete_in with (v:=v); auto. - -unfold pmap_delete_preferences. -set (f := (fun (y : VertexSet.elt) (m' : VertexMap.t VertexSet.t) => - VertexMap.add y (VertexSet.remove v (adj_set y (pmap g))) m')) in *. -set (m := VertexMap.add v VertexSet.empty (pmap g)). -rewrite VertexSet.fold_1. -induction (VertexSet.elements (adj_set v (pmap g))). simpl. -unfold m. -rewrite MapFacts.in_find_iff. -destruct (Vertex.eq_dec x v). -rewrite MapFacts.add_eq_o. -congruence. intuition. -rewrite MapFacts.add_neq_o. -generalize (proj2 (extremities_pmap g x) H0). -rewrite MapFacts.in_find_iff. auto. -auto. -set (h := (fun (a0 : VertexMap.t VertexSet.t) (e : VertexSet.elt) => f e a0)) in *. - -cut (EqualSetMap (fold_left h (a :: l) m) (h (fold_left h l m) a)). intro. -generalize (H1 x). clear H1. intro H1. -inversion H1. -set (tmp := fold_left h l m) in *. -unfold h in H4. -unfold f in H4. -destruct (Vertex.eq_dec x a). -rewrite MapFacts.add_eq_o in H4. -congruence. intuition. -rewrite MapFacts.add_neq_o in H4. -rewrite MapFacts.in_find_iff in IHl. rewrite <-H4 in IHl. -congruence. intuition. -simpl. rewrite MapFacts.in_find_iff. rewrite <-H2. congruence. - -apply fold_left_assoc_map. -unfold EqualSetMap. unfold h. unfold f. intros. -destruct (Vertex.eq_dec x0 z). -rewrite InterfFacts.add_eq_o. -destruct (Vertex.eq_dec x0 y). -rewrite InterfFacts.add_eq_o. -constructor. -generalize (Vertex.eq_trans (Vertex.eq_sym e0) e). intro HHH. -unfold adj_set. -rewrite (InterfFacts.find_o _ HHH). apply VertexSet.eq_refl. -intuition. - -rewrite InterfFacts.add_neq_o. -rewrite InterfFacts.add_eq_o. -constructor. apply VertexSet.eq_refl. -intuition. -auto. -intuition. - -rewrite InterfFacts.add_neq_o. -destruct (Vertex.eq_dec x0 y). -rewrite InterfFacts.add_eq_o. -rewrite InterfFacts.add_eq_o. -constructor. apply VertexSet.eq_refl. -intuition. -intuition. - -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -destruct (VertexMap.find x0 s); constructor. -apply VertexSet.eq_refl. -auto. -auto. -auto. -auto. - -intros. -unfold EqualSetMap. unfold h. unfold f. intros. -destruct (Vertex.eq_dec x0 a0). -rewrite InterfFacts.add_eq_o. -rewrite InterfFacts.add_eq_o. -constructor. apply VertexSet.eq_refl. -intuition. -intuition. -rewrite InterfFacts.add_neq_o. -rewrite InterfFacts.add_neq_o. -apply H1. auto. auto. -Qed. - -(* -Lemma extremities_in_delete_preferences v g : -forall x, -VertexMap.In x (imap g) \/ -VertexMap.In x (pmap_delete_preferences v g) <-> -VertexSet.In x (V g). - -Proof. -split; intros. -destruct (Vertex.eq_dec v x). -apply (proj1 (extremities g x)). -destruct H;[left|right]. -assumption. -apply in_pmap_delete_in with (v:=v); auto. - -Qed. -*) - -Lemma simple_graph_delete_preferences v g : forall x y, -VertexSet.In x (adj_set y (imap g)) /\ -VertexSet.In x (adj_set y (pmap_delete_preferences v g)) -> False. - -Proof. -intros. -destruct H. -generalize (delete_preference_sub _ _ _ _ H0). clear H0. intro H0. -apply (simple_graph g x y). intuition. -Qed. - -Lemma sym_imap_delete_preferences (v : Vertex.t) g : -forall x y, -VertexSet.In x (adj_set y (imap g)) -> -VertexSet.In y (adj_set x (imap g)). - -Proof. -exact (fun v g => sym_imap g). -Qed. - -Lemma in_adj_delete_preference_not_eq_1 : forall x y v g, -VertexSet.In x (adj_set y (pmap_delete_preferences v g)) -> -~Vertex.eq y v. - -Proof. -intros. -unfold pmap_delete_preferences in H. -rewrite VertexSet.fold_1 in H. -set (f:= (fun (a : VertexMap.t VertexSet.t) (e : VertexSet.elt) => - VertexMap.add e (VertexSet.remove v (adj_set e (pmap g))) a)) in *. -set (l := VertexSet.elements (adj_set v (pmap g))) in *. -generalize VertexSet.elements_2. intro HH. -generalize (HH (adj_set v (pmap g))). clear HH. intro HH. -fold l in HH. -assert (forall z, In z l -> ~Vertex.eq v z) as Hneq. -clear H. -intros. intro H0. -induction l. inversion H. -simpl in H. destruct H. subst. -assert (VertexSet.In v (adj_set v (pmap g))). -apply HH. left. auto. -elim (not_eq_extremities g v v). -right. assumption. -auto. -apply IHl. -intros. apply HH. auto. -assumption. - -induction l. simpl in H. -unfold adj_set in H. intro Helim. -rewrite MapFacts.add_eq_o in H. -elim (VertexSet.empty_1 H). intuition. -set (s := VertexMap.add v VertexSet.empty (pmap g)) in *. - -assert (EqualSetMap (fold_left f (a :: l) s) (f (fold_left f l s) a)). -apply fold_left_assoc_map. - -unfold f. unfold EqualSetMap. intros. -destruct (Vertex.eq_dec x0 z). -rewrite MapFacts.add_eq_o. -destruct (Vertex.eq_dec x0 y0). -rewrite MapFacts.add_eq_o. -constructor. -apply Props.Equal_remove. -unfold adj_set. -rewrite (MapFacts.find_o _ (Vertex.eq_trans (Vertex.eq_sym e) e0)). -apply VertexSet.eq_refl. intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_eq_o. -constructor. -apply VertexSet.eq_refl. -intuition. -auto. -intuition. -rewrite MapFacts.add_neq_o. -destruct (Vertex.eq_dec x0 y0). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. -apply VertexSet.eq_refl. -intuition. -intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -destruct (VertexMap.find x0 s0); constructor. -apply VertexSet.eq_refl. -auto. -auto. -auto. -auto. - -unfold f. unfold EqualSetMap. intros. -destruct (Vertex.eq_dec x0 a0). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. -apply VertexSet.eq_refl. -intuition. -intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -apply H0. -auto. -auto. - -intro Helim. -cut (VertexSet.In x (adj_set y (f (fold_left f l s) a))). -set (tmp := fold_left f l s) in *. -intro H1. -unfold f in H1. -generalize (H0 x). clear H0. intro H0. -unfold adj_set in H1. inversion H0. -rewrite <-H4 in *. -destruct (Vertex.eq_dec y a). -elim (Hneq a). left. auto. -apply (Vertex.eq_trans (Vertex.eq_sym Helim) e). -rewrite MapFacts.add_neq_o in H1. -apply IHl. -assumption. -intros. apply HH. auto. -intros. apply Hneq. right. auto. -assumption. -auto. - -rewrite <-H3 in *. -destruct (Vertex.eq_dec y a). -elim (Hneq a). left. auto. -apply (Vertex.eq_trans (Vertex.eq_sym Helim) e). -rewrite MapFacts.add_neq_o in H1. -apply IHl. -assumption. -intros. apply HH. auto. -intros. apply Hneq. right. auto. -assumption. -auto. - -clear HH Hneq IHl. -unfold adj_set in H. unfold adj_set. -generalize (H0 y). clear H0. intro H0. -simpl in H. -inversion H0; subst. -rewrite <-H3 in *. -simpl in H. rewrite <-H2 in *. -elim (VertexSet.empty_1 H). -rewrite <-H1 in *. -rewrite <-H2 in *. -rewrite <-H3. assumption. -Qed. - -Lemma in_adj_delete_preference_not_eq_2 : forall x y v g, -VertexSet.In x (adj_set y (pmap_delete_preferences v g)) -> -~Vertex.eq x v. - -Proof. -intros. intro Helim. generalize H. intro Hcopy. -unfold pmap_delete_preferences in H. -rewrite VertexSet.fold_1 in H. -set (f:= (fun (a : VertexMap.t VertexSet.t) (e : VertexSet.elt) => - VertexMap.add e (VertexSet.remove v (adj_set e (pmap g))) a)) in *. -generalize VertexSet.elements_1. intro. -generalize (H0 (adj_set v (pmap g)) y). clear H0. intro HH. -set (l := VertexSet.elements (adj_set v (pmap g))) in *. -induction l. simpl in H. -unfold adj_set in H. -rewrite MapFacts.add_neq_o in H. -generalize (sym_pmap g). intro Hsym. -generalize (Hsym _ _ H). intro H0. -unfold adj_set in H0. -rewrite (MapFacts.find_o _ Helim) in H0. -generalize (HH H0). intro H1. inversion H1. -generalize (in_adj_delete_preference_not_eq_1 x y v g Hcopy). auto. -set (s := VertexMap.add v VertexSet.empty (pmap g)) in *. - -assert (EqualSetMap (fold_left f (a :: l) s) (f (fold_left f l s) a)). -apply fold_left_assoc_map. - -unfold f. unfold EqualSetMap. intros. -destruct (Vertex.eq_dec x0 z). -rewrite MapFacts.add_eq_o. -destruct (Vertex.eq_dec x0 y0). -rewrite MapFacts.add_eq_o. -constructor. -apply Props.Equal_remove. -unfold adj_set. -rewrite (MapFacts.find_o _ (Vertex.eq_trans (Vertex.eq_sym e) e0)). -apply VertexSet.eq_refl. intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_eq_o. -constructor. -apply VertexSet.eq_refl. -intuition. -auto. -intuition. -rewrite MapFacts.add_neq_o. -destruct (Vertex.eq_dec x0 y0). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. -apply VertexSet.eq_refl. -intuition. -intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -destruct (VertexMap.find x0 s0); constructor. -apply VertexSet.eq_refl. -auto. -auto. -auto. -auto. - -unfold f. unfold EqualSetMap. intros. -destruct (Vertex.eq_dec x0 a0). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. -apply VertexSet.eq_refl. -intuition. -intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -apply H0. -auto. -auto. - -clear Hcopy. -cut (VertexSet.In x (adj_set y (f (fold_left f l s) a))). -set (tmp := fold_left f l s) in *. -intro H1. -unfold f in H1. unfold adj_set in H1. -destruct (Vertex.eq_dec y a). -rewrite MapFacts.add_eq_o in H1. -elim (VertexSet.remove_1 (Vertex.eq_sym Helim) H1). intuition. -rewrite MapFacts.add_neq_o in H1. -apply IHl. -assumption. -intro. generalize (HH H2). intro H3. -inversion H3; subst. -elim (n H5). -assumption. -auto. - -unfold adj_set. -generalize (H0 y). clear H0. intro H0. -inversion H0; subst. -simpl in H. unfold adj_set in H. rewrite <-H2 in H. -assumption. -simpl in H. unfold adj_set in H. rewrite <-H1 in H. -rewrite <-H3. assumption. -Qed. - -Lemma sym_pmap_delete_preferences (v : Vertex.t) g : -forall x y, -VertexSet.In x (adj_set y (pmap_delete_preferences v g)) -> -VertexSet.In y (adj_set x (pmap_delete_preferences v g)). - -Proof. -intros. -generalize (in_adj_delete_preference_not_eq_1 x y v g H). intro Hneqy. -generalize (in_adj_delete_preference_not_eq_2 x y v g H). intro Hneqx. -generalize (delete_preference_sub y v g x H). intro HH. -generalize (sym_pmap g _ _ HH). clear HH. intro HH. -unfold pmap_delete_preferences in *. -rewrite VertexSet.fold_1 in *. -set (f := fun (a : VertexMap.t VertexSet.t) (e : VertexSet.elt) => - VertexMap.add e (VertexSet.remove v (adj_set e (pmap g))) a) in *. -induction (VertexSet.elements (adj_set v (pmap g))). -simpl in *. -destruct (Vertex.eq_dec x v). -destruct (Vertex.eq_dec y v). -unfold adj_set in H. rewrite MapFacts.add_eq_o in H. -elim (VertexSet.empty_1 H). -case_eq (VertexMap.find y (VertexMap.add v VertexSet.empty (pmap g))). intros. -rewrite H0 in H. -rewrite MapFacts.add_eq_o in H0. inversion H0. subst. -elim (VertexSet.empty_1 H). intuition. -rewrite MapFacts.add_eq_o. congruence. intuition. - -elim (Hneqx e). - -unfold adj_set. rewrite MapFacts.add_neq_o. assumption. intuition. - -set (s := VertexMap.add v VertexSet.empty (pmap g)) in *. -assert (EqualSetMap (fold_left f (a :: l) s) (f (fold_left f l s) a)). -apply fold_left_assoc_map. - -unfold f. unfold EqualSetMap. intros. -destruct (Vertex.eq_dec x0 z). -rewrite MapFacts.add_eq_o. -destruct (Vertex.eq_dec x0 y0). -rewrite MapFacts.add_eq_o. -constructor. -apply Props.Equal_remove. -unfold adj_set. -rewrite (MapFacts.find_o _ (Vertex.eq_trans (Vertex.eq_sym e) e0)). -apply VertexSet.eq_refl. intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_eq_o. -constructor. -apply VertexSet.eq_refl. -intuition. -auto. -intuition. -rewrite MapFacts.add_neq_o. -destruct (Vertex.eq_dec x0 y0). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. -apply VertexSet.eq_refl. -intuition. -intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -destruct (VertexMap.find x0 s0); constructor. -apply VertexSet.eq_refl. -auto. -auto. -auto. -auto. - -unfold f. unfold EqualSetMap. intros. -destruct (Vertex.eq_dec x0 a0). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. -apply VertexSet.eq_refl. -intuition. -intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -apply H0. -auto. -auto. - -unfold EqualSetMap in H0. -unfold adj_set. -case_eq (VertexMap.find x (fold_left f (a :: l) s)); intros. -generalize (H0 x). intro HH0. -rewrite H1 in HH0. inversion HH0. subst. -set (tmp := fold_left f l s) in *. -unfold adj_set in H. -unfold f in H3. -destruct (Vertex.eq_dec x a). -rewrite MapFacts.add_eq_o in H3. -inversion H3. subst. clear H3. -rewrite H4. apply VertexSet.remove_2. -auto. -unfold adj_set. -rewrite (MapFacts.find_o _ (Vertex.eq_sym e)). assumption. -intuition. - -rewrite MapFacts.add_neq_o in H3. -rewrite H4. -cut (VertexSet.In y (adj_set x tmp)). -intro. -unfold adj_set in H2. -rewrite <-H3 in H2. assumption. -apply IHl. -assert (VertexSet.In x (adj_set y (pmap g)) -> VertexSet.In x (adj_set y tmp)). -clear IHl H0 H1 HH0 H4 H3 HH H. -intros. -unfold tmp. -induction l. simpl. unfold s. -unfold adj_set. -rewrite MapFacts.add_neq_o. assumption. -auto. - -assert (EqualSetMap (fold_left f (a0 :: l) s) (f (fold_left f l s) a0)). -apply fold_left_assoc_map. - -unfold f. unfold EqualSetMap. intros. -destruct (Vertex.eq_dec x0 z). -rewrite MapFacts.add_eq_o. -destruct (Vertex.eq_dec x0 y0). -rewrite MapFacts.add_eq_o. -constructor. -apply Props.Equal_remove. -unfold adj_set. -rewrite (MapFacts.find_o _ (Vertex.eq_trans (Vertex.eq_sym e) e0)). -apply VertexSet.eq_refl. -intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_eq_o. -constructor. -apply VertexSet.eq_refl. -intuition. -auto. -intuition. -rewrite MapFacts.add_neq_o. -destruct (Vertex.eq_dec x0 y0). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. -apply VertexSet.eq_refl. -intuition. -intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -destruct (VertexMap.find x0 s0); constructor. -apply VertexSet.eq_refl. -auto. -auto. -auto. -auto. - -unfold f. unfold EqualSetMap. intros. -destruct (Vertex.eq_dec x0 a1). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. -apply VertexSet.eq_refl. -intuition. -intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -apply H0. -auto. -auto. - -generalize (H0 y). clear H0. intro H0. -inversion H0; subst. -unfold adj_set. simpl. rewrite <-H2. -set (tmp' := fold_left f l s) in *. -unfold f in H3. -destruct (Vertex.eq_dec y a0). -rewrite MapFacts.add_eq_o in H3. inversion H3. -intuition. -rewrite MapFacts.add_neq_o in H3. -unfold adj_set in IHl. rewrite <-H3 in IHl. -assumption. -auto. -unfold adj_set. simpl. rewrite <-H1. -rewrite H3. -set (tmp' := fold_left f l s) in *. -unfold f in H2. -destruct (Vertex.eq_dec y a0). -rewrite MapFacts.add_eq_o in H2. -inversion H2. subst. clear H2. -apply VertexSet.remove_2. -auto. -unfold adj_set. -rewrite (MapFacts.find_o _ (Vertex.eq_sym e)). -assumption. -intuition. -rewrite MapFacts.add_neq_o in H2. -assert (VertexSet.In x (adj_set y tmp')). -apply IHl. -unfold adj_set in H4. -rewrite <-H2 in H4. assumption. -auto. -apply H2. -apply (sym_pmap g). assumption. -auto. - -generalize (H0 x). clear H0. intro H0. -rewrite H1 in H0. inversion H0. subst. -set (tmp := fold_left f l s) in *. -unfold f in H2. -destruct (Vertex.eq_dec x a). -rewrite MapFacts.add_eq_o in H2. -inversion H2. -intuition. -rewrite MapFacts.add_neq_o in H2. - -assert (VertexSet.In y (adj_set x (pmap g)) -> VertexSet.In y (adj_set x tmp)). -clear IHl H H0 HH H2 H1. -intros. -unfold tmp. -induction l. simpl. unfold s. -unfold adj_set. -rewrite MapFacts.add_neq_o. assumption. -intuition. - -assert (EqualSetMap (fold_left f (a0 :: l) s) (f (fold_left f l s) a0)). -apply fold_left_assoc_map. - -unfold f. unfold EqualSetMap. intros. -destruct (Vertex.eq_dec x0 z). -rewrite MapFacts.add_eq_o. -destruct (Vertex.eq_dec x0 y0). -rewrite MapFacts.add_eq_o. -constructor. -apply Props.Equal_remove. -unfold adj_set. -rewrite (MapFacts.find_o _ (Vertex.eq_trans (Vertex.eq_sym e) e0)). -apply VertexSet.eq_refl. intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_eq_o. -constructor. -apply VertexSet.eq_refl. -intuition. -auto. -intuition. -rewrite MapFacts.add_neq_o. -destruct (Vertex.eq_dec x0 y0). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. -apply VertexSet.eq_refl. -intuition. -intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -destruct (VertexMap.find x0 s0); constructor. -apply VertexSet.eq_refl. -auto. -auto. -auto. -auto. - -unfold f. unfold EqualSetMap. intros. -destruct (Vertex.eq_dec x0 a1). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. -apply VertexSet.eq_refl. -intuition. -intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -apply H0. -auto. -auto. - -generalize (H0 x). clear H0. intro H0. -inversion H0; subst. -unfold adj_set. simpl. rewrite <-H2. -set (tmp' := fold_left f l s) in *. -unfold f in H3. -destruct (Vertex.eq_dec x a0). -rewrite MapFacts.add_eq_o in H3. inversion H3. -intuition. -rewrite MapFacts.add_neq_o in H3. -unfold adj_set in IHl. rewrite <-H3 in IHl. -assumption. -auto. -unfold adj_set. simpl. rewrite <-H1. -rewrite H3. -set (tmp' := fold_left f l s) in *. -unfold f in H2. -destruct (Vertex.eq_dec x a0). -rewrite MapFacts.add_eq_o in H2. -inversion H2. subst. clear H2. -apply VertexSet.remove_2. -auto. -unfold adj_set. -rewrite (MapFacts.find_o _ (Vertex.eq_sym e)). -assumption. -intuition. -rewrite MapFacts.add_neq_o in H2. -assert (VertexSet.In y (adj_set x tmp')). -apply IHl. -unfold adj_set in H4. -rewrite <-H2 in H4. assumption. -auto. -assert (VertexSet.In y (adj_set x tmp)). -apply H3. assumption. -unfold adj_set in H4. -rewrite <-H2 in H4. -assumption. -auto. -Qed. - -Lemma not_eq_extremities_delete_preferences v g : -forall x y, -VertexSet.In x (adj_set y (imap g)) \/ -VertexSet.In x (adj_set y (pmap_delete_preferences v g)) -> -~Vertex.eq x y. - -Proof. -intros. -apply (not_eq_extremities g). -destruct H;[left|right]. -assumption. -apply (delete_preference_sub _ _ _ _ H). -Qed. - -Definition delete_preference_edges v g H := -Make_Graph (V g) - (imap g) - (pmap_delete_preferences v g) - (extremities_in_delete_preferences_imap v g) - (extremities_in_delete_preferences_pmap v g H) - (simple_graph_delete_preferences v g) - (sym_imap_delete_preferences v g) - (sym_pmap_delete_preferences v g) - (not_eq_extremities_delete_preferences v g). - -Definition Prefere x y g := -exists w, In_graph_edge (x,y,Some w) g. - -Definition Interfere x y g := -In_graph_edge (x,y,None) g. - -Definition precolored g := -VertexSet.filter (fun v => is_precolored v g) (V g). - -Parameter mreg_ext : forall x y, -Vertex.eq x y -> is_mreg x = is_mreg y. - -Lemma mem_ext : forall x y s, -Vertex.eq x y -> VertexSet.mem x s = VertexSet.mem y s. - -Proof. -intros. -case_eq (VertexSet.mem x s); intros. -generalize (VertexSet.mem_2 H0). intro H1. -rewrite H in H1. generalize (VertexSet.mem_1 H1). auto. -case_eq (VertexSet.mem y s); intros. -generalize (VertexSet.mem_2 H1). intro H2. -rewrite <-H in H2. generalize (VertexSet.mem_1 H2). intro H3. -rewrite H3 in H0. inversion H0. -reflexivity. -Qed. - -Lemma compat_bool_is_precolored : forall g, -compat_bool Vertex.eq (fun x => is_precolored x g). - -Proof. -unfold compat_bool. intros. -unfold is_precolored. -apply mreg_ext. auto. -Qed. - -(* Equivalence of predicate and function *) -Lemma precolored_equiv : forall x g, -VertexSet.In x (precolored g) <-> (is_precolored x g = true /\ In_graph x g). - -Proof. -split; intros. -split. -apply (VertexSet.filter_2 (compat_bool_is_precolored _) H). -apply (VertexSet.filter_1 (compat_bool_is_precolored _) H). -apply VertexSet.filter_3. apply compat_bool_is_precolored. -unfold is_precolored in H. -case_eq (VertexSet.mem x (V g)); intros. -apply VertexSet.mem_2. assumption. intuition. intuition. -Qed. - -Lemma In_graph_aff_edge_in_AE : forall e g, -EdgeSet.In e (AE g) <-> aff_edge e /\ In_graph_edge e g. - -Proof. -intros e g. split; intro H. -split. -unfold aff_edge. exists N0. exact (AE_weights g e H). -left. assumption. -destruct H as [H H0]. -destruct H0. -assumption. -unfold aff_edge in H. destruct H as [w H]. -rewrite (IE_weights g e H0) in H. inversion H. -Qed. - -Lemma In_graph_interf_edge_in_IE : forall e g, -EdgeSet.In e (IE g) <-> interf_edge e /\ In_graph_edge e g. - -Proof. -intros e g. split; intro H. -split. -exact (IE_weights g e H). -right. assumption. -destruct H as [H H0]. -destruct H0. -unfold interf_edge in H. -rewrite (AE_weights g e H0) in H. inversion H. -assumption. -Qed. - -(* Spec of Prefere *) -Lemma Prefere_1 : forall e g, -aff_edge e -> -In_graph_edge e g -> -Prefere (fst_ext e) (snd_ext e) g. - -Proof. -intros. -unfold Prefere. -destruct H0. -exists N0. rewrite <-(AE_weights g e H0). rewrite <-edge_eq. left. assumption. -generalize (proj1 (In_graph_interf_edge_in_IE _ _) H0). intro H1. -destruct H1 as [H1 _]. unfold interf_edge in H1. unfold aff_edge in H. -rewrite H1 in H. inversion H. inversion H2. -Qed. - -Lemma Prefere_2 : forall x y g, -Prefere x y g -> -In_graph_edge (x,y,Some N0) g. - -Proof. -intros. -unfold Prefere in H. -destruct H. -assert (EdgeSet.In (x,y,Some x0) (AE g)). -rewrite In_graph_aff_edge_in_AE. split. exists x0. auto. auto. -generalize (AE_weights _ _ H0). intro. simpl in H1. rewrite H1 in H. auto. -Qed. - -(* spec of Interfere *) -Lemma Interfere_1 : forall e g, -interf_edge e -> -In_graph_edge e g -> -Interfere (fst_ext e) (snd_ext e) g. - -Proof. -intros. -unfold Interfere. -destruct H0. -generalize (proj1 (In_graph_aff_edge_in_AE _ _) H0). intro H1. -destruct H1 as [H1 _]. unfold interf_edge in H. unfold aff_edge in H1. -rewrite H in H1. inversion H1. inversion H2. -rewrite <-(IE_weights g e H0). rewrite <-edge_eq. right. assumption. -Qed. - -Lemma Interfere_2 : forall x y g, -Interfere x y g -> -In_graph_edge (x,y,None) g. - -Proof. -auto. -Qed. - -(* Machine registers are registers *) -Lemma In_precolored : forall g , -VertexSet.Subset (precolored g) (V g). - -Proof. -intros. -unfold VertexSet.Subset; intros. -apply (VertexSet.filter_1 (compat_bool_is_precolored _) H). -Qed. - -(* specification of remove_vertex *) -Lemma In_remove : forall x r g, -In_graph x g -> -~Vertex.eq x r -> -In_graph x (remove_vertex r g). - -Proof. -intros. -unfold remove_vertex. simpl. -apply VertexSet.remove_2; auto. -Qed. - -(* A precolored vertex cannot be removed from the graph *) -Lemma not_in_remove : forall r g, -~In_graph r (remove_vertex r g). - -Proof. -intros. -unfold remove_vertex. -apply VertexSet.remove_1. apply Vertex.eq_refl. -Qed. - -Lemma in_remove_in : forall x r g, -In_graph x (remove_vertex r g) -> -In_graph x g. - -Proof. -intros. -unfold remove_vertex in H. -unfold In_graph in H. simpl in H. -apply (VertexSet.remove_3 H). -Qed. - -Add Morphism In_graph : In_graph_m. - -Proof. -intros. -unfold In_graph. -rewrite H. reflexivity. -Qed. - -(* Probably redundant, TODO get out of interface *) -Lemma precolored_remove_vertex2 : forall x y g, -VertexSet.In x (VertexSet.remove y (precolored g)) <-> -VertexSet.In x (precolored (remove_vertex y g)). - -Proof. -intros. -split; intros. -generalize (VertexSet.remove_3 H). intro H0. -generalize (VertexSet.filter_1 (compat_bool_is_precolored _) H0). intro. -generalize (VertexSet.filter_2 (compat_bool_is_precolored _) H0). clear H0. intro. -apply VertexSet.filter_3. -apply compat_bool_is_precolored. -unfold remove_vertex. simpl. -apply VertexSet.remove_2. intro. apply (VertexSet.remove_1 H2 H). -assumption. -unfold is_precolored. assumption. -generalize (proj1 (precolored_equiv _ _) H). clear H. intro H. destruct H. -apply VertexSet.remove_2. -intro H1. rewrite <-H1 in H0. elim (not_in_remove _ _ H0). -apply (proj2 (precolored_equiv _ _)). -split. -assumption. -apply in_remove_in with (r:=y). assumption. -Qed. - -Lemma In_remove_edge_ : forall e r g, -In_graph_edge e g -> -~incident e r -> -In_graph_edge e (remove_vertex r g). - -Proof. -intros. -destruct H;[left|right]. -generalize (AE_weights _ _ H). intro Hw. -unfold AE in *. -unfold remove_vertex. -simpl. -rewrite (edge_eq e) in *. -simpl in Hw. rewrite Hw. -apply (proj2 (edgemap_to_edgeset_charac _ _ _ _ (sym_pmap_remove_vertex r g))). -apply imap_remove_1. -intro H1. elim H0. right. auto. -intro H1. elim H0. left. auto. -rewrite Hw in H. -apply (proj1 (edgemap_to_edgeset_charac _ _ _ _ (sym_pmap g)) H). - -generalize (IE_weights _ _ H). intro Hw. -unfold IE in *. -unfold remove_vertex. -simpl. -rewrite (edge_eq e) in *. -simpl in Hw. rewrite Hw. -apply (proj2 (edgemap_to_edgeset_charac _ _ _ _ (sym_imap_remove_vertex r g))). -apply imap_remove_1. -intro H1. elim H0. right. auto. -intro H1. elim H0. left. auto. -rewrite Hw in H. -apply (proj1 (edgemap_to_edgeset_charac _ _ _ _ (sym_imap g)) H). -Qed. - -Lemma in_remove_in_edge : forall e r g, -In_graph_edge e (remove_vertex r g) -> -In_graph_edge e g. - -Proof. -intros. -destruct H;[left|right]. -generalize (AE_weights _ _ H). intro Hw. -unfold AE in *. -unfold remove_vertex in H. -simpl in H. -rewrite (edge_eq e) in *. -simpl in Hw. rewrite Hw. -apply (proj2 (edgemap_to_edgeset_charac _ _ _ _ (sym_pmap g))). -apply imap_remove_3 with (r:=r). -rewrite Hw in H. -apply (proj1 (edgemap_to_edgeset_charac _ _ _ _(sym_pmap_remove_vertex r g)) H). - -generalize (IE_weights _ _ H). intro Hw. -unfold IE in *. -unfold remove_vertex in H. -simpl in H. -rewrite (edge_eq e) in *. -simpl in Hw. rewrite Hw. -apply (proj2 (edgemap_to_edgeset_charac _ _ _ _ (sym_imap g))). -apply imap_remove_3 with (r:=r). -rewrite Hw in H. -apply (proj1 (edgemap_to_edgeset_charac _ _ _ _(sym_imap_remove_vertex r g)) H). -Qed. - -Lemma not_in_remove_edge : forall e r g, -incident e r -> -~In_graph_edge e (remove_vertex r g). - -Proof. -intros. generalize H. intro H0. -destruct (In_graph_edge_dec e g). -assert (In_graph r g) as Hin. -destruct H. -rewrite H. apply (proj1 (In_graph_edge_in_ext _ _ i)). -rewrite H. apply (proj2 (In_graph_edge_in_ext _ _ i)). -intro H1. destruct H1. -generalize (AE_weights _ _ H1). intro Hw. -unfold AE in *. -unfold remove_vertex in H1. -simpl in H1. -rewrite (edge_eq e) in H1. -rewrite Hw in H1. rewrite edge_comm in H1. -generalize (proj1 (edgemap_to_edgeset_charac _ _ _ _ (sym_pmap_remove_vertex r g)) H1). -intro H3. -apply (imap_remove_2 _ _ (pmap g) _ H (sym_pmap g) H3). - -generalize (IE_weights _ _ H1). intro Hw. -unfold IE in *. -unfold remove_vertex in H1. -simpl in H1. -rewrite (edge_eq e) in H1. -rewrite Hw in H1. rewrite edge_comm in H1. -generalize (proj1 (edgemap_to_edgeset_charac _ _ _ _ (sym_imap_remove_vertex r g)) H1). -intro H3. -apply (imap_remove_2 _ _ (imap g) _ H (sym_imap g) H3). -intro H1. elim (n (in_remove_in_edge _ _ _ H1)). -Qed. - - -(* spec of merge *) -Lemma In_merge_in : forall g e x p q, -In_graph x (merge e g p q) -> In_graph x g. - -Proof. -intros. -unfold merge in H. -destruct (aff_edge_dec e). -destruct (In_graph_edge_dec e g). -unfold In_graph in H. simpl in H. -apply (VertexSet.remove_3 H). -elim (n p). -elim (n q). -Qed. - -(* the second extremity must not be precolored, - cause it is removed from the graph *) -Lemma merge_1 : forall e g p q, -~In_graph (snd_ext e) (merge e g p q). - -Proof. -intros. -unfold merge. unfold In_graph. simpl. -apply VertexSet.remove_1. apply Vertex.eq_refl. -Qed. - -Lemma merge_2 : forall x e g p q, -~Vertex.eq x (snd_ext e) -> -In_graph x g -> -In_graph x (merge e g p q). - -Proof. -intros. unfold merge. -unfold In_graph. simpl. apply VertexSet.remove_2 with (x:=(snd_ext e)); auto. -Qed. - -Lemma merge_3 : forall e e' g p q, -In_graph_edge e' g -> -interf_edge e' -> -In_graph_edge (redirect (snd_ext e) (fst_ext e) e') (merge e g p q). - -Proof. -intros e e' g p q H0 H1. generalize I. intro H. -assert (EdgeSet.In (fst_ext e, snd_ext e, Some N0) (AE g)) as He. -destruct p. rewrite (edge_eq e) in H2. -generalize (AE_weights _ _ H2). intro. simpl in H3. -rewrite H3 in H2. assumption. -generalize (IE_weights _ _ H2). inversion q. congruence. -assert (VertexSet.In (snd_ext e) (adj_set (fst_ext e) (pmap g))) as Hee. -apply (proj1 (edgemap_to_edgeset_charac _ _ _ (Some N0) (sym_pmap g))). -assumption. -assert (EdgeSet.In (fst_ext e', snd_ext e', None) (IE g)) as He'. -destruct H0. generalize (AE_weights _ _ H0). intro H2. inversion H2. congruence. -rewrite (edge_eq e') in H0. generalize (IE_weights _ _ H0). intro. simpl in H2. -rewrite H2 in H0. assumption. -assert (VertexSet.In (snd_ext e') (adj_set (fst_ext e') (imap g))) as Hee'. -apply (proj1 (edgemap_to_edgeset_charac _ _ _ None (sym_imap g))). -assumption. -assert (forall a b, VertexSet.In a (adj_set b (imap_merge e g)) -> - VertexSet.In b (adj_set a (imap_merge e g))) as Hsym. -apply sym_imap_merge_map; auto. - -right. -unfold merge. unfold IE. simpl. -unfold redirect. -destruct (OTFacts.eq_dec (fst_ext e') (snd_ext e)). -rewrite H1. apply (proj2 (edgemap_to_edgeset_charac (imap_merge e g) _ _ None Hsym)). - -unfold imap_merge. -unfold map_merge. -set (s := (VertexSet.union - (VertexSet.remove (snd_ext e) (adj_set (fst_ext e) (imap g))) - (VertexSet.remove (fst_ext e) (adj_set (snd_ext e) (imap g))))) in *. -set (m := imap g) in *. -set (f := (fun (y : VertexSet.elt) (m' : VertexMap.t VertexSet.t) => - VertexMap.add y - (VertexSet.add (fst_ext e) - (VertexSet.remove (snd_ext e) (adj_set y m))) m')) in *. -set (s' := VertexSet.remove (fst_ext e) (adj_set (snd_ext e) m)) in *. -unfold adj_set. -rewrite MapFacts.remove_neq_o. -rewrite MapFacts.add_eq_o. -unfold s. -apply VertexSet.union_3. -unfold s'. -apply VertexSet.remove_2. -intro. -elim (simple_graph g (fst_ext e) (snd_ext e)). -split. -rewrite H2. unfold adj_set. rewrite (MapFacts.find_o _ (Vertex.eq_sym r)). -fold (adj_set (fst_ext e') (imap g)). -assumption. -apply (sym_pmap g). assumption. -unfold adj_set. rewrite (MapFacts.find_o _ (Vertex.eq_sym r)). -fold (adj_set (fst_ext e') m). -assumption. -intuition. -apply (In_graph_edge_diff_ext _ _ p). - -destruct (OTFacts.eq_dec (snd_ext e') (snd_ext e)). -rewrite H1. -apply (proj2 (edgemap_to_edgeset_charac (imap_merge e g) _ _ None Hsym)). -unfold imap_merge. -unfold map_merge. -set (s := (VertexSet.union - (VertexSet.remove (snd_ext e) (adj_set (fst_ext e) (imap g))) - (VertexSet.remove (fst_ext e) (adj_set (snd_ext e) (imap g))))) in *. -set (m := imap g) in *. -set (f := (fun (y : VertexSet.elt) (m' : VertexMap.t VertexSet.t) => - VertexMap.add y - (VertexSet.add (fst_ext e) - (VertexSet.remove (snd_ext e) (adj_set y m))) m')) in *. -set (s' := VertexSet.remove (fst_ext e) (adj_set (snd_ext e) m)) in *. -unfold adj_set. -rewrite MapFacts.remove_neq_o. -rewrite MapFacts.add_neq_o. -rewrite VertexSet.fold_1. -set (f' := fun a e => f e a). -generalize VertexSet.elements_1. intro HH. -generalize (HH s' (fst_ext e')). clear HH. intro HH. -induction (VertexSet.elements s'). simpl. -assert (InA Vertex.eq (fst_ext e') nil). -apply HH. -unfold s'. apply VertexSet.remove_2. -intro. -elim (simple_graph g (fst_ext e) (snd_ext e)). -split. -rewrite H2. unfold adj_set. rewrite (MapFacts.find_o _ (Vertex.eq_sym r)). -fold (adj_set (fst_ext e') (imap g)). -apply (sym_imap g). assumption. -apply (sym_pmap g). assumption. -unfold adj_set. rewrite (MapFacts.find_o _ (Vertex.eq_sym r)). -apply (sym_imap g). assumption. -inversion H2. - -cut (EqualSetMap (fold_left f' (a :: l) m) (f' (fold_left f' l m) a)). intro. -generalize (H2 (fst_ext e')). clear H2. intro H2. simpl. inversion H2;clear H2. -set (tmp := fold_left f' l m) in *. -unfold f' in H5. unfold f in H5. -destruct (Vertex.eq_dec (fst_ext e') a). -rewrite MapFacts.add_eq_o in H5. congruence. intuition. -rewrite MapFacts.add_neq_o in H5. rewrite <-H5 in IHl. -apply IHl. -intro. generalize (HH H2). clear HH. intro HH. -inversion HH; subst. -elim (n0 H6). -assumption. -auto. - -set (tmp := fold_left f' l m) in *. -unfold f' in H4. unfold f in H4. -destruct (Vertex.eq_dec (fst_ext e') a). -rewrite MapFacts.add_eq_o in H4. -rewrite H5. inversion H4; subst; clear H4. -apply VertexSet.add_1. intuition. intuition. -rewrite MapFacts.add_neq_o in H4. -rewrite <-H4 in IHl. -rewrite H5. apply IHl. -intro. generalize (HH H2). clear HH. intro HH. inversion HH; subst. -elim (n0 H7). -assumption. -auto. - -apply fold_left_assoc_map. -unfold EqualSetMap. unfold f'. unfold f. -intros. -destruct (Vertex.eq_dec x z). -rewrite MapFacts.add_eq_o. -destruct (Vertex.eq_dec x y). -rewrite MapFacts.add_eq_o. -constructor. -unfold adj_set. -rewrite (MapFacts.find_o _ (Vertex.eq_sym e1)). -rewrite (MapFacts.find_o _ (Vertex.eq_sym e0)). -apply VertexSet.eq_refl. intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_eq_o. -constructor. apply VertexSet.eq_refl. -intuition. -auto. -intuition. -rewrite MapFacts.add_neq_o. -destruct (Vertex.eq_dec x y). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. apply VertexSet.eq_refl. -intuition. -intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -destruct (VertexMap.find x s0); constructor; apply VertexSet.eq_refl. -auto. -auto. -auto. -auto. - -unfold EqualSetMap. intros. -unfold f'. unfold f. -destruct (Vertex.eq_dec x a0). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. -apply VertexSet.eq_refl. -intuition. -intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -apply H2. auto. auto. - -intro. -elim (simple_graph g (fst_ext e) (snd_ext e)). -split. -rewrite H2. unfold adj_set. rewrite (MapFacts.find_o _ (Vertex.eq_sym r)). -fold (adj_set (fst_ext e') (imap g)). -apply (sym_imap g). assumption. -apply (sym_pmap g). assumption. - -intuition. -rewrite (edge_eq e'). rewrite H1. -apply (proj2 (edgemap_to_edgeset_charac (imap_merge e g) _ _ None Hsym)). -unfold imap_merge. -unfold map_merge. -set (s := (VertexSet.union - (VertexSet.remove (snd_ext e) (adj_set (fst_ext e) (imap g))) - (VertexSet.remove (fst_ext e) (adj_set (snd_ext e) (imap g))))) in *. -set (m := imap g) in *. -set (f := (fun (y : VertexSet.elt) (m' : VertexMap.t VertexSet.t) => - VertexMap.add y - (VertexSet.add (fst_ext e) - (VertexSet.remove (snd_ext e) (adj_set y m))) m')) in *. -set (s' := VertexSet.remove (fst_ext e) (adj_set (snd_ext e) m)) in *. -unfold adj_set. -rewrite MapFacts.remove_neq_o. -destruct (Vertex.eq_dec (fst_ext e') (fst_ext e)). -rewrite MapFacts.add_eq_o. -unfold s. -apply VertexSet.union_2. -apply VertexSet.remove_2. intuition. -unfold adj_set. rewrite (MapFacts.find_o _ (Vertex.eq_sym e0)). -assumption. -intuition. - -rewrite MapFacts.add_neq_o. -rewrite VertexSet.fold_1. -set (f' := fun a e => f e a). -(* -generalize VertexSet.elements_1. intro HH. -generalize (HH s' (fst_ext e')). clear HH. intro HH. -*) -induction (VertexSet.elements s'). simpl. assumption. -cut (EqualSetMap (fold_left f' (a :: l) m) (f' (fold_left f' l m) a)). intro. -generalize (H2 (fst_ext e')). clear H2. intro H2. simpl. inversion H2;clear H2. -set (tmp := fold_left f' l m) in *. -unfold f' in H5. unfold f in H5. -destruct (Vertex.eq_dec (fst_ext e') a). -rewrite MapFacts.add_eq_o in H5. congruence. intuition. -rewrite MapFacts.add_neq_o in H5. rewrite <-H5 in IHl. -apply IHl. auto. - -set (tmp := fold_left f' l m) in *. -unfold f' in H4. unfold f in H4. -destruct (Vertex.eq_dec (fst_ext e') a). -rewrite MapFacts.add_eq_o in H4. -rewrite H5. inversion H4; subst; clear H4. -apply VertexSet.add_2. apply VertexSet.remove_2. intuition. -unfold adj_set. rewrite (MapFacts.find_o _ (Vertex.eq_sym e0)). assumption. -intuition. - -rewrite MapFacts.add_neq_o in H4. -rewrite <-H4 in IHl. -rewrite H5. apply IHl. -auto. - -apply fold_left_assoc_map. -unfold EqualSetMap. unfold f'. unfold f. -intros. -destruct (Vertex.eq_dec x z). -rewrite MapFacts.add_eq_o. -destruct (Vertex.eq_dec x y). -rewrite MapFacts.add_eq_o. -constructor. -unfold adj_set. -rewrite (MapFacts.find_o _ (Vertex.eq_sym e1)). -rewrite (MapFacts.find_o _ (Vertex.eq_sym e0)). -apply VertexSet.eq_refl. intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_eq_o. -constructor. apply VertexSet.eq_refl. -intuition. -auto. -intuition. -rewrite MapFacts.add_neq_o. -destruct (Vertex.eq_dec x y). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. apply VertexSet.eq_refl. -intuition. -intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -destruct (VertexMap.find x s0); constructor; apply VertexSet.eq_refl. -auto. -auto. -auto. -auto. - -unfold EqualSetMap. intros. -unfold f'. unfold f. -destruct (Vertex.eq_dec x a0). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. -apply VertexSet.eq_refl. -intuition. -intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -apply H2. auto. auto. - -auto. -intuition. -Qed. - -Lemma merge_4 : forall e e' g p q, -In_graph_edge e' (merge e g p q) -> -exists a, In_graph_edge a g /\ weak_eq e' (redirect (snd_ext e) (fst_ext e) a) - /\ same_type a e'. - -Proof. -intros e e' g p q H0. generalize I. intro H. -generalize H0. intro Hin. -unfold merge in H0. destruct H0. -generalize (AE_weights _ _ H0). intro Hw. -unfold AE in H0. simpl in H0. -rewrite (edge_eq e') in H0. rewrite Hw in H0. -generalize (proj1 (edgemap_to_edgeset_charac _ _ _ (Some N0) (sym_pmap_merge_map _ _ q p)) H0). -clear H0. intro H0. -generalize (pmap_merge_sub _ _ _ _ H0). clear H0. intro H0. -unfold map_merge in H0. -set (s := (VertexSet.union - (VertexSet.remove (snd_ext e) (adj_set (fst_ext e) (pmap g))) - (VertexSet.remove (fst_ext e) (adj_set (snd_ext e) (pmap g))))) in *. -set (m := pmap g) in *. -set (f := (fun (y : VertexSet.elt) (m' : VertexMap.t VertexSet.t) => - VertexMap.add y - (VertexSet.add (fst_ext e) - (VertexSet.remove (snd_ext e) (adj_set y m))) m')) in *. -set (s' := VertexSet.remove (fst_ext e) (adj_set (snd_ext e) m)) in *. -unfold adj_set in H0. -destruct (Vertex.eq_dec (fst_ext e') (snd_ext e)). -rewrite MapFacts.remove_eq_o in H0. -elim (VertexSet.empty_1 H0). intuition. -rewrite MapFacts.remove_neq_o in H0. -destruct (Vertex.eq_dec (fst_ext e') (fst_ext e)). -rewrite MapFacts.add_eq_o in H0. -unfold s in H0. -destruct (VertexSet.union_1 H0). -generalize (VertexSet.remove_3 H1). clear H1. intro H1. -exists (fst_ext e', snd_ext e', Some N0). split. -left. unfold AE. -apply (proj2 (edgemap_to_edgeset_charac _ _ _ (Some N0) (sym_pmap g))). -unfold adj_set. rewrite (MapFacts.find_o _ e0). assumption. -unfold redirect. change_rewrite. -destruct (OTFacts.eq_dec (fst_ext e') (snd_ext e)). -elim (n r). -destruct (OTFacts.eq_dec (snd_ext e') (snd_ext e)). -elim (merge_1 e g p q). -rewrite <-r. apply (proj2 (In_graph_edge_in_ext _ _ Hin)). -split. -unfold weak_eq. change_rewrite. -left. split; auto. -unfold same_type. left. split. exists N0. auto. exists N0. auto. -unfold s' in H1. -generalize (VertexSet.remove_3 H1). intro H2. -exists (snd_ext e', snd_ext e, Some N0). -split. -left. unfold AE. -apply (proj2 (edgemap_to_edgeset_charac _ _ _ (Some N0) (sym_pmap g))). -apply (sym_pmap g). assumption. -unfold redirect. change_rewrite. -destruct (OTFacts.eq_dec (snd_ext e') (snd_ext e)). -unfold weak_eq. change_rewrite. -split. -left. split; auto. -unfold same_type. left. split; exists N0; auto. -destruct (OTFacts.eq_dec (snd_ext e) (snd_ext e)). -unfold weak_eq. change_rewrite. -split. -right. split; auto. -unfold same_type. left. split; exists N0; auto. -elim n1. intuition. intuition. -rewrite MapFacts.add_neq_o in H0. - -rewrite VertexSet.fold_1 in H0. -set (f' := fun a e => f e a) in *. -generalize VertexSet.elements_2. intro HH. -generalize (HH s'). clear HH. intro HH. -induction (VertexSet.elements s'). simpl in H0. -case_eq (VertexMap.find (fst_ext e') m); intros. rewrite H1 in H0. -exists (fst_ext e', snd_ext e', Some N0). -split. -left. apply (proj2 (edgemap_to_edgeset_charac _ _ _ (Some N0) (sym_pmap g))). -unfold adj_set. fold m. rewrite H1. assumption. -unfold redirect. change_rewrite. -destruct (OTFacts.eq_dec (fst_ext e') (snd_ext e)). -elim (n r). -destruct (OTFacts.eq_dec (snd_ext e') (snd_ext e)). -elim (merge_1 e g p q). -rewrite <-r. apply (proj2 (In_graph_edge_in_ext _ _ Hin)). -unfold weak_eq. change_rewrite. -split. -left. split ; auto. -left. split; exists N0; auto. -rewrite H1 in H0. elim (VertexSet.empty_1 H0). -simpl in H0. -cut (EqualSetMap (fold_left f' (a :: l) m) (f' (fold_left f' l m) a)). intro. -generalize (H1 (fst_ext e')). clear H1. intro H1. inversion H1; clear H1. -rewrite <-H3 in *. elim (VertexSet.empty_1 H0). -set (tmp := fold_left f' l m) in *. -unfold f' in H3. unfold f in H3. -destruct (Vertex.eq_dec (fst_ext e') a). -rewrite MapFacts.add_eq_o in H3. inversion H3; subst; clear H3. -rewrite <-H2 in H0. -rewrite H4 in H0. clear H4. -destruct (proj1 (Props.Dec.F.add_iff _ _ _ ) H0). -exists (fst_ext e', snd_ext e, Some N0). -split. -left. apply (proj2 (edgemap_to_edgeset_charac _ _ _ (Some N0) (sym_pmap g))). -apply (sym_pmap g). -assert (VertexSet.In (fst_ext e') s'). -apply HH. left. auto. -unfold s' in H3. apply (VertexSet.remove_3 H3). -unfold redirect. change_rewrite. -destruct (OTFacts.eq_dec (fst_ext e') (snd_ext e)). -unfold weak_eq. change_rewrite. -split. -right. split; auto. -unfold same_type. left. split; exists N0; auto. -destruct (OTFacts.eq_dec (snd_ext e) (snd_ext e)). -unfold weak_eq. change_rewrite. -split. -left. split; auto. -left. split; exists N0; auto. -elim n2. intuition. -exists (fst_ext e', snd_ext e', Some N0). -split. -left. -apply (proj2 (edgemap_to_edgeset_charac _ _ _ (Some N0) (sym_pmap g))). -generalize (VertexSet.remove_3 H1). clear H1. intro H1. -unfold adj_set. rewrite (MapFacts.find_o _ e0). assumption. -unfold redirect. change_rewrite. -destruct (OTFacts.eq_dec (fst_ext e') (snd_ext e)). -elim (merge_1 e g p q). -rewrite <-r. apply (proj1 (In_graph_edge_in_ext _ _ Hin)). -destruct (OTFacts.eq_dec (snd_ext e') (snd_ext e)). -elim (merge_1 e g p q). -rewrite <-r. apply (proj2 (In_graph_edge_in_ext _ _ Hin)). -unfold weak_eq. change_rewrite. -split. -left. split; auto. -left. split; exists N0; auto. -intuition. -auto. - -rewrite MapFacts.add_neq_o in H3. rewrite <-H3 in IHl. -apply IHl. rewrite <-H2 in H0. rewrite H4 in H0. auto. -intros. apply HH. auto. -auto. - -apply fold_left_assoc_map. -unfold EqualSetMap. unfold f'. unfold f. -intros. -destruct (Vertex.eq_dec x z). -rewrite MapFacts.add_eq_o. -destruct (Vertex.eq_dec x y). -rewrite MapFacts.add_eq_o. -constructor. -unfold adj_set. -rewrite (MapFacts.find_o _ (Vertex.eq_sym e1)). -rewrite (MapFacts.find_o _ (Vertex.eq_sym e0)). -apply VertexSet.eq_refl. -intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_eq_o. -constructor. apply VertexSet.eq_refl. -intuition. -auto. -intuition. -rewrite MapFacts.add_neq_o. -destruct (Vertex.eq_dec x y). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. apply VertexSet.eq_refl. -intuition. -intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -destruct (VertexMap.find x s0); constructor; apply VertexSet.eq_refl. -auto. -auto. -auto. -auto. - -unfold EqualSetMap. intros. -unfold f'. unfold f. -destruct (Vertex.eq_dec x a0). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. -apply VertexSet.eq_refl. -intuition. -intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -apply H1. auto. auto. - -auto. -auto. - -generalize (IE_weights _ _ H0). intro Hw. -unfold IE in H0. simpl in H0. -rewrite (edge_eq e') in H0. rewrite Hw in H0. -generalize (proj1 (edgemap_to_edgeset_charac _ _ _ None (sym_imap_merge_map _ _ q p)) H0). -clear H0. intro H0. -unfold imap_merge in H0. unfold map_merge in H0. -set (s := (VertexSet.union - (VertexSet.remove (snd_ext e) (adj_set (fst_ext e) (imap g))) - (VertexSet.remove (fst_ext e) (adj_set (snd_ext e) (imap g))))) in *. -set (m := imap g) in *. -set (f := (fun (y : VertexSet.elt) (m' : VertexMap.t VertexSet.t) => - VertexMap.add y - (VertexSet.add (fst_ext e) - (VertexSet.remove (snd_ext e) (adj_set y m))) m')) in *. -set (s' := VertexSet.remove (fst_ext e) (adj_set (snd_ext e) m)) in *. -unfold adj_set in H0. -destruct (Vertex.eq_dec (fst_ext e') (snd_ext e)). -rewrite MapFacts.remove_eq_o in H0. -elim (VertexSet.empty_1 H0). intuition. -rewrite MapFacts.remove_neq_o in H0. -destruct (Vertex.eq_dec (fst_ext e') (fst_ext e)). -rewrite MapFacts.add_eq_o in H0. -unfold s in H0. -destruct (VertexSet.union_1 H0). -generalize (VertexSet.remove_3 H1). clear H1. intro H1. -exists (fst_ext e', snd_ext e', None). split. -right. unfold IE. -apply (proj2 (edgemap_to_edgeset_charac _ _ _ None (sym_imap g))). -unfold adj_set. rewrite (MapFacts.find_o _ e0). assumption. -unfold redirect. change_rewrite. -destruct (OTFacts.eq_dec (fst_ext e') (snd_ext e)). -elim (n r). -destruct (OTFacts.eq_dec (snd_ext e') (snd_ext e)). -elim (merge_1 e g p q). -rewrite <-r. apply (proj2 (In_graph_edge_in_ext _ _ Hin)). -unfold weak_eq. change_rewrite. -split. left; split; auto. -right; split; unfold interf_edge; simpl; auto. -unfold s' in H1. -generalize (VertexSet.remove_3 H1). intro H2. -exists (snd_ext e', snd_ext e, None). -split. -right. unfold IE. -apply (proj2 (edgemap_to_edgeset_charac _ _ _ None (sym_imap g))). -apply (sym_imap g). assumption. -unfold redirect. change_rewrite. -destruct (OTFacts.eq_dec (snd_ext e') (snd_ext e)). -unfold weak_eq. change_rewrite. -split. -left. split; auto. -right. unfold interf_edge;split; simpl; auto. -destruct (OTFacts.eq_dec (snd_ext e) (snd_ext e)). -unfold weak_eq. change_rewrite. -split. -right. split; auto. -right. unfold interf_edge; split; simpl; auto. -elim n1. intuition. -intuition. -rewrite MapFacts.add_neq_o in H0. - -rewrite VertexSet.fold_1 in H0. -set (f' := fun a e => f e a) in *. -generalize VertexSet.elements_2. intro HH. -generalize (HH s'). clear HH. intro HH. -induction (VertexSet.elements s'). simpl in H0. -case_eq (VertexMap.find (fst_ext e') m); intros. rewrite H1 in H0. -exists (fst_ext e', snd_ext e', None). -split. -right. apply (proj2 (edgemap_to_edgeset_charac _ _ _ None (sym_imap g))). -unfold adj_set. fold m. rewrite H1. assumption. -unfold redirect. change_rewrite. -destruct (OTFacts.eq_dec (fst_ext e') (snd_ext e)). -elim (n r). -destruct (OTFacts.eq_dec (snd_ext e') (snd_ext e)). -elim (merge_1 e g p q). -rewrite <-r. apply (proj2 (In_graph_edge_in_ext _ _ Hin)). -unfold weak_eq. change_rewrite. -split. -left. split; auto. -right. unfold interf_edge; split; simpl; auto. -rewrite H1 in H0. elim (VertexSet.empty_1 H0). -simpl in H0. -cut (EqualSetMap (fold_left f' (a :: l) m) (f' (fold_left f' l m) a)). intro. -generalize (H1 (fst_ext e')). clear H1. intro H1. inversion H1; clear H1. -rewrite <-H3 in *. elim (VertexSet.empty_1 H0). -set (tmp := fold_left f' l m) in *. -unfold f' in H3. unfold f in H3. -destruct (Vertex.eq_dec (fst_ext e') a). -rewrite MapFacts.add_eq_o in H3. inversion H3; subst; clear H3. -rewrite <-H2 in H0. -rewrite H4 in H0. clear H4. -destruct (proj1 (Props.Dec.F.add_iff _ _ _ ) H0). -exists (fst_ext e', snd_ext e, None). -split. -right. apply (proj2 (edgemap_to_edgeset_charac _ _ _ None (sym_imap g))). -apply (sym_imap g). -assert (VertexSet.In (fst_ext e') s'). -apply HH. left. auto. -unfold s' in H3. apply (VertexSet.remove_3 H3). -unfold redirect. change_rewrite. -destruct (OTFacts.eq_dec (fst_ext e') (snd_ext e)). -unfold weak_eq. change_rewrite. -split. -right. split; auto. -right. unfold interf_edge; split; simpl; auto. -destruct (OTFacts.eq_dec (snd_ext e) (snd_ext e)). -unfold weak_eq. change_rewrite. -split. -left. split; auto. -right. unfold interf_edge; split; simpl; auto. -elim n2. intuition. -exists (fst_ext e', snd_ext e', None). -split. -right. apply (proj2 (edgemap_to_edgeset_charac _ _ _ None (sym_imap g))). -generalize (VertexSet.remove_3 H1). clear H1. intro H1. -unfold adj_set. rewrite (MapFacts.find_o _ e0). assumption. -unfold redirect. change_rewrite. -destruct (OTFacts.eq_dec (fst_ext e') (snd_ext e)). -elim (merge_1 e g p q). -rewrite <-r. apply (proj1 (In_graph_edge_in_ext _ _ Hin)). -destruct (OTFacts.eq_dec (snd_ext e') (snd_ext e)). -elim (merge_1 e g p q). -rewrite <-r. apply (proj2 (In_graph_edge_in_ext _ _ Hin)). -unfold weak_eq. change_rewrite. -split. -left. split; auto. -right. unfold interf_edge; split; simpl; auto. -intuition. - -rewrite MapFacts.add_neq_o in H3. rewrite <-H3 in IHl. -apply IHl. rewrite <-H2 in H0. rewrite H4 in H0. auto. -intros. apply HH. auto. -auto. - -apply fold_left_assoc_map. -unfold EqualSetMap. unfold f'. unfold f. -intros. -destruct (Vertex.eq_dec x z). -rewrite MapFacts.add_eq_o. -destruct (Vertex.eq_dec x y). -rewrite MapFacts.add_eq_o. -constructor. -unfold adj_set. -rewrite (MapFacts.find_o _ (Vertex.eq_sym e1)). -rewrite (MapFacts.find_o _ (Vertex.eq_sym e0)). -apply VertexSet.eq_refl. -intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_eq_o. -constructor. apply VertexSet.eq_refl. -intuition. -auto. -intuition. -rewrite MapFacts.add_neq_o. -destruct (Vertex.eq_dec x y). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. apply VertexSet.eq_refl. -intuition. -intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -destruct (VertexMap.find x s0); constructor; apply VertexSet.eq_refl. -auto. -auto. -auto. -auto. - -unfold EqualSetMap. intros. -unfold f'. unfold f. -destruct (Vertex.eq_dec x a0). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. -apply VertexSet.eq_refl. -intuition. -intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -apply H1. auto. auto. - -auto. -auto. -Qed. - -Lemma resolve_conflicts_map_5 : forall x y e g, -aff_edge e -> -In_graph_edge e g -> -VertexSet.In x (adj_set y (map_merge e (pmap g))) -> -~VertexSet.In x (adj_set y (imap_merge e g)) -> -VertexSet.In x (adj_set y (resolve_conflicts (fst_ext e) (map_merge e (pmap g)) - (adj_set (fst_ext e) (map_merge e (pmap g))) - (adj_set (fst_ext e) (imap_merge e g)))). - -Proof. -intros x y e g p q H H0. -unfold resolve_conflicts. -set (f := (fun (x0 : VertexSet.elt) (m : VertexMap.t VertexSet.t) => - VertexMap.add x0 - (VertexSet.remove (fst_ext e) - (adj_set x0 (map_merge e (pmap g)))) m)) in *. -set (m := map_merge e (pmap g)) in *. -set (s' := (VertexSet.diff (adj_set (fst_ext e) m) - (adj_set (fst_ext e) (imap_merge e g)))) in *. -set (s := (VertexSet.inter (adj_set (fst_ext e) m) - (adj_set (fst_ext e) (imap_merge e g)))) in *. -rewrite VertexSet.fold_1. -set (f' := fun a e => f e a) in *. -generalize VertexSet.elements_1. intro. -generalize (H1 s y). clear H1. intro HH. -generalize VertexSet.elements_2. intro. -generalize (H1 s y). clear H1. intro HHH. -induction (VertexSet.elements s). simpl. -destruct (Vertex.eq_dec y (fst_ext e)). -unfold adj_set. rewrite (MapFacts.find_o _ e0). -rewrite MapFacts.add_eq_o. -apply VertexSet.diff_3. -unfold adj_set. rewrite (MapFacts.find_o _ (Vertex.eq_sym e0)). assumption. -unfold adj_set. rewrite (MapFacts.find_o _ (Vertex.eq_sym e0)). assumption. -intuition. -unfold adj_set. rewrite MapFacts.add_neq_o. assumption. -auto. - -cut (EqualSetMap (fold_left f' (a :: l) m) (f' (fold_left f' l m) a)). -intro. generalize (H1 y). clear H1. intro H1. simpl. inversion H1; clear H1. -set (tmp := fold_left f' l m) in *. -unfold f' in H4. unfold f in H4. -destruct (Vertex.eq_dec y a). -rewrite MapFacts.add_eq_o in H4. congruence. intuition. -destruct (Vertex.eq_dec y (fst_ext e)). unfold adj_set. -rewrite MapFacts.add_eq_o. -unfold adj_set in IHl. -rewrite MapFacts.add_eq_o in IHl. -apply IHl. intro. -generalize (HH H1). intro. -inversion H2; subst. -elim (n H6). -auto. -intro. apply HHH. right. auto. -intuition. -intuition. - -rewrite MapFacts.add_neq_o in H4. -unfold adj_set in IHl. rewrite MapFacts.add_neq_o in IHl. -rewrite <-H4 in IHl. -assert (VertexSet.In x VertexSet.empty). -apply IHl. intro. -generalize (HH H1). intro. -inversion H2; subst. -elim (n H6). -auto. -intro. apply HHH. right. auto. -elim (VertexSet.empty_1 H1). -auto. -auto. -auto. - -set (tmp := fold_left f' l m) in *. -unfold f' in H3. unfold f in H3. -destruct (Vertex.eq_dec y a). -rewrite MapFacts.add_eq_o in H3. -unfold adj_set. destruct (Vertex.eq_dec y (fst_ext e)). -rewrite MapFacts.add_eq_o. -apply VertexSet.diff_3. -unfold adj_set. rewrite (MapFacts.find_o _ (Vertex.eq_sym e1)). assumption. -unfold adj_set. rewrite (MapFacts.find_o _ (Vertex.eq_sym e1)). assumption. -intuition. - -rewrite MapFacts.add_neq_o. rewrite <-H2. rewrite H4. -inversion H3; subst; clear H3. -apply VertexSet.remove_2. - -intro H5. -assert (VertexSet.In y s). -apply HHH. left. auto. -generalize (VertexSet.inter_2 H1). intro. -unfold adj_set in H3. rewrite (MapFacts.find_o _ H5) in H3. -generalize (sym_imap_merge_map e g p q _ _ H3). intro. -elim H0. assumption. -unfold adj_set. rewrite (MapFacts.find_o _ (Vertex.eq_sym e0)). -auto. -auto. -intuition. - -rewrite MapFacts.add_neq_o in H3. -destruct (Vertex.eq_dec y (fst_ext e)). -unfold adj_set in IHl. -rewrite MapFacts.add_eq_o in IHl. -unfold adj_set. rewrite MapFacts.add_eq_o. -apply IHl. - -intro. generalize (HH H1). intro. -inversion H5; subst. -elim (n H7). -auto. - -intro. apply HHH. right. auto. -intuition. -intuition. - -unfold adj_set in IHl. rewrite MapFacts.add_neq_o in IHl. -unfold adj_set. rewrite MapFacts.add_neq_o. -rewrite <-H2. -rewrite <-H3 in IHl. -rewrite H4. -apply IHl. - -intro. generalize (HH H1). intro. -inversion H5; subst. -elim (n H7). -auto. - -intro. apply HHH. right. auto. -auto. -auto. -auto. - -apply fold_left_assoc_map. -unfold EqualSetMap. unfold f'. unfold f. -intros. -destruct (Vertex.eq_dec x0 z). -rewrite MapFacts.add_eq_o. -destruct (Vertex.eq_dec x0 y0). -rewrite MapFacts.add_eq_o. -constructor. -unfold adj_set. -rewrite (MapFacts.find_o _ (Vertex.eq_sym e1)). -rewrite (MapFacts.find_o _ (Vertex.eq_sym e0)). -apply VertexSet.eq_refl. intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_eq_o. -constructor. apply VertexSet.eq_refl. -intuition. -auto. -intuition. -rewrite MapFacts.add_neq_o. -destruct (Vertex.eq_dec x0 y0). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. apply VertexSet.eq_refl. -intuition. -intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -destruct (VertexMap.find x0 s0); constructor; apply VertexSet.eq_refl. -auto. -auto. -auto. -auto. - -unfold EqualSetMap. intros. -unfold f'. unfold f. -destruct (Vertex.eq_dec x0 a0). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. -apply VertexSet.eq_refl. -intuition. -intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -apply H1. auto. auto. -Qed. - -Lemma merge_3_aux : forall e e' g, -aff_edge e -> -In_graph_edge e g -> -In_graph_edge e' g -> -aff_edge e' -> -~Edge.eq e e' -> -VertexSet.In (snd_ext (redirect (snd_ext e) (fst_ext e) e')) - (adj_set (fst_ext (redirect (snd_ext e) (fst_ext e) e')) (map_merge e (pmap g))). - -Proof. -intros e e' g q p H0 H1 Hdiff. generalize I. intro H. -assert (get_weight e = Some N0) as Hw. -destruct p. apply (AE_weights _ _ H2). -generalize (IE_weights _ _ H2). inversion q. congruence. -assert (get_weight e' = Some N0) as Hw'. -destruct H0. apply (AE_weights _ _ H0). -generalize (IE_weights _ _ H0). inversion H1. congruence. -assert (EdgeSet.In (fst_ext e, snd_ext e, Some N0) (AE g)) as He. -destruct p. rewrite (edge_eq e) in H2. -generalize (AE_weights _ _ H2). intro. simpl in H3. -rewrite H3 in H2. assumption. -generalize (IE_weights _ _ H2). inversion q. congruence. -assert (VertexSet.In (snd_ext e) (adj_set (fst_ext e) (pmap g))) as Hee. -apply (proj1 (edgemap_to_edgeset_charac _ _ _ (Some N0) (sym_pmap g))). -assumption. -assert (EdgeSet.In (fst_ext e', snd_ext e', Some N0) (AE g)) as He'. -destruct H0. rewrite (edge_eq e') in H0. -generalize (AE_weights _ _ H0). intro. simpl in H2. -rewrite H2 in H0. assumption. -generalize (IE_weights _ _ H0). intro H2. inversion H1. congruence. -assert (VertexSet.In (snd_ext e') (adj_set (fst_ext e') (pmap g))) as Hee'. -apply (proj1 (edgemap_to_edgeset_charac _ _ _ (Some N0) (sym_pmap g))). -assumption. -assert (forall a b, VertexSet.In a (adj_set b (map_merge e (pmap g))) -> - VertexSet.In b (adj_set a (map_merge e (pmap g)))) as Hsym. -intros; apply sym_map_merge_pmap; auto. - -unfold redirect. -destruct (OTFacts.eq_dec (fst_ext e') (snd_ext e)). -unfold map_merge. -set (s := (VertexSet.union - (VertexSet.remove (snd_ext e) (adj_set (fst_ext e) (pmap g))) - (VertexSet.remove (fst_ext e) (adj_set (snd_ext e) (pmap g))))) in *. -set (m := pmap g) in *. -set (f := (fun (y : VertexSet.elt) (m' : VertexMap.t VertexSet.t) => - VertexMap.add y - (VertexSet.add (fst_ext e) - (VertexSet.remove (snd_ext e) (adj_set y m))) m')) in *. -set (s' := VertexSet.remove (fst_ext e) (adj_set (snd_ext e) m)) in *. -unfold adj_set. -rewrite MapFacts.remove_neq_o. -rewrite MapFacts.add_eq_o. -unfold s. -apply VertexSet.union_3. -unfold s'. -apply VertexSet.remove_2. -intro. change_rewrite. -elim Hdiff. -rewrite (edge_eq e). rewrite edge_comm. apply eq_ordered_eq. -rewrite (edge_eq e'). unfold E.eq; change_rewrite; simpl; intuition. -rewrite Hw. rewrite Hw'. apply OptionN_as_OT.eq_refl. -unfold adj_set. rewrite (MapFacts.find_o _ (Vertex.eq_sym r)). -fold (adj_set (fst_ext e') m). -assumption. change_rewrite. intuition. -change_rewrite. -apply (In_graph_edge_diff_ext _ _ p). - -destruct (OTFacts.eq_dec (snd_ext e') (snd_ext e)); change_rewrite. -unfold map_merge. -set (s := (VertexSet.union - (VertexSet.remove (snd_ext e) (adj_set (fst_ext e) (pmap g))) - (VertexSet.remove (fst_ext e) (adj_set (snd_ext e) (pmap g))))) in *. -set (m := pmap g) in *. -set (f := (fun (y : VertexSet.elt) (m' : VertexMap.t VertexSet.t) => - VertexMap.add y - (VertexSet.add (fst_ext e) - (VertexSet.remove (snd_ext e) (adj_set y m))) m')) in *. -set (s' := VertexSet.remove (fst_ext e) (adj_set (snd_ext e) m)) in *. -unfold adj_set. -rewrite MapFacts.remove_neq_o. -rewrite MapFacts.add_neq_o. -rewrite VertexSet.fold_1. -set (f' := fun a e => f e a). -generalize VertexSet.elements_1. intro HH. -generalize (HH s' (fst_ext e')). clear HH. intro HH. -induction (VertexSet.elements s'). simpl. -assert (InA Vertex.eq (fst_ext e') nil). -apply HH. -unfold s'. apply VertexSet.remove_2. -intro. elim Hdiff. -apply eq_ordered_eq. -rewrite (edge_eq e). rewrite (edge_eq e'). -unfold E.eq; change_rewrite; simpl; intuition. -rewrite Hw. rewrite Hw'. apply OptionN_as_OT.eq_refl. -unfold adj_set. rewrite (MapFacts.find_o _ (Vertex.eq_sym r)). -apply (sym_pmap g). assumption. -inversion H2. - -cut (EqualSetMap (fold_left f' (a :: l) m) (f' (fold_left f' l m) a)). intro. -generalize (H2 (fst_ext e')). clear H2. intro H2. simpl. inversion H2;clear H2. -set (tmp := fold_left f' l m) in *. -unfold f' in H5. unfold f in H5. -destruct (Vertex.eq_dec (fst_ext e') a). -rewrite MapFacts.add_eq_o in H5. congruence. intuition. -rewrite MapFacts.add_neq_o in H5. rewrite <-H5 in IHl. -apply IHl. -intro. generalize (HH H2). clear HH. intro HH. -inversion HH; subst. -elim (n0 H6). -assumption. -auto. - -set (tmp := fold_left f' l m) in *. -unfold f' in H4. unfold f in H4. -destruct (Vertex.eq_dec (fst_ext e') a). -rewrite MapFacts.add_eq_o in H4. -rewrite H5. inversion H4; subst; clear H4. -apply VertexSet.add_1. intuition. intuition. -rewrite MapFacts.add_neq_o in H4. -rewrite <-H4 in IHl. -rewrite H5. apply IHl. -intro. generalize (HH H2). clear HH. intro HH. inversion HH; subst. -elim (n0 H7). -assumption. -auto. - -apply fold_left_assoc_map. -unfold EqualSetMap. unfold f'. unfold f. -intros. -destruct (Vertex.eq_dec x z). -rewrite MapFacts.add_eq_o. -destruct (Vertex.eq_dec x y). -rewrite MapFacts.add_eq_o. -constructor. -unfold adj_set. -rewrite (MapFacts.find_o _ (Vertex.eq_sym e1)). -rewrite (MapFacts.find_o _ (Vertex.eq_sym e0)). -apply VertexSet.eq_refl. intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_eq_o. -constructor. apply VertexSet.eq_refl. -intuition. -auto. -intuition. -rewrite MapFacts.add_neq_o. -destruct (Vertex.eq_dec x y). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. apply VertexSet.eq_refl. -intuition. -intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -destruct (VertexMap.find x s0); constructor; apply VertexSet.eq_refl. -auto. -auto. -auto. -auto. - -unfold EqualSetMap. intros. -unfold f'. unfold f. -destruct (Vertex.eq_dec x a0). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. -apply VertexSet.eq_refl. -intuition. -intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -apply H2. auto. auto. - -intro. elim Hdiff. -apply eq_ordered_eq. -rewrite (edge_eq e). rewrite (edge_eq e'). -unfold E.eq; change_rewrite; simpl; intuition. -rewrite Hw. rewrite Hw'. apply OptionN_as_OT.eq_refl. - -intuition. -unfold map_merge. -set (s := (VertexSet.union - (VertexSet.remove (snd_ext e) (adj_set (fst_ext e) (pmap g))) - (VertexSet.remove (fst_ext e) (adj_set (snd_ext e) (pmap g))))) in *. -set (m := pmap g) in *. -set (f := (fun (y : VertexSet.elt) (m' : VertexMap.t VertexSet.t) => - VertexMap.add y - (VertexSet.add (fst_ext e) - (VertexSet.remove (snd_ext e) (adj_set y m))) m')) in *. -set (s' := VertexSet.remove (fst_ext e) (adj_set (snd_ext e) m)) in *. -unfold adj_set. -rewrite MapFacts.remove_neq_o. -destruct (Vertex.eq_dec (fst_ext e') (fst_ext e)). -rewrite MapFacts.add_eq_o. -unfold s. -apply VertexSet.union_2. -apply VertexSet.remove_2. intuition. -unfold adj_set. rewrite (MapFacts.find_o _ (Vertex.eq_sym e0)). -assumption. -intuition. - -rewrite MapFacts.add_neq_o. -rewrite VertexSet.fold_1. -set (f' := fun a e => f e a). -(* -generalize VertexSet.elements_1. intro HH. -generalize (HH s' (fst_ext e')). clear HH. intro HH. -*) -induction (VertexSet.elements s'). simpl. assumption. -cut (EqualSetMap (fold_left f' (a :: l) m) (f' (fold_left f' l m) a)). intro. -generalize (H2 (fst_ext e')). clear H2. intro H2. simpl. inversion H2;clear H2. -set (tmp := fold_left f' l m) in *. -unfold f' in H5. unfold f in H5. -destruct (Vertex.eq_dec (fst_ext e') a). -rewrite MapFacts.add_eq_o in H5. congruence. intuition. -rewrite MapFacts.add_neq_o in H5. rewrite <-H5 in IHl. -apply IHl. auto. - -set (tmp := fold_left f' l m) in *. -unfold f' in H4. unfold f in H4. -destruct (Vertex.eq_dec (fst_ext e') a). -rewrite MapFacts.add_eq_o in H4. -rewrite H5. inversion H4; subst; clear H4. -apply VertexSet.add_2. apply VertexSet.remove_2. intuition. -unfold adj_set. rewrite (MapFacts.find_o _ (Vertex.eq_sym e0)). assumption. -intuition. - -rewrite MapFacts.add_neq_o in H4. -rewrite <-H4 in IHl. -rewrite H5. apply IHl. -auto. - -apply fold_left_assoc_map. -unfold EqualSetMap. unfold f'. unfold f. -intros. -destruct (Vertex.eq_dec x z). -rewrite MapFacts.add_eq_o. -destruct (Vertex.eq_dec x y). -rewrite MapFacts.add_eq_o. -constructor. -unfold adj_set. -rewrite (MapFacts.find_o _ (Vertex.eq_sym e1)). -rewrite (MapFacts.find_o _ (Vertex.eq_sym e0)). -apply VertexSet.eq_refl. intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_eq_o. -constructor. apply VertexSet.eq_refl. -intuition. -auto. -intuition. -rewrite MapFacts.add_neq_o. -destruct (Vertex.eq_dec x y). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. apply VertexSet.eq_refl. -intuition. -intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -destruct (VertexMap.find x s0); constructor; apply VertexSet.eq_refl. -auto. -auto. -auto. -auto. - -unfold EqualSetMap. intros. -unfold f'. unfold f. -destruct (Vertex.eq_dec x a0). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. -apply VertexSet.eq_refl. -intuition. -intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -apply H2. auto. auto. - -auto. -intuition. -Qed. - -Lemma merge_5 : forall e e' g p q, -In_graph_edge e' g -> -~Edge.eq e e' -> -aff_edge e' -> -~Interfere (fst_ext (redirect (snd_ext e) (fst_ext e) e')) - (snd_ext (redirect (snd_ext e) (fst_ext e) e')) - (merge e g p q) -> -Prefere (fst_ext (redirect (snd_ext e) (fst_ext e) e')) - (snd_ext (redirect (snd_ext e) (fst_ext e) e')) - (merge e g p q). - -Proof. -intros e e' g p q H0 H1 H2 H3. generalize I. intro H. -unfold Prefere. exists N0. left. -unfold merge. unfold AE. simpl. -apply (proj2 (edgemap_to_edgeset_charac _ _ _ (Some N0) (sym_pmap_merge_map e g q p))). -apply resolve_conflicts_map_5. -assumption. -assumption. -apply merge_3_aux; assumption. -intro. elim H3. unfold Interfere. right. unfold merge. -unfold IE. simpl. -apply (proj2 (edgemap_to_edgeset_charac _ _ _ None (sym_imap_merge_map e g q p))). -assumption. -Qed. -(* -unfold change. -destruct (OTFacts.eq_dec (fst_ext e') (snd_ext e)). change_rewrite. -apply VertexSet.remove_2. -apply (In_graph_edge_diff_ext e g p). -apply (proj1 (In_graph_edge_in_ext _ _ p)). -destruct (OTFacts.eq_dec (snd_ext e') (snd_ext e)). change_rewrite. -apply VertexSet.remove_2. -intro. elim (In_graph_edge_diff_ext e' g H0). -apply Vertex.eq_trans with (y := snd_ext e); auto. -apply (proj1 (In_graph_edge_in_ext _ _ H0)). -apply VertexSet.remove_2. -auto. -apply (proj1 (In_graph_edge_in_ext _ _ H0)). -Qed. -*) - -Lemma precolored_merge2 : forall x e g p q, -(VertexSet.In x (VertexSet.remove (snd_ext e) (precolored g)) <-> - VertexSet.In x (precolored (merge e g p q))). - -Proof. -intros x e g HH q. split; intros. -unfold merge. -unfold precolored. simpl. unfold is_precolored. simpl. -apply VertexSet.filter_3. -unfold compat_bool;intros. -rewrite (mreg_ext _ _ H0). auto. -apply VertexSet.remove_2. -intro H1. elim (VertexSet.remove_1 H1 H). -apply (VertexSet.filter_1 (compat_bool_is_precolored _) (VertexSet.remove_3 H)). -generalize (VertexSet.filter_2 (compat_bool_is_precolored _) (VertexSet.remove_3 H)). intro H0. -assumption. - -unfold precolored, is_precolored in *. unfold merge in H. simpl in H. -apply VertexSet.remove_2. intro. -generalize (VertexSet.filter_1 (compat_bool_is_precolored (merge e g HH q)) H). intro. -elim (VertexSet.remove_1 H0 H1). -apply VertexSet.filter_3. -unfold compat_bool. apply mreg_ext. -generalize (VertexSet.filter_1 (compat_bool_is_precolored (merge e g HH q)) H). intros. -apply (VertexSet.remove_3 H0). -apply (VertexSet.filter_2 (compat_bool_is_precolored (merge e g HH q)) H). -Qed. - -(* spec of delete_preference_edges *) -Lemma delete_preference_edges_prec : forall r g Hin, -VertexSet.Equal (precolored (delete_preference_edges r g Hin)) (precolored g). - -Proof. -intros. -unfold delete_preference_edges. -unfold precolored. simpl. -unfold is_precolored. simpl. -apply VertexSet.eq_refl. -Qed. - -Lemma delete_preference_edges_1 : forall e r g Hin, -In_graph_edge e (delete_preference_edges r g Hin) -> In_graph_edge e g. - -Proof. -intros. -unfold delete_preference_edges in H. -destruct H;[left; generalize (AE_weights _ _ H); unfold AE in *| - right; generalize (IE_weights _ _ H); unfold IE in *]; simpl in H; intros. -rewrite (edge_eq e) in H. -rewrite H0 in H. generalize (proj1 (edgemap_to_edgeset_charac _ _ _ _(sym_pmap_delete_preferences r g)) H). intro. -rewrite (edge_eq e). -rewrite H0. apply (proj2 (edgemap_to_edgeset_charac _ _ _ _ (sym_pmap g))). -apply (delete_preference_sub _ _ _ _ H1). -assumption. -Qed. - -Lemma IE_delete_preference_eq : forall x g H, -EdgeSet.Equal (IE g) (IE (delete_preference_edges x g H)). - -Proof. -intros. -unfold delete_preference_edges. -destruct (In_graph_dec x g). -unfold IE. simpl. apply EdgeSet.eq_refl. -apply EdgeSet.eq_refl. -Qed. - -Lemma delete_preference_edges_2 : forall e r g Hin, -In_graph_edge e g -> -~incident e r -> -In_graph_edge e (delete_preference_edges r g Hin). - -Proof. -intros. -destruct H;[left|right]. -generalize (AE_weights _ _ H). intro Hw. -unfold AE in *. -rewrite (edge_eq e). rewrite Hw. -unfold delete_preference_edges. -apply (proj2 (edgemap_to_edgeset_charac _ _ _ _ (sym_pmap_delete_preferences r g))). -unfold pmap_delete_preferences. -rewrite VertexSet.fold_1. -induction (VertexSet.elements (adj_set r (pmap g))). simpl. -unfold adj_set. rewrite InterfFacts.add_neq_o. -rewrite (edge_eq e) in H. rewrite Hw in H. -apply (proj1 (edgemap_to_edgeset_charac _ _ _ _ (sym_pmap g)) H). -intro H1. elim H0. left. auto. - -cut (EqualSetMap - (fold_left - (fun (a : VertexMap.t VertexSet.t) (e : VertexSet.elt) => - VertexMap.add e (VertexSet.remove r (adj_set e (pmap g))) a) (a :: l) - (VertexMap.add r VertexSet.empty (pmap g))) - (VertexMap.add a - (VertexSet.remove r - (adj_set a (pmap g))) - (fold_left - (fun (a : VertexMap.t VertexSet.t) (e : VertexSet.elt) => - VertexMap.add e (VertexSet.remove r (adj_set e (pmap g))) a) l - (VertexMap.add r VertexSet.empty (pmap g))))). intro. -unfold EqualSetMap in H1. generalize (H1 (fst_ext e)). clear H1. intro H1. -inversion H1. simpl. unfold adj_set in *. rewrite <-H3. -destruct (Vertex.eq_dec (fst_ext e) a). -rewrite InterfFacts.add_eq_o in H4. inversion H4. -intuition. -rewrite InterfFacts.add_neq_o in H4. rewrite <-H4 in *. -elim (VertexSet.empty_1 IHl). -auto. -simpl. unfold adj_set in *. rewrite <-H2. -rewrite H4. clear H1. clear H2. clear H4. -destruct (Vertex.eq_dec (fst_ext e) a). -rewrite InterfFacts.add_eq_o in H3. -inversion H3. clear H2. -apply VertexSet.remove_2. -intro. elim H0. right. auto. -clear H3. -rewrite (edge_eq e) in H. rewrite Hw in H. -rewrite (InterfFacts.find_o _ (Vertex.eq_sym e0)). -apply (proj1 (edgemap_to_edgeset_charac _ _ _ _(sym_pmap g)) H). -intuition. -rewrite (InterfFacts.add_neq_o) in H3. -case_eq (VertexMap.find (elt:=VertexSet.t) (fst_ext e) - (fold_left - (fun (a : VertexMap.t VertexSet.t) (e : VertexSet.elt) => - VertexMap.add e - (VertexSet.remove r - match VertexMap.find (elt:=VertexSet.t) e (pmap g) with - | Some x => x - | None => VertexSet.empty - end) a) l (VertexMap.add r VertexSet.empty (pmap g)))); -intros; rewrite H1 in *. -inversion H3. assumption. -inversion H3. -auto. - -apply fold_left_assoc_map. -intros. -unfold EqualSetMap. intros. -destruct (Vertex.eq_dec x z). -rewrite InterfFacts.add_eq_o. -destruct (Vertex.eq_dec x y). -rewrite InterfFacts.add_eq_o. -constructor. -generalize (Vertex.eq_trans (Vertex.eq_sym e0) e1). intro HHH. -unfold adj_set. -rewrite (InterfFacts.find_o _ HHH). apply VertexSet.eq_refl. -intuition. - -rewrite InterfFacts.add_neq_o. -rewrite InterfFacts.add_eq_o. -constructor. apply VertexSet.eq_refl. -intuition. -auto. -intuition. - -rewrite InterfFacts.add_neq_o. -destruct (Vertex.eq_dec x y). -rewrite InterfFacts.add_eq_o. -rewrite InterfFacts.add_eq_o. -constructor. apply VertexSet.eq_refl. -intuition. -intuition. - -intros. -unfold EqualSetMap. intros. -destruct (Vertex.eq_dec x y). -rewrite InterfFacts.add_eq_o. -rewrite InterfFacts.add_eq_o. -constructor. apply VertexSet.eq_refl. -intuition. -intuition. -rewrite InterfFacts.add_neq_o. -rewrite InterfFacts.add_neq_o. -rewrite InterfFacts.add_neq_o. -apply EqualSetMap_refl. -auto. -auto. -auto. -auto. - -unfold EqualSetMap. intros. -destruct (Vertex.eq_dec x a0). -rewrite InterfFacts.add_eq_o. -rewrite InterfFacts.add_eq_o. -constructor. apply VertexSet.eq_refl. -intuition. -intuition. -rewrite InterfFacts.add_neq_o. -rewrite InterfFacts.add_neq_o. -apply (H1 x). -auto. -auto. - -rewrite <-IE_delete_preference_eq. auto. -Qed. - -Lemma V_delete_preference_eq : forall x g Hin, -VertexSet.Equal (V g) (V (delete_preference_edges x g Hin)). - -Proof. -intros. -unfold delete_preference_edges. -destruct (In_graph_dec x g); simpl; apply VertexSet.eq_refl. -Qed. - -Lemma in_delete_preference_not_incident : forall e r g Hdep, -EdgeSet.In e (AE (delete_preference_edges r g Hdep)) -> ~incident e r. - -Proof. -intros. -generalize (AE_weights _ _ H). intro Hw. -generalize (proj1 (In_graph_aff_edge_in_AE _ _) H). intro Hin. -unfold AE in H. -rewrite (edge_eq e) in H. rewrite Hw in H. -unfold delete_preference_edges in H. -generalize (proj1 (edgemap_to_edgeset_charac _ _ _ _(sym_pmap_delete_preferences r g)) H). clear H. intro H. -unfold pmap_delete_preferences in H. -rewrite VertexSet.fold_1 in H. -generalize VertexSet.elements_1. intro HH. -generalize (HH (adj_set r (pmap g)) (fst_ext e)). clear HH. intro HH. -generalize VertexSet.elements_2. intro HH1. -generalize (HH1 (adj_set r (pmap g))). clear HH1. intro HH1. - -assert (forall z, In z (VertexSet.elements (adj_set r (pmap g))) -> ~Vertex.eq r z) as Hneq. -clear H HH Hin Hw Hdep. -intros. intro H0. -induction (VertexSet.elements (adj_set r (pmap g))). inversion H. -simpl in H. destruct H. subst. -assert (VertexSet.In r (adj_set r (pmap g))). -apply HH1. left. auto. -elim (not_eq_extremities g r r). -right. assumption. -auto. -apply IHl. -intros. apply HH1. auto. -assumption. - -induction (VertexSet.elements (adj_set r (pmap g))). simpl in H. -unfold adj_set in H. -destruct (Vertex.eq_dec (fst_ext e) r). -rewrite InterfFacts.add_eq_o in H. -elim (VertexSet.empty_1 H). intuition. -rewrite InterfFacts.add_neq_o in H. -destruct (Vertex.eq_dec (snd_ext e) r). -assert (InA Vertex.eq (fst_ext e) nil). -apply HH. -apply (proj1 (edgemap_to_edgeset_charac (pmap g) r (fst_ext e) (Some N0) (sym_pmap g))). -assert (eq (r,fst_ext e, Some N0) e). -rewrite edge_comm. -apply eq_ordered_eq. -unfold E.eq; simpl; intuition. apply Regs.eq_refl. -rewrite <-Hw. apply OptionN_as_OT.eq_refl. -rewrite H0. -destruct Hin. -generalize (delete_preference_edges_1 e r g Hdep H2). clear H2. intro H2. -destruct H2. -unfold AE in H2. assumption. -generalize (IE_weights _ _ H2). intro Hw'. rewrite Hw' in Hw. inversion Hw. -inversion H0. -intro H0. destruct H0. -elim n. auto. -elim n0. auto. -auto. - -set (f := (fun (a : VertexMap.t VertexSet.t) (e : VertexSet.elt) => - VertexMap.add e (VertexSet.remove r (adj_set e (pmap g))) a)) in *. -set (s := VertexMap.add r VertexSet.empty (pmap g)) in *. - -assert (EqualSetMap (fold_left f (a :: l) s) (f (fold_left f l s) a)). -apply fold_left_assoc_map. - -unfold f. unfold EqualSetMap. intros. -destruct (Vertex.eq_dec x z). -rewrite MapFacts.add_eq_o. -destruct (Vertex.eq_dec x y). -rewrite MapFacts.add_eq_o. -constructor. -apply Props.Equal_remove. -unfold adj_set. -rewrite (MapFacts.find_o _ (Vertex.eq_trans (Vertex.eq_sym e0) e1)). -apply VertexSet.eq_refl. intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_eq_o. -constructor. -apply VertexSet.eq_refl. -intuition. -auto. -intuition. -rewrite MapFacts.add_neq_o. -destruct (Vertex.eq_dec x y). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. -apply VertexSet.eq_refl. -intuition. -intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -destruct (VertexMap.find x s0); constructor. -apply VertexSet.eq_refl. -auto. -auto. -auto. -auto. - -unfold f. unfold EqualSetMap. intros. -destruct (Vertex.eq_dec x a0). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. -apply VertexSet.eq_refl. -intuition. -intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -apply H0. -auto. -auto. - -generalize (H0 (fst_ext e)). clear H0. intro H0. -simpl in H. unfold adj_set in H. -inversion H0. subst. -rewrite <-H2 in H. -elim (VertexSet.empty_1 H). -rewrite <-H1 in H. -set (tmp := fold_left f l s) in *. -unfold f in H2. -destruct (Vertex.eq_dec (fst_ext e) a). -rewrite MapFacts.add_eq_o in H2. -intro. -inversion H2. subst. clear H2. -rewrite H3 in H. -destruct H4. - -elim (Hneq a). -left. auto. -apply (Vertex.eq_trans H2 e0). -elim (VertexSet.remove_1 H2 H). -intuition. - -rewrite MapFacts.add_neq_o in H2. -inversion H2. subst. clear H2. -apply IHl. -clear IHl. -unfold adj_set. rewrite <-H5. -rewrite <-H3. assumption. -intro. -assert (InA Vertex.eq (fst_ext e) (a :: l)). -apply HH. assumption. -inversion H4; subst. -elim (n H7). -assumption. -intros. apply HH1. auto. -intros. apply Hneq. right. auto. -auto. -Qed. - -Add Morphism In_graph_edge : In_graph_edge_m. - -Proof. -unfold In_graph_edge;intros x y H g. -fold (eq x y) in H. -split;intro H0;[rewrite <-H|rewrite H];assumption. -Qed. - -(* There cannot exist both an interference and - a preference between two vertices *) -Lemma interf_pref_conflict : forall x y g, -Prefere x y g /\ Interfere x y g -> False. - -Proof. -intros. -unfold Prefere in H. unfold Interfere in H. -destruct H. -apply (simple_graph g x y). -split. -destruct H0. -generalize (proj1 (In_graph_aff_edge_in_AE _ _) H0). intro H1. -destruct H1 as [H1 _]. -unfold aff_edge in H1. -destruct H1. inversion H1. -unfold IE in H0. -rewrite edge_comm in H0. apply (proj1 (edgemap_to_edgeset_charac _ _ _ _ (sym_imap g)) H0). -destruct H. destruct H. generalize (AE_weights _ _ H). simpl. intro. -rewrite H1 in H. -rewrite edge_comm in H. unfold AE in H. apply (proj1 (edgemap_to_edgeset_charac _ _ _ _(sym_pmap g)) H). -generalize (proj1 (In_graph_interf_edge_in_IE _ _) H). intro H1. -destruct H1 as [H1 _]. -unfold interf_edge in H1. inversion H1. -Qed. - -Lemma is_simple_graph : forall e e' g, -In_graph_edge e g -> -In_graph_edge e' g -> -weak_eq e e' -> -eq e e'. - -Proof. -intros. -unfold weak_eq in H1. -destruct H; destruct H0. -generalize (AE_weights _ _ H); intros. -generalize (AE_weights _ _ H0); intros. -destruct H1; destruct H1. -rewrite (edge_eq e). rewrite (edge_eq e'). rewrite H2. rewrite H3. Eq_eq. -rewrite (edge_eq e). rewrite (edge_eq e'). rewrite H2. rewrite H3. Eq_comm_eq. -elim (interf_pref_conflict (fst_ext e') (snd_ext e') g). -unfold Prefere, Interfere; split. -destruct H1; destruct H1. exists N0. -rewrite <-(AE_weights _ _ H). -assert (eq (fst_ext e', snd_ext e', get_weight e) (fst_ext e, snd_ext e, get_weight e)) by Eq_eq. -rewrite H3. rewrite <-(edge_eq e). rewrite In_graph_aff_edge_in_AE in H. destruct H. intuition. -exists N0. rewrite <-(AE_weights _ _ H). -assert (eq (fst_ext e', snd_ext e', get_weight e) (fst_ext e, snd_ext e, get_weight e)) by Eq_comm_eq. -rewrite H3. rewrite <-(edge_eq e). rewrite In_graph_aff_edge_in_AE in H. destruct H. intuition. -rewrite <-(IE_weights _ _ H0). rewrite In_graph_interf_edge_in_IE in H0. destruct H0. rewrite (edge_eq e') in H2. intuition. -elim (interf_pref_conflict (fst_ext e') (snd_ext e') g). -unfold Prefere, Interfere; split. -exists N0. rewrite <-(AE_weights _ _ H0). -rewrite (edge_eq e') in H0. rewrite In_graph_aff_edge_in_AE in H0. destruct H0. auto. -rewrite <-(IE_weights _ _ H). -destruct H1; destruct H1. -assert (eq (fst_ext e', snd_ext e', get_weight e) (fst_ext e, snd_ext e, get_weight e)) by Eq_eq. -rewrite H3. rewrite <-(edge_eq e). rewrite In_graph_interf_edge_in_IE in H. destruct H. auto. -assert (eq (fst_ext e', snd_ext e', get_weight e) (fst_ext e, snd_ext e, get_weight e)) by Eq_comm_eq. -rewrite H3. rewrite <-(edge_eq e). rewrite In_graph_interf_edge_in_IE in H. destruct H. auto. -generalize (IE_weights _ _ H); intros. -generalize (IE_weights _ _ H0); intros. -destruct H1; destruct H1. -rewrite (edge_eq e). rewrite (edge_eq e'). rewrite H2. rewrite H3. Eq_eq. -rewrite (edge_eq e). rewrite (edge_eq e'). rewrite H2. rewrite H3. Eq_comm_eq. -Qed. - -Lemma is_precolored_ext : forall x y g, -Vertex.eq x y -> -is_precolored x g = is_precolored y g. - -Proof. -intros. apply compat_bool_is_precolored. assumption. -Qed. - -(* This module respects the interface *) - -(* A vertex x is in (remove_vertex r g) iff it is in g - and it is different from r*) -Lemma In_remove_vertex : forall x r g, -In_graph x (remove_vertex r g) <-> (In_graph x g /\ ~Vertex.eq x r). - -Proof. -split; intros. -split. apply in_remove_in with (r:=r). auto. -intro. rewrite H0 in H. elim (not_in_remove r g H). -destruct H. apply In_remove; auto. -Qed. - -(* The precolored vertices of (remove_vertex r g) are - the precolored vertices of g, minus r *) -Lemma precolored_remove_vertex : forall r g, -VertexSet.Equal (precolored (remove_vertex r g)) - (VertexSet.remove r (precolored g)). - -Proof. -split; intros. -rewrite precolored_remove_vertex2; auto. -rewrite <-precolored_remove_vertex2; auto. -Qed. - -(* An edge e is in (remove_vertex r g) iff it is in g - and is not incident to r *) -Lemma In_remove_edge : forall e r g, -In_graph_edge e (remove_vertex r g) <-> (In_graph_edge e g /\ ~incident e r). - -Proof. -split; intros. -split. apply in_remove_in_edge with (r:=r). auto. -intro. elim (not_in_remove_edge _ _ _ H0 H). -destruct H. apply In_remove_edge_; auto. -Qed. - -(* Specification of merge *) - -(* A vertex is in (merge e g p q) iff x is in g and - x is not equal to the second endpoint of g *) -Lemma In_merge_vertex : forall g e x p q, -In_graph x (merge e g p q) <-> (In_graph x g /\ ~Vertex.eq x (snd_ext e)). - -Proof. -split; intros. -split. apply (In_merge_in g e x p q H). -intro. rewrite H0 in H. elim (merge_1 e g p q H). -destruct H. apply merge_2; auto. -Qed. - -(* If an interference edge e' is in the graph g then - its redirection from the second endpoint of e - to the first endpoint of e is in (merge e g p q) *) -Lemma In_merge_interf_edge : forall e e' g p q, -In_graph_edge e' g -> -interf_edge e' -> -In_graph_edge (redirect (snd_ext e) (fst_ext e) e') (merge e g p q). - -Proof. -intros. apply merge_3; auto. -Qed. - -(* If a preference edge e' different from e is in the graph g, - and iff the endpoints of its redirection from the second - endpoint of e to the first endpoint of e do not interfere, - then these endpoints are linked with an affinity edge, whose - weight may be different from the one of e' *) -Lemma In_merge_pref_edge : forall e e' g p q, -In_graph_edge e' g -> -~Edge.eq e e' -> -aff_edge e' -> -~Interfere (fst_ext (redirect (snd_ext e) (fst_ext e) e')) - (snd_ext (redirect (snd_ext e) (fst_ext e) e')) - (merge e g p q) -> - Prefere (fst_ext (redirect (snd_ext e) (fst_ext e) e')) - (snd_ext (redirect (snd_ext e) (fst_ext e) e')) - (merge e g p q). - -Proof. -intros. apply merge_5; auto. -Qed. - -(* Inversely, if e' is an edge of (merge e g p q) then there exists - an edge a of g, such that e' is weakly equal to the redirection - of a from the second endpoint of e to the first endpoint of e *) -Lemma In_merge_edge_inv : forall e e' g p q, -In_graph_edge e' (merge e g p q) -> -exists a, In_graph_edge a g /\ weak_eq e' (redirect (snd_ext e) (fst_ext e) a) /\ - same_type a e'. - -Proof. -intros. apply (merge_4 e e' g p q); auto. -Qed. - -(* The precolored vertices of (merge e g p q) are the ones of g, - minus the second endpoint of e *) -Lemma precolored_merge : forall e g p q, -VertexSet.Equal (precolored (merge e g p q)) - (VertexSet.remove (snd_ext e) (precolored g)). - -Proof. -split; intros. -rewrite <-precolored_merge2 in H. auto. -rewrite <-precolored_merge2. auto. -Qed. - -(* Specification of delete_preference_edges *) - -(* A vertex is in (delete_preference_edges r g p) iff it is in g *) -Lemma In_delete_preference_edges_vertex : forall x r g p, -In_graph x (delete_preference_edges r g p) <-> In_graph x g. - -Proof. -unfold In_graph. split; intros. -rewrite <-V_delete_preference_eq in H. auto. -rewrite <-V_delete_preference_eq. auto. -Qed. - -(* The precolored vertices of (delete_preference_edges r g p) - iff it is precolored in g *) -Lemma precolored_delete_preference_edges : forall r g p, -VertexSet.Equal (precolored (delete_preference_edges r g p)) - (precolored g). - -Proof. -intros. apply delete_preference_edges_prec. -Qed. - -(* An edge e is in (delete_preference_edges r g p) iff - if is in g and it is not an affinity edge incident to r *) -Lemma In_delete_preference_edges_edge : forall e r g p, -In_graph_edge e (delete_preference_edges r g p) <-> -(In_graph_edge e g /\ ~(aff_edge e /\ incident e r)). - -Proof. -split; intros. -split. apply (delete_preference_edges_1 _ _ _ _ H). -intro. destruct H0. -assert (~incident e r). -apply (in_delete_preference_not_incident e r g p). -rewrite In_graph_aff_edge_in_AE. split; auto. -elim (H2 H1). -destruct H. destruct H. -rewrite In_graph_aff_edge_in_AE in H. destruct H. -apply delete_preference_edges_2; auto. -right. rewrite <-IE_delete_preference_eq. auto. -Qed. - -Definition interference_adj v g := -adj_set v (imap g). - -Definition preference_adj v g := -adj_set v (pmap g). - -(* Definition of the interference and preference degrees *) -Definition interf_degree g v := VertexSet.cardinal (interference_adj v g). -Definition pref_degree g v := VertexSet.cardinal (preference_adj v g). - -(* Definition of the low-degree function, - returns true iff the interference degree of v in g is strictly lower than K *) -Definition has_low_degree g K v := -if le_lt_dec K (interf_degree g v) then false else true. - -(* Definition of the move-related function, - returns true iff the vertex is move-related *) -Definition move_related g x := negb (VertexSet.is_empty (preference_adj x g)). - -Lemma in_pref_pref : forall x y g, -VertexSet.In x (preference_adj y g) -> -exists w, In_graph_edge (x,y,Some w) g. - -Proof. -intros. -unfold preference_adj in H. -exists N0. -left. unfold AE. -rewrite edge_comm. apply (proj2 (edgemap_to_edgeset_charac _ _ _ _ (sym_pmap g))). assumption. -Qed. - -Lemma pref_in_pref : forall x y g w, -In_graph_edge (x,y,Some w) g -> -VertexSet.In x (preference_adj y g). - -Proof. -intros. -unfold preference_adj. -destruct H. -generalize (AE_weights _ _ H). intro. -unfold AE in H. simpl in H0. rewrite H0 in H. -rewrite edge_comm in H. apply (proj1 (edgemap_to_edgeset_charac _ _ _ _(sym_pmap g)) H). -generalize (proj1 (In_graph_interf_edge_in_IE _ _) H). intro. -destruct H0. -unfold interf_edge in H0. inversion H0. -Qed. - -Lemma in_interf_interf : forall x y g, -VertexSet.In x (interference_adj y g) -> -In_graph_edge (x,y,None) g. - -Proof. -intros. -unfold preference_adj in H. -right. unfold IE. -rewrite edge_comm. apply (proj2 (edgemap_to_edgeset_charac _ _ _ _ (sym_imap g))). assumption. -Qed. - -Lemma interf_in_interf : forall x y g, -In_graph_edge (x,y,None) g -> -VertexSet.In x (interference_adj y g). - -Proof. -intros. -unfold interference_adj. -destruct H. -generalize (AE_weights _ _ H). intro. inversion H0. -rewrite edge_comm in H. apply (proj1 (edgemap_to_edgeset_charac _ _ _ _(sym_imap g)) H). -Qed. - -Lemma compat_interference_adj : forall x y g, -Vertex.eq x y -> -interference_adj x g = interference_adj y g. - -Proof. -intros. -unfold interference_adj. -unfold adj_set. -rewrite (InterfFacts.find_o _ H). reflexivity. -Qed. - -Lemma compat_preference_adj : forall x y g, -Vertex.eq x y -> -preference_adj x g = preference_adj y g. - -Proof. -intros. -unfold preference_adj. -unfold adj_set. -rewrite (InterfFacts.find_o _ H). reflexivity. -Qed. - -Lemma compat_bool_move : forall g, -compat_bool Vertex.eq (move_related g). - -Proof. -intros. -unfold compat_bool. -intros. -unfold move_related. -rewrite (compat_preference_adj _ _ _ H). -reflexivity. -Qed. - -(* characterisation of move relation *) - -Lemma move_related_charac : forall x g, -move_related g x = true -> -exists e, aff_edge e /\ In_graph_edge e g /\ incident e x. - -Proof. -intros. -unfold move_related in H. -case_eq (VertexSet.is_empty (preference_adj x g)); intros. -rewrite H0 in H. inversion H. generalize H0. clear H H0. intro H. -case_eq (VertexSet.choose (preference_adj x g)); intros. -exists (x,e,Some N0). -split. -exists N0. auto. -generalize (VertexSet.choose_1 H0). clear H0. intro H0. -split. -left. unfold AE. apply (proj2 (edgemap_to_edgeset_charac _ _ _ _(sym_pmap g))). -assumption. -left. auto. -generalize (VertexSet.choose_2 H0). clear H0. intro H0. -rewrite (VertexSet.is_empty_1 H0) in H. inversion H. -Qed. - -(* the inversion characterisation of move related *) - -Lemma move_related_charac2 : forall x g e, -aff_edge e -> -In_graph_edge e g -> -incident e x -> -move_related g x = true. - -Proof. -intros. -unfold move_related. -case_eq (VertexSet.is_empty (preference_adj x g)); intros. -generalize (VertexSet.is_empty_2 H2). clear H2. intro H2. -destruct H1. -elim (H2 (snd_ext e)). -apply pref_in_pref with (w:=N0). -rewrite edge_comm. -assert (eq (x,snd_ext e, Some N0) e). -apply eq_ordered_eq. -unfold E.eq; simpl; intuition. apply Regs.eq_refl. -destruct H0. generalize (AE_weights _ _ H0). intro. -rewrite <-H3. apply OptionN_as_OT.eq_refl. -generalize (proj1 (In_graph_interf_edge_in_IE _ _) H0). intros. -destruct H3. -unfold aff_edge in H. rewrite H3 in H. destruct H. inversion H. -rewrite H3. assumption. - -elim (H2 (fst_ext e)). -apply pref_in_pref with (w:=N0). -assert (eq (fst_ext e, x, Some N0) e). -apply eq_ordered_eq. -unfold E.eq; simpl; intuition. apply Regs.eq_refl. -destruct H0. generalize (AE_weights _ _ H0). intro. -rewrite <-H3. apply OptionN_as_OT.eq_refl. -generalize (proj1 (In_graph_interf_edge_in_IE _ _) H0). intros. -destruct H3. -unfold aff_edge in H. rewrite H3 in H. destruct H. inversion H. -rewrite H3. assumption. -auto. -Qed. -Definition WS := (VertexSet.t*VertexSet.t*VertexSet.t*EdgeSet.t)%type. - -Definition get_spillWL (w : WS) := fst (fst (fst w)). -Definition get_freezeWL (w : WS) := snd (fst (fst w)). -Definition get_simplifyWL (w : WS) := snd (fst w). -Definition get_movesWL (w : WS) := snd w. - -Lemma compat_bool_low : forall g palette, -compat_bool Vertex.eq (has_low_degree g palette). - -Proof. -unfold compat_bool. intros. -unfold has_low_degree, interf_degree. -rewrite (compat_interference_adj _ _ _ H). -reflexivity. -Qed. - -Definition WS_properties g K (WL : WS) : Prop := -(forall x, VertexSet.In x (get_spillWL WL) <-> has_low_degree g K x = false /\ In_graph x g /\ ~VertexSet.In x (precolored g)) /\ -(forall x, VertexSet.In x (get_freezeWL WL) <-> has_low_degree g K x = true /\ (move_related g) x = true /\ ~VertexSet.In x (precolored g)) /\ -(forall x, VertexSet.In x (get_simplifyWL WL) <-> has_low_degree g K x = true /\ (move_related g) x = false /\ In_graph x g /\ ~VertexSet.In x (precolored g)) /\ -(forall e, EdgeSet.In e (get_movesWL WL) <-> aff_edge e /\ In_graph_edge e g). - -Definition get_WL g palette := -let not_pre := VertexSet.diff (V g) (precolored g) in -let (low, spill) := VertexSet.partition (has_low_degree g palette) not_pre in -let (free, simp) := VertexSet.partition (move_related g) low in -(spill, free, simp, AE g). - -Module Import RegOTFacts := MyOTFacts Vertex. - -Lemma move_related_in : forall g x, -move_related g x = true -> -In_graph x g. - -Proof. -intros. -generalize (move_related_charac _ _ H). clear H. intro H. -destruct H as [e H]. destruct H. destruct H0. -apply (proj1 (extremities_pmap g x)). -destruct H0. -generalize (AE_weights _ _ H0). intro Hw. -unfold AE in *. -destruct H1. -rewrite (edge_eq e) in H0. -simpl in Hw. rewrite Hw in H0. -generalize (proj1 (edgemap_to_edgeset_charac _ _ _ _(sym_pmap g)) H0). intro H2. -apply (proj2 (InterfFacts.in_find_iff _ _)). -unfold adj_set in H2. -case_eq (VertexMap.find (fst_ext e) (pmap g)); intros; rewrite H3 in H2. -intro Helim. -rewrite (InterfFacts.find_o _ H1) in Helim. -rewrite H3 in Helim. inversion Helim. -elim (VertexSet.empty_1 H2). - -rewrite (edge_eq e) in H0. rewrite edge_comm in H0. -simpl in Hw. rewrite Hw in H0. -generalize (proj1 (edgemap_to_edgeset_charac _ _ _ _(sym_pmap g)) H0). intro H2. -apply (proj2 (InterfFacts.in_find_iff _ _)). -unfold adj_set in H2. -case_eq (VertexMap.find (snd_ext e) (pmap g)); intros; rewrite H3 in H2. -intro Helim. -rewrite (InterfFacts.find_o _ H1) in Helim. -rewrite H3 in Helim. inversion Helim. -elim (VertexSet.empty_1 H2). - -generalize (proj1 (In_graph_interf_edge_in_IE _ _) H0). intros. -destruct H2. unfold aff_edge in H. rewrite H2 in H. inversion H. inversion H4. -Qed. - -Lemma WS_prop_get : forall g palette, -WS_properties g palette (get_WL g palette). - -Proof. -intros. -unfold get_WL. -set (not_pre := VertexSet.diff (V g) (precolored g)) in *. -case_eq (VertexSet.partition (has_low_degree g palette) not_pre). -intros low spill H. -case_eq (VertexSet.partition (move_related g) low). -intros free simp H0. -unfold WS_properties. -unfold get_spillWL. unfold get_simplifyWL. unfold get_freezeWL. unfold get_movesWL. simpl. -assert (VertexSet.Equal low (VertexSet.filter (has_low_degree g palette) not_pre)). -assert (low = fst (VertexSet.partition (has_low_degree g palette) not_pre)). -rewrite H. auto. -rewrite H1. apply VertexSet.partition_1. apply compat_bool_low. -assert (VertexSet.Equal spill (VertexSet.filter (fun x => negb (has_low_degree g palette x)) not_pre)). -assert (spill = snd (VertexSet.partition (has_low_degree g palette) not_pre)). -rewrite H. auto. -rewrite H2. apply VertexSet.partition_2. apply compat_bool_low. -assert (VertexSet.Equal free (VertexSet.filter (move_related g) low)). -assert (free = fst (VertexSet.partition (move_related g) low)). -rewrite H0. auto. -rewrite H3. apply VertexSet.partition_1. apply compat_bool_move. -assert (VertexSet.Equal simp (VertexSet.filter (fun x => negb (move_related g x)) low)). -assert (simp = snd (VertexSet.partition (move_related g) low)). -rewrite H0. auto. -rewrite H4. apply VertexSet.partition_2. apply compat_bool_move. -split; intros. -split; intros. -rewrite H2 in H5. -split. -generalize (VertexSet.filter_2 (compat_not_compat (compat_bool_low _ _)) H5). -destruct (has_low_degree g palette x); intuition. -split. -generalize (VertexSet.filter_1 (compat_not_compat (compat_bool_low _ _)) H5). intro. -apply (VertexSet.diff_1 H6). -generalize (VertexSet.filter_1 (compat_not_compat (compat_bool_low _ _)) H5). intro. -apply (VertexSet.diff_2 H6). -rewrite H2. -apply VertexSet.filter_3. -apply compat_not_compat. apply compat_bool_low. -apply VertexSet.diff_3; intuition. -destruct H5. rewrite H5. auto. - -split; intros. -split; intros. -rewrite H3 in H5. -generalize (VertexSet.filter_1 (compat_bool_move _) H5). intro H6. -generalize (VertexSet.filter_2 (compat_bool_move _) H5). clear H5. intro H5. -rewrite H1 in H6. -generalize (VertexSet.filter_1 (compat_bool_low _ _) H6). intro H7. -generalize (VertexSet.filter_2 (compat_bool_low _ _) H6). clear H6. intro H6. -generalize (VertexSet.diff_2 H7). intuition. - -rewrite H3. apply VertexSet.filter_3. -apply compat_bool_move. -rewrite H1. -apply VertexSet.filter_3. -apply compat_bool_low. -apply VertexSet.diff_3. -apply move_related_in. intuition. -intuition. -intuition. -intuition. - -split;intros. -split;intros. -rewrite H4 in H5. -generalize (VertexSet.filter_1 (compat_not_compat (compat_bool_move _)) H5). intro H6. -generalize (VertexSet.filter_2 (compat_not_compat (compat_bool_move _)) H5). clear H5. intro H5. -rewrite H1 in H6. -generalize (VertexSet.filter_1 (compat_bool_low _ _) H6). intro H7. -generalize (VertexSet.filter_2 (compat_bool_low _ _) H6). clear H6. intro H6. -generalize (VertexSet.diff_2 H7). intuition. -destruct (move_related g x); intuition. -apply (VertexSet.diff_1 H7). - -rewrite H4. -apply VertexSet.filter_3. -apply compat_not_compat. apply compat_bool_move. -rewrite H1. apply VertexSet.filter_3. -apply compat_bool_low. -apply VertexSet.diff_3. -intuition. -intuition. -intuition. -destruct (move_related g x); intuition. -exact (In_graph_aff_edge_in_AE _ _). -Qed. - -Definition not_incident_edges x s g := -VertexSet.fold (fun y s' => EdgeSet.remove (x,y,Some N0) s') - (adj_set x (pmap g)) - s. - -Lemma not_incident_edges_1 : forall x e s g, -(forall y, EdgeSet.In y s -> aff_edge y /\ In_graph_edge y g) -> -(EdgeSet.In e (not_incident_edges x s g) <-> - EdgeSet.In e s /\ ~incident e x). - -Proof. -split; intros. -unfold not_incident_edges in H0. -rewrite VertexSet.fold_1 in H0. -assert (EdgeSet.In e s) as Hin. -induction (VertexSet.elements (adj_set x (pmap g))). simpl in H0. -assumption. -rewrite MEdgeFacts.fold_left_assoc in H0. -apply IHl. -apply (EdgeSet.remove_3 H0). - -intros. -split; intros. -apply EdgeSet.remove_2. intro. -generalize (EdgeSet.remove_3 H1). intro. -elim (EdgeSet.remove_1 H2 H3). -apply EdgeSet.remove_2. intro. -elim (EdgeSet.remove_1 H2 H1). -apply (EdgeSet.remove_3 (EdgeSet.remove_3 H1)). -apply EdgeSet.remove_2. intro. -generalize (EdgeSet.remove_3 H1). intro. -elim (EdgeSet.remove_1 H2 H3). -apply EdgeSet.remove_2. intro. -elim (EdgeSet.remove_1 H2 H1). -apply (EdgeSet.remove_3 (EdgeSet.remove_3 H1)). -intros. apply RegRegProps.Equal_remove. assumption. - -split. -assumption. - -generalize VertexSet.elements_1. intro HH. -intro Helim. destruct Helim. -generalize (HH (adj_set x (pmap g)) (snd_ext e)). clear HH. intro HH. -induction (VertexSet.elements (adj_set x (pmap g))). simpl in H0. - -assert (VertexSet.In (snd_ext e) (adj_set x (pmap g))). -apply (proj1 (edgemap_to_edgeset_charac _ _ _ (Some N0) (sym_pmap g))). -assert (eq e (x, snd_ext e, Some N0)). -assert (get_weight e = Some N0). - -generalize (H _ H0). intro. -destruct H2. unfold In_graph_edge in H3. -destruct H3. -apply (AE_weights g). assumption. -generalize (IE_weights g _ H3). intro. -destruct H2. congruence. -rewrite <-H2. Eq_eq. apply Regs.eq_refl. -rewrite <-H2. - -generalize (H _ H0). intro. -destruct H3. destruct H4. -assumption. -generalize (IE_weights g _ H4). intro. -destruct H3. congruence. - -generalize (HH H2). intro. inversion H3. - -rewrite MEdgeFacts.fold_left_assoc in H0. -apply IHl. -apply (EdgeSet.remove_3 H0). - -intro. generalize (HH H2). clear HH H2. intro H2. -inversion H2; subst. -assert (eq e (x, a, Some N0)). -rewrite (edge_eq e). -apply eq_ordered_eq. -constructor. simpl. split; intuition. -simpl. - -generalize (H _ Hin). intro H5. -destruct H5. -destruct H5. -rewrite (AE_weights g _ H5). apply OptionN_as_OT.eq_refl. -generalize (IE_weights g _ H5). intro. destruct H3. congruence. -elim (EdgeSet.remove_1 (eq_sym H3) H0). -assumption. - -intros. -split; intros. -apply EdgeSet.remove_2. intro. -generalize (EdgeSet.remove_3 H2). intro. -elim (EdgeSet.remove_1 H3 H4). -apply EdgeSet.remove_2. intro. -elim (EdgeSet.remove_1 H3 H2). -apply (EdgeSet.remove_3 (EdgeSet.remove_3 H2)). -apply EdgeSet.remove_2. intro. -generalize (EdgeSet.remove_3 H2). intro. -elim (EdgeSet.remove_1 H3 H4). -apply EdgeSet.remove_2. intro. -elim (EdgeSet.remove_1 H3 H2). -apply (EdgeSet.remove_3 (EdgeSet.remove_3 H2)). -intros. apply RegRegProps.Equal_remove. assumption. - -generalize (HH (adj_set x (pmap g)) (fst_ext e)). clear HH. intro HH. -induction (VertexSet.elements (adj_set x (pmap g))). simpl in H0. - -assert (VertexSet.In (fst_ext e) (adj_set x (pmap g))). -apply (proj1 (edgemap_to_edgeset_charac _ _ _ (Some N0) (sym_pmap g))). -assert (eq e (x, fst_ext e, Some N0)). -assert (get_weight e = Some N0). - -generalize (H _ H0). intro. -destruct H2. unfold In_graph_edge in H3. -destruct H3. -apply (AE_weights g). assumption. -generalize (IE_weights g _ H3). intro. -destruct H2. congruence. -rewrite <-H2. Eq_comm_eq. apply Regs.eq_refl. -rewrite <-H2. - -generalize (H _ H0). intro. -destruct H3. destruct H4. -assumption. -generalize (IE_weights g _ H4). intro. -destruct H3. congruence. - -generalize (HH H2). intro. inversion H3. - -rewrite MEdgeFacts.fold_left_assoc in H0. -apply IHl. -apply (EdgeSet.remove_3 H0). - -intro. generalize (HH H2). clear HH H2. intro H2. -inversion H2; subst. -assert (eq e (x, a, Some N0)). -rewrite (edge_eq e). -rewrite edge_comm. apply eq_ordered_eq. -constructor. simpl. split; intuition. -simpl. - -generalize (H _ Hin). intro H5. -destruct H5. -destruct H5. -rewrite (AE_weights g _ H5). apply OptionN_as_OT.eq_refl. -generalize (IE_weights g _ H5). intro. destruct H3. congruence. -elim (EdgeSet.remove_1 (eq_sym H3) H0). -assumption. - -intros. -split; intros. -apply EdgeSet.remove_2. intro. -generalize (EdgeSet.remove_3 H2). intro. -elim (EdgeSet.remove_1 H3 H4). -apply EdgeSet.remove_2. intro. -elim (EdgeSet.remove_1 H3 H2). -apply (EdgeSet.remove_3 (EdgeSet.remove_3 H2)). -apply EdgeSet.remove_2. intro. -generalize (EdgeSet.remove_3 H2). intro. -elim (EdgeSet.remove_1 H3 H4). -apply EdgeSet.remove_2. intro. -elim (EdgeSet.remove_1 H3 H2). -apply (EdgeSet.remove_3 (EdgeSet.remove_3 H2)). -intros. apply RegRegProps.Equal_remove. assumption. - -unfold not_incident_edges. -rewrite VertexSet.fold_1. -induction (VertexSet.elements (adj_set x (pmap g))). simpl. -intuition. -rewrite MEdgeFacts.fold_left_assoc. -apply EdgeSet.remove_2. -destruct H0. -intro H2. elim H1. -destruct (eq_charac _ _ H2); destruct H3; change_rewrite. -left. auto. -right. auto. -assumption. - -intros. -split; intros. -apply EdgeSet.remove_2. intro. -generalize (EdgeSet.remove_3 H1). intro. -elim (EdgeSet.remove_1 H2 H3). -apply EdgeSet.remove_2. intro. -elim (EdgeSet.remove_1 H2 H1). -apply (EdgeSet.remove_3 (EdgeSet.remove_3 H1)). -apply EdgeSet.remove_2. intro. -generalize (EdgeSet.remove_3 H1). intro. -elim (EdgeSet.remove_1 H2 H3). -apply EdgeSet.remove_2. intro. -elim (EdgeSet.remove_1 H2 H1). -apply (EdgeSet.remove_3 (EdgeSet.remove_3 H1)). -intros. apply RegRegProps.Equal_remove. assumption. -Qed. - -Definition not_incident_merge x s es := -VertexSet.fold (fun y s' => EdgeSet.remove (x,y,Some N0) s') - es s. - -Lemma not_incident_merge_1 : forall x s es e, -EdgeSet.In e (not_incident_merge x es s) -> -EdgeSet.In e es. - -Proof. -unfold not_incident_merge; intros. -set (f := (fun (y : VertexSet.elt) (s' : EdgeSet.t) => - EdgeSet.remove (x, y, Some 0%N) s')) in *. -rewrite VertexSet.fold_1 in H. -set (f' := fun a e => f e a) in *. -induction (VertexSet.elements s). simpl in H. assumption. -rewrite MEdgeFacts.fold_left_assoc in H. -set (tmp := fold_left f' l es) in H. -unfold f' in H. unfold f in H. -apply (IHl (EdgeSet.remove_3 H)). - -unfold f'. unfold f. intros. -split; intros. -apply EdgeSet.remove_2. -intro H1. elim (EdgeSet.remove_1 H1 (EdgeSet.remove_3 H0)). -apply EdgeSet.remove_2. -intro H1. elim (EdgeSet.remove_1 H1 H0). -apply (EdgeSet.remove_3 (EdgeSet.remove_3 H0)). -apply EdgeSet.remove_2. -intro H1. elim (EdgeSet.remove_1 H1 (EdgeSet.remove_3 H0)). -apply EdgeSet.remove_2. -intro H1. elim (EdgeSet.remove_1 H1 H0). -apply (EdgeSet.remove_3 (EdgeSet.remove_3 H0)). - -intros. -unfold f'. unfold f. -apply RegRegProps.Equal_remove. auto. -Qed. - - -Definition AE_merge_up newadj adj adjsnd e es := -let diff1 := VertexSet.diff adj newadj in -let diff2 := VertexSet.diff newadj adj in -let new_es := not_incident_merge (snd_ext e) - (not_incident_merge (fst_ext e) es diff1) - adjsnd in -VertexSet.fold (fun y s'' => EdgeSet.add (fst_ext e,y,Some N0) s'') - diff2 new_es. - -Lemma not_incident_merge_2 : forall x y s s', -EdgeSet.In (x,y,Some N0) (not_incident_merge x s s') -> -~VertexSet.In y s'. - -Proof. -intros. -unfold not_incident_merge in H. -set (f := (fun (y : VertexSet.elt) (s' : EdgeSet.t) => - EdgeSet.remove (x, y, Some 0%N) s')) in *. -rewrite VertexSet.fold_1 in H. -set (f' := fun a e => f e a) in *. -generalize VertexSet.elements_1. intro HH. -generalize (HH s' y). clear HH. intro HH. -induction (VertexSet.elements s'). simpl in H. -intro. generalize (HH H0). intro H1. inversion H1. -rewrite MEdgeFacts.fold_left_assoc in H. -set (tmp := fold_left f' l s) in *. -unfold f' in H. unfold f in H. -destruct (Vertex.eq_dec y a). -cut (eq (x,y,Some N0) (x,a, Some N0)). intro H0. -elim (EdgeSet.remove_1 (Edge.eq_sym H0) H). -apply eq_ordered_eq. constructor;simpl; try split; intuition. -apply OptionN_as_OT.eq_refl. -apply IHl. apply (EdgeSet.remove_3 H). -intro. generalize (HH H0). intro H1. -inversion H1; subst. -elim (n H3). -auto. - -unfold f'. unfold f. intros. -split; intros. -apply EdgeSet.remove_2. -intro H5. elim (EdgeSet.remove_1 H5 (EdgeSet.remove_3 H0)). -apply EdgeSet.remove_2. -intro H5. elim (EdgeSet.remove_1 H5 H0). -apply (EdgeSet.remove_3 (EdgeSet.remove_3 H0)). - -apply EdgeSet.remove_2. -intro H5. elim (EdgeSet.remove_1 H5 (EdgeSet.remove_3 H0)). -apply EdgeSet.remove_2. -intro H5. elim (EdgeSet.remove_1 H5 H0). -apply (EdgeSet.remove_3 (EdgeSet.remove_3 H0)). - -unfold f'. unfold f. intros. -apply RegRegProps.Equal_remove. auto. -Qed. - -Lemma AE_merge_wl_1 : forall x y e g, -aff_edge e -> -In_graph_edge e g -> -(VertexSet.In x (adj_set y (pmap_merge e g (imap_merge e g))) <-> -(VertexSet.In x (adj_set y (map_merge e (pmap g))) /\ -~VertexSet.In x (adj_set y (imap_merge e g)))). - -Proof. -intros x y e g p q. split; intros. -split. -apply pmap_merge_sub. assumption. -intro. apply (simple_graph (merge e g q p) x y). intuition. -destruct H. -unfold merge. simpl. apply resolve_conflicts_map_5; auto. -Qed. - -Lemma not_in_diff_equiv : forall x s s', -~VertexSet.In x (VertexSet.diff s s') -> -VertexSet.In x s' \/ ~VertexSet.In x s. - -Proof. -intros. destruct (Props.In_dec x s). - destruct (Props.In_dec x s'). -left. auto. -elim H. apply VertexSet.diff_3; auto. -right. auto. -Qed. - -Lemma AE_aux_2 : forall e g s p q, -(forall a, EdgeSet.In a s <-> aff_edge a /\ In_graph_edge a g) -> -(forall b, EdgeSet.In b (AE_merge_up (preference_adj (fst_ext e) (merge e g p q)) - (preference_adj (fst_ext e) g) - (preference_adj (snd_ext e) g) - e s) - <-> - (~incident b (snd_ext e) /\ - ((EdgeSet.In b s /\ ~Interfere (fst_ext b) (snd_ext b) (merge e g p q)) \/ - EdgeSet.In b (VertexSet.fold - (fun y s' => EdgeSet.add (fst_ext e, y, Some N0) s') - (preference_adj (fst_ext e) (merge e g p q)) - EdgeSet.empty)))). - -Proof. -intros e g s p q; split; intros. -unfold AE_merge_up in H0. -set (f := (fun (y : VertexSet.elt) (s'' : EdgeSet.t) => - EdgeSet.add (fst_ext e, y, Some 0%N) s'')) in *. -set (diff1 := (VertexSet.diff (preference_adj (fst_ext e) (merge e g p q)) - (preference_adj (fst_ext e) g))) in *. -set (diff2 := (VertexSet.diff (preference_adj (fst_ext e) g) - (preference_adj (fst_ext e) (merge e g p q)))) in *. -cut (get_weight b = Some N0). intro Hw. -cut (~incident b (snd_ext e)). intro Hcut. -split. -assumption. -rewrite VertexSet.fold_1 in H0. -set (f' := fun a e => f e a) in *. -generalize VertexSet.elements_2. intro HH. -generalize (HH diff1). clear HH. intro HH. -induction (VertexSet.elements diff1). simpl in H0. -left. split. -apply (not_incident_merge_1 _ _ _ _ (not_incident_merge_1 _ _ _ _ H0)). -intro H1. unfold Interfere in H1. -generalize (merge_4 e (fst_ext b, snd_ext b, None) _ p q H1). intro H2. -destruct H2. destruct H2. -unfold redirect in H3. -destruct (OTFacts.eq_dec (fst_ext x) (snd_ext e)). destruct H3. -assert (eq (fst_ext b, snd_ext b, None) (fst_ext e, snd_ext x, get_weight x)). -unfold weak_eq in H3. change_rewrite. destruct H3. -apply eq_ordered_eq; split; simpl; auto. -unfold same_type in H4. destruct H4. -destruct H4. destruct H5. simpl in H5. inversion H5. -destruct H4. rewrite <-H4. apply OptionN_as_OT.eq_refl. -destruct H3. -unfold same_type in H4. destruct H4. -destruct H4. destruct H6. simpl in H6. inversion H6. -destruct H4. rewrite <-H4. rewrite edge_comm. apply eq_ordered_eq. -split; auto. simpl. apply OptionN_as_OT.eq_refl. -generalize H5. clear H3 H4 H5. intro H3. -destruct (eq_charac _ _ H3); destruct H4; change_rewrite. -generalize (not_incident_merge_1 _ _ _ _ H0). intro H6. -cut (eq b (fst_ext e, snd_ext x, Some N0)). intro H7. -rewrite H7 in H6. clear H7. -generalize (not_incident_merge_2 _ _ _ _ H6). intro H7. -unfold diff2 in H7. -generalize (not_in_diff_equiv _ _ _ H7). clear H7. intro H7. destruct H7. -rewrite <-H5 in H7. generalize (sym_pmap_merge_map e g q p _ _ H7). intro H8. -rewrite <-H4 in H8. generalize (sym_pmap_merge_map e g q p _ _ H8). intro H9. -elim (simple_graph (merge e g p q) (snd_ext b) (fst_ext b)). split. -apply (proj1 (edgemap_to_edgeset_charac _ _ _ None (sym_imap_merge_map e g q p))). -destruct H1. -generalize (AE_weights _ _ H1). simpl. congruence. -assumption. -assumption. -rewrite <-H5 in H7. -elim H7. apply (sym_pmap g). -rewrite <-H4. -cut (In_graph_edge b g). -intro H8. -destruct H8. -apply (proj1 (edgemap_to_edgeset_charac _ _ _ (Some N0) (sym_pmap g))). -cut (eq b (snd_ext b, fst_ext b, Some N0)). intro H9. -rewrite <-H9. assumption. -rewrite edge_comm. rewrite (edge_eq b). change_rewrite. -rewrite Hw. apply eq_refl. -generalize (IE_weights _ _ H8). congruence. -generalize (proj1 (H _) (not_incident_merge_1 _ _ _ _ (not_incident_merge_1 _ _ _ _ H0))). -intuition. -apply eq_ordered_eq. -constructor; simpl. split. auto. auto. -fold (get_weight b). rewrite Hw. apply OptionN_as_OT.eq_refl. - -generalize (not_incident_merge_1 _ _ _ _ H0). intro H6. -cut (eq b (fst_ext e, snd_ext x, Some N0)). intro H7. -rewrite H7 in H6. clear H7. -generalize (not_incident_merge_2 _ _ _ _ H6). intro H7. -unfold diff2 in H7. -generalize (not_in_diff_equiv _ _ _ H7). clear H7. intro H7. -destruct H7. -rewrite <-H4 in H7. generalize (sym_pmap_merge_map e g q p _ _ H7). intro H8. -rewrite <-H5 in H8. generalize (sym_pmap_merge_map e g q p _ _ H8). intro H9. -elim (simple_graph (merge e g p q) (snd_ext b) (fst_ext b)). split. -apply (proj1 (edgemap_to_edgeset_charac _ _ _ None (sym_imap_merge_map e g q p))). -destruct H1. -generalize (AE_weights _ _ H1). simpl. congruence. -assumption. -assumption. -rewrite <-H4 in H7. -elim H7. apply (sym_pmap g). -rewrite <-H5. -cut (In_graph_edge b g). -intro H8. -destruct H8. -apply (proj1 (edgemap_to_edgeset_charac _ _ _ (Some N0) (sym_pmap g))). -cut (eq b (snd_ext b, fst_ext b, Some N0)). intro H9. -rewrite edge_comm. rewrite <-H9. assumption. -rewrite edge_comm. rewrite (edge_eq b). change_rewrite. -rewrite Hw. apply eq_refl. -generalize (IE_weights _ _ H8). congruence. -generalize (proj1 (H _) (not_incident_merge_1 _ _ _ _ (not_incident_merge_1 _ _ _ _ H0))). -intuition. -rewrite edge_comm. apply eq_ordered_eq. -constructor; simpl. split. auto. auto. -fold (get_weight b). rewrite Hw. apply OptionN_as_OT.eq_refl. - -destruct (OTFacts.eq_dec (snd_ext x) (snd_ext e)). -assert (eq (fst_ext b, snd_ext b, None) (fst_ext x, fst_ext e, get_weight x)). -destruct H3. unfold same_type in H4. destruct H4; destruct H4. -destruct H5. simpl in H5. inversion H5. rewrite <-H4. -unfold weak_eq in H3. change_rewrite. -destruct H3; destruct H3. -apply eq_ordered_eq. split; auto. simpl. apply OptionN_as_OT.eq_refl. -rewrite edge_comm. apply eq_ordered_eq. split; auto. simpl. apply OptionN_as_OT.eq_refl. -generalize H4. clear H3 H4. intro H3. -destruct (eq_charac _ _ H3); change_rewrite. destruct H4. -generalize (not_incident_merge_1 _ _ _ _ H0). intro H6. -cut (eq b (fst_ext e, fst_ext x, Some N0)). intro H7. -rewrite H7 in H6. clear H7. -generalize (not_incident_merge_2 _ _ _ _ H6). intro H7. -unfold diff2 in H7. -generalize (not_in_diff_equiv _ _ _ H7). clear H7. intro H7. -destruct H7. -rewrite <-H4 in H7. generalize (sym_pmap_merge_map e g q p _ _ H7). intro H8. -rewrite <-H5 in H8. generalize (sym_pmap_merge_map e g q p _ _ H8). intro H9. -elim (simple_graph (merge e g p q) (snd_ext b) (fst_ext b)). split. -apply (proj1 (edgemap_to_edgeset_charac _ _ _ None (sym_imap_merge_map e g q p))). -destruct H1. -generalize (AE_weights _ _ H1). simpl. congruence. -assumption. -assumption. -rewrite <-H4 in H7. -elim H7. apply (sym_pmap g). -rewrite <-H5. -cut (In_graph_edge b g). -intro H8. -destruct H8. -apply (proj1 (edgemap_to_edgeset_charac _ _ _ (Some N0) (sym_pmap g))). -cut (eq b (fst_ext b, snd_ext b, Some N0)). intro H9. -rewrite <-H9. assumption. -rewrite (edge_eq b). change_rewrite. -rewrite Hw. apply eq_refl. -generalize (IE_weights _ _ H8). congruence. -generalize (proj1 (H _) (not_incident_merge_1 _ _ _ _ (not_incident_merge_1 _ _ _ _ H0))). -intuition. -rewrite edge_comm. apply eq_ordered_eq. -constructor; simpl. split. auto. auto. -fold (get_weight b). rewrite Hw. apply OptionN_as_OT.eq_refl. - -destruct H4. -generalize (not_incident_merge_1 _ _ _ _ H0). intro H6. -cut (eq b (fst_ext e, fst_ext x, Some N0)). intro H7. -rewrite H7 in H6. clear H7. -generalize (not_incident_merge_2 _ _ _ _ H6). intro H7. -unfold diff2 in H7. -generalize (not_in_diff_equiv _ _ _ H7). clear H7. intro H7. -destruct H7. -rewrite <-H5 in H7. generalize (sym_pmap_merge_map e g q p _ _ H7). intro H8. -rewrite <-H4 in H8. generalize (sym_pmap_merge_map e g q p _ _ H8). intro H9. -elim (simple_graph (merge e g p q) (snd_ext b) (fst_ext b)). split. -apply (proj1 (edgemap_to_edgeset_charac _ _ _ None (sym_imap_merge_map e g q p))). -destruct H1. -generalize (AE_weights _ _ H1). simpl. congruence. -assumption. -assumption. -rewrite <-H5 in H7. -elim H7. apply (sym_pmap g). -rewrite <-H4. -cut (In_graph_edge b g). -intro H8. -destruct H8. -apply (proj1 (edgemap_to_edgeset_charac _ _ _ (Some N0) (sym_pmap g))). -cut (eq b (fst_ext b, snd_ext b, Some N0)). intro H9. -rewrite edge_comm. rewrite <-H9. assumption. -rewrite (edge_eq b). change_rewrite. -rewrite Hw. apply eq_refl. -generalize (IE_weights _ _ H8). congruence. -generalize (proj1 (H _) (not_incident_merge_1 _ _ _ _ (not_incident_merge_1 _ _ _ _ H0))). -intuition. -apply eq_ordered_eq. -constructor; simpl. split. auto. auto. -fold (get_weight b). rewrite Hw. apply OptionN_as_OT.eq_refl. - -assert (eq (fst_ext b, snd_ext b, None) x). -destruct H3. unfold same_type in H4. destruct H4; destruct H4. -destruct H5. simpl in H5. inversion H5. -rewrite <-H4. -unfold weak_eq in H3; change_rewrite. destruct H3; destruct H3. -apply eq_ordered_eq; split; auto. apply OptionN_as_OT.eq_refl. -rewrite edge_comm. apply eq_ordered_eq; split; auto. apply OptionN_as_OT.eq_refl. -generalize H4. clear H3 H4. intro H3. -elim (simple_graph g (fst_ext b) (snd_ext b)). -split. -apply (proj1 (edgemap_to_edgeset_charac _ _ _ None (sym_imap g))). -rewrite edge_comm. rewrite H3. -destruct H2. -rewrite <-H3 in H2. generalize (AE_weights _ _ H2). simpl. congruence. -assumption. -apply (proj1 (edgemap_to_edgeset_charac _ _ _ (Some N0) (sym_pmap g))). -assert (In_graph_edge b g). -generalize (proj1 (H _) (not_incident_merge_1 _ _ _ _ (not_incident_merge_1 _ _ _ _ H0))). -intuition. -destruct H4. -assert (eq b (fst_ext b, snd_ext b, Some N0)). -rewrite (edge_eq b). change_rewrite. -rewrite Hw. apply eq_refl. -rewrite edge_comm. rewrite <-H5. assumption. -generalize (IE_weights _ _ H4). congruence. -rewrite MEdgeFacts.fold_left_assoc in H0. -set (set := (not_incident_merge (snd_ext e) - (not_incident_merge (fst_ext e) s diff2) - (preference_adj (snd_ext e) g))) in *. -set (tmp := fold_left f' l set) in *. -unfold f' in H0. unfold f in H0. -destruct (proj1 (RegRegProps.Dec.F.add_iff _ _ _) H0). -fold (eq (fst_ext e, a, Some N0) b) in H1. -right. -rewrite <-H1. -rewrite VertexSet.fold_1. -fold f'. -set (s' := preference_adj (fst_ext e) (merge e g p q)) in *. -generalize VertexSet.elements_1. intro HHH. -generalize (HHH s' a). clear HHH. intro HHH. -induction (VertexSet.elements s'). simpl. -assert (InA Vertex.eq a nil). -apply HHH. -unfold s'. -assert (VertexSet.In a diff1). -apply HH. left. intuition. -apply (VertexSet.diff_1 H2). inversion H2. -rewrite MEdgeFacts.fold_left_assoc. -set (tmp' := fold_left f' l0 EdgeSet.empty) in *. -unfold f'. unfold f. -destruct (Vertex.eq_dec a a0). -apply EdgeSet.add_1. -apply eq_ordered_eq. constructor;simpl. -split; intuition. -apply OptionN_as_OT.eq_refl. -apply EdgeSet.add_2. -apply IHl0. intros. -generalize (HHH H2). intro. -inversion H3; subst. -elim (n H5). -auto. - -unfold f'. unfold f. intros. -apply RegRegProps.add_add. - -unfold f'. unfold f. intros. -apply RegRegProps.Dec.F.add_m. apply eq_refl. auto. - -apply IHl. -assumption. -intros. -apply HH. -right. auto. - -unfold f'. unfold f. intros. -apply RegRegProps.add_add. - -unfold f'. unfold f. intros. -apply RegRegProps.Dec.F.add_m. apply eq_refl. auto. - -rewrite VertexSet.fold_1 in H0. -set (f' := fun a e => f e a) in *. -generalize VertexSet.elements_2. intro HH. -generalize (HH diff1). clear HH. intro HH. -induction (VertexSet.elements diff1). simpl in H0. -intro Hinc. destruct Hinc. -assert (eq b (snd_ext e, snd_ext b, Some N0)). -rewrite (edge_eq b). change_rewrite. rewrite Hw. -apply eq_ordered_eq. -constructor;simpl. split;intuition. -apply OptionN_as_OT.eq_refl. - -rewrite H2 in H0. -generalize (not_incident_merge_2 _ _ _ _ H0). intro H3. -elim H3. -apply (proj1 (edgemap_to_edgeset_charac _ _ _ (Some N0) (sym_pmap g))). rewrite <-H2. -assert (In_graph_edge b g). -generalize (proj1 (H _) (not_incident_merge_1 _ _ _ _ (not_incident_merge_1 _ _ _ _ H0))). -intro. destruct H4. rewrite <-H2 in H5. assumption. -destruct H4. -assumption. -generalize (IE_weights _ _ H4). congruence. - -assert (eq b (fst_ext b, snd_ext e, Some N0)). -rewrite (edge_eq b). change_rewrite. rewrite Hw. -apply eq_ordered_eq. -constructor;simpl. split; intuition. -apply OptionN_as_OT.eq_refl. - -rewrite H2 in H0. rewrite edge_comm in H0. -generalize (not_incident_merge_2 _ _ _ _ H0). intro H3. -elim H3. -apply (proj1 (edgemap_to_edgeset_charac _ _ _ (Some N0) (sym_pmap g))). rewrite edge_comm. rewrite <-H2. -assert (In_graph_edge b g). -generalize (proj1 (H _) (not_incident_merge_1 _ _ _ _ (not_incident_merge_1 _ _ _ _ H0))). -intro. destruct H4. rewrite edge_comm in H2. rewrite <-H2 in H5. assumption. -destruct H4. -assumption. -generalize (IE_weights _ _ H4). congruence. - -rewrite MEdgeFacts.fold_left_assoc in H0. -set (set := (not_incident_merge (snd_ext e) - (not_incident_merge (fst_ext e) s diff2) - (preference_adj (snd_ext e) g))) in *. -set (tmp := fold_left f' l set) in *. -unfold f' in H0. unfold f in H0. -destruct (proj1 (RegRegProps.Dec.F.add_iff _ _ _) H0). -fold (eq (fst_ext e, a, Some N0) b) in H1. -intro. destruct H2. -destruct (eq_charac _ _ H1); change_rewrite. destruct H3. -elim (In_graph_edge_diff_ext _ _ p). -apply Vertex.eq_trans with (y := fst_ext b); auto. -destruct H3. -assert (VertexSet.In a diff1). apply HH. -left. intuition. -unfold diff1 in H5. generalize (VertexSet.diff_2 H5). clear H5. intro H5. -elim H5. -rewrite H4. rewrite <-H2. -apply (proj1 (edgemap_to_edgeset_charac _ _ _ (Some N0) (sym_pmap g))). -generalize p. intro Hin. -destruct Hin. -assert (eq (fst_ext e, snd_ext e, Some N0) e). -apply eq_ordered_eq. -constructor. simpl. split; apply Regs.eq_refl. -simpl. fold (get_weight e). rewrite (AE_weights _ _ H6). -apply OptionN_as_OT.eq_refl. -rewrite H7. assumption. -generalize (IE_weights _ _ H6). generalize q. intro Haff. destruct Haff. congruence. - -destruct (eq_charac _ _ H1); change_rewrite. destruct H3. -assert (VertexSet.In a diff1). apply HH. -left. intuition. -unfold diff1 in H5. generalize (VertexSet.diff_2 H5). clear H5. intro H5. -elim H5. -rewrite H4. rewrite <-H2. -apply (proj1 (edgemap_to_edgeset_charac _ _ _ (Some N0) (sym_pmap g))). -generalize p. intro Hin. -destruct Hin. -assert (eq (fst_ext e, snd_ext e, Some N0) e). -apply eq_ordered_eq. -constructor. simpl. split; apply Regs.eq_refl. -simpl. fold (get_weight e). rewrite (AE_weights _ _ H6). -apply OptionN_as_OT.eq_refl. -rewrite H7. assumption. -generalize (IE_weights _ _ H6). generalize q. intro Haff. destruct Haff. congruence. -elim (In_graph_edge_diff_ext _ _ p). -apply Vertex.eq_trans with (y := snd_ext b); auto. -destruct H3. auto. -apply IHl. auto. -intros. apply HH. right. auto. - -unfold f'. unfold f. intros. -apply RegRegProps.add_add. - -unfold f'. unfold f. intros. -apply RegRegProps.Dec.F.add_m. apply eq_refl. auto. - -rewrite VertexSet.fold_1 in H0. -set (f' := fun a e => f e a) in *. -induction (VertexSet.elements diff1). simpl in H0. -assert (aff_edge b /\ In_graph_edge b g). -rewrite <-H. -apply (not_incident_merge_1 _ _ _ _ (not_incident_merge_1 _ _ _ _ H0)). -destruct H1. -destruct H2. -apply (AE_weights _ _ H2). -destruct H1. generalize (IE_weights _ _ H2). congruence. - -rewrite MEdgeFacts.fold_left_assoc in H0. -set (set := (not_incident_merge (snd_ext e) - (not_incident_merge (fst_ext e) s diff2) - (preference_adj (snd_ext e) g))) in *. -set (tmp := fold_left f' l set) in *. -unfold f' in H0. unfold f in H0. -destruct (proj1 (RegRegProps.Dec.F.add_iff _ _ _) H0). -fold (eq (fst_ext e, a, Some N0) b) in H1. -rewrite <-H1. auto. -apply IHl. auto. - -unfold f'. unfold f. intros. -apply RegRegProps.add_add. - -unfold f'. unfold f. intros. -apply RegRegProps.Dec.F.add_m. apply eq_refl. auto. - -destruct H0. -destruct H1. -destruct H1. -unfold AE_merge_up. -set (f := (fun (y : VertexSet.elt) (s'' : EdgeSet.t) => - EdgeSet.add (fst_ext e, y, Some 0%N) s'')) in *. -set (diff1 := (VertexSet.diff (preference_adj (fst_ext e) (merge e g p q)) - (preference_adj (fst_ext e) g))) in *. -set (diff2 := (VertexSet.diff (preference_adj (fst_ext e) g) - (preference_adj (fst_ext e) (merge e g p q)))) in *. -cut (EdgeSet.In b (not_incident_merge (snd_ext e) - (not_incident_merge (fst_ext e) s diff2) - (preference_adj (snd_ext e) g))). intro H3. -rewrite VertexSet.fold_1. -set (f' := fun a e => f e a) in *. -induction (VertexSet.elements diff1). simpl. assumption. -rewrite MEdgeFacts.fold_left_assoc. -set (set := (fold_left f' l - (not_incident_merge (snd_ext e) - (not_incident_merge (fst_ext e) s diff2) - (preference_adj (snd_ext e) g)))) in *. -unfold f'. unfold f. -apply EdgeSet.add_2. assumption. - -unfold f'. unfold f. intros. -apply RegRegProps.add_add. - -unfold f'. unfold f. intros. -apply RegRegProps.Dec.F.add_m. apply eq_refl. auto. - -cut (EdgeSet.In b (not_incident_merge (fst_ext e) s diff2)). intro. -set (set := (not_incident_merge (fst_ext e) s diff2)) in *. -unfold not_incident_merge. -set (h := (fun (y : VertexSet.elt) (s' : EdgeSet.t) => - EdgeSet.remove (snd_ext e, y, Some 0%N) s')) in *. -set (adjsnd := preference_adj (snd_ext e) g) in *. -rewrite VertexSet.fold_1. -set (h' := fun a e => h e a) in *. -induction (VertexSet.elements adjsnd). simpl. assumption. -rewrite MEdgeFacts.fold_left_assoc. -set (tmp := fold_left h' l set) in *. -unfold h'. unfold h. -apply EdgeSet.remove_2. -intro. elim H0. -destruct (eq_charac _ _ H4); change_rewrite; destruct H5. -left. auto. -right. auto. assumption. - -unfold h'. unfold h. intros. -split; intros. -apply EdgeSet.remove_2. -intro H5. elim (EdgeSet.remove_1 H5 (EdgeSet.remove_3 H4)). -apply EdgeSet.remove_2. -intro H5. elim (EdgeSet.remove_1 H5 H4). -apply (EdgeSet.remove_3 (EdgeSet.remove_3 H4)). - -apply EdgeSet.remove_2. -intro H5. elim (EdgeSet.remove_1 H5 (EdgeSet.remove_3 H4)). -apply EdgeSet.remove_2. -intro H5. elim (EdgeSet.remove_1 H5 H4). -apply (EdgeSet.remove_3 (EdgeSet.remove_3 H4)). - -unfold h'. unfold h. intros. -apply RegRegProps.Equal_remove. auto. - -unfold not_incident_merge. -set (h := (fun (y : VertexSet.elt) (s' : EdgeSet.t) => - EdgeSet.remove (fst_ext e, y, Some 0%N) s')) in *. -rewrite VertexSet.fold_1. -set (h' := fun a e => h e a) in *. -generalize VertexSet.elements_2. intro. -generalize (H3 diff2). clear H3. intro HH. -induction (VertexSet.elements diff2). simpl. assumption. -rewrite MEdgeFacts.fold_left_assoc. -set (tmp := fold_left h' l s) in *. -unfold h'. unfold h. -apply EdgeSet.remove_2. -intro. -assert (VertexSet.In a diff2). -apply HH. left. intuition. -unfold diff2 in H4. -elim (VertexSet.diff_2 H4). clear H4. -apply (proj1 (edgemap_to_edgeset_charac _ _ _ (Some N0) (sym_pmap_merge_map e g q p))). -assert (eq (fst_ext e, a, Some N0) b). auto. generalize H4. clear H3 H4. intro H3. -rewrite H3. -cut (eq b (fst_ext b, snd_ext b, Some N0)). intro. -rewrite H4. -cut (In_graph_edge (fst_ext b, snd_ext b, Some N0) (merge e g p q)). intro. -destruct H5. assumption. -generalize (IE_weights _ _ H5). simpl. congruence. -rewrite <-H4. -cut (eq b (redirect (snd_ext e) (fst_ext e) b)). intro. -rewrite H5. rewrite (edge_eq _). -cut (get_weight (redirect (snd_ext e) (fst_ext e) b) = Some N0). intro. -rewrite H6. -assert (exists w, In_graph_edge - (fst_ext (redirect (snd_ext e) (fst_ext e) b), - snd_ext (redirect (snd_ext e) (fst_ext e) b), Some w) (merge e g p q)). -apply merge_5. - -rewrite H in H1. intuition. -intro. -elim H0. -destruct (eq_charac _ _ H7);change_rewrite; destruct H8. -right. auto. -left. auto. -unfold aff_edge. exists N0. rewrite <-H3. auto. - -cut (eq (fst_ext (redirect (snd_ext e) (fst_ext e) b), - snd_ext (redirect (snd_ext e) (fst_ext e) b), - None) (fst_ext b, snd_ext b, None)). intro. -unfold Interfere. rewrite H7. assumption. - -unfold redirect. destruct (OTFacts.eq_dec (fst_ext b) (snd_ext e)). -elim H0. left. auto. -destruct (OTFacts.eq_dec (snd_ext b) (snd_ext e)). -elim H0. right. auto. -apply eq_refl. - -destruct H7. -destruct H7. generalize (AE_weights _ _ H7). simpl. intro. -rewrite H8 in H7. left. auto. -generalize (IE_weights _ _ H7). simpl. congruence. - -rewrite <-H5. rewrite H4. auto. - -unfold redirect. destruct (OTFacts.eq_dec (fst_ext b) (snd_ext e)). -elim H0. left. auto. -destruct (OTFacts.eq_dec (snd_ext b) (snd_ext e)). -elim H0. right. auto. -apply eq_refl. - -apply eq_ordered_eq. constructor; simpl; try split. -apply Regs.eq_refl. -apply Regs.eq_refl. -fold (get_weight b). rewrite <-H3. simpl. apply OptionN_as_OT.eq_refl. - -apply IHl. intros. apply HH. right. auto. - -unfold h'. unfold h. intros. -split; intros. -apply EdgeSet.remove_2. -intro H5. elim (EdgeSet.remove_1 H5 (EdgeSet.remove_3 H3)). -apply EdgeSet.remove_2. -intro H5. elim (EdgeSet.remove_1 H5 H3). -apply (EdgeSet.remove_3 (EdgeSet.remove_3 H3)). - -apply EdgeSet.remove_2. -intro H5. elim (EdgeSet.remove_1 H5 (EdgeSet.remove_3 H3)). -apply EdgeSet.remove_2. -intro H5. elim (EdgeSet.remove_1 H5 H3). -apply (EdgeSet.remove_3 (EdgeSet.remove_3 H3)). - -unfold h'. unfold h. intros. -apply RegRegProps.Equal_remove. auto. - -(* last part !!!!!!! *) - -unfold AE_merge_up. -set (f := (fun (y : VertexSet.elt) (s'' : EdgeSet.t) => - EdgeSet.add (fst_ext e, y, Some 0%N) s'')) in *. -set (diff1 := (VertexSet.diff (preference_adj (fst_ext e) (merge e g p q)) - (preference_adj (fst_ext e) g))) in *. -set (diff2 := (VertexSet.diff (preference_adj (fst_ext e) g) - (preference_adj (fst_ext e) (merge e g p q)))) in *. - -set (pref := preference_adj (fst_ext e) (merge e g p q)) in *. -rewrite VertexSet.fold_1 in H1. -generalize VertexSet.elements_2. intro HH. -generalize (HH pref). clear HH. intro HH. -set (f' := fun a e => f e a) in *. induction (VertexSet.elements pref). -simpl in H1. elim (EdgeSet.empty_1 H1). - -set (set := (not_incident_merge (snd_ext e) - (not_incident_merge (fst_ext e) s diff2) - (preference_adj (snd_ext e) g))) in *. -rewrite MEdgeFacts.fold_left_assoc in H1. -set (tmp := fold_left f' l EdgeSet.empty) in *. -unfold f' in H1. unfold f in H1. -destruct (proj1 (RegRegProps.Dec.F.add_iff _ _ _) H1). -fold (eq (fst_ext e, a, Some N0) b) in H2. -rewrite <-H2. -destruct (Props.In_dec a (preference_adj (fst_ext e) g)). -cut (EdgeSet.In b set). intro Hcut. -rewrite VertexSet.fold_1. fold f'. -assert (~InA Vertex.eq a (VertexSet.elements diff1)). -intro. generalize (VertexSet.elements_2 H3). intro H4. -unfold diff1 in H4. elim (VertexSet.diff_2 H4 i). -induction (VertexSet.elements diff1). simpl. -rewrite H2. assumption. -rewrite MEdgeFacts.fold_left_assoc. -set (tmp' := fold_left f' l0 set) in *. -unfold f'. unfold f. -destruct (Vertex.eq_dec a a0). -apply EdgeSet.add_1. apply eq_ordered_eq. constructor; simpl. split; intuition. -apply OptionN_as_OT.eq_refl. -apply EdgeSet.add_2. -apply IHl0. intro. elim H3. right. auto. - -unfold f'. unfold f. intros. -apply RegRegProps.add_add. - -unfold f'. unfold f. intros. -apply RegRegProps.Dec.F.add_m. apply eq_refl. auto. - -unfold set. cut (EdgeSet.In b (not_incident_merge (fst_ext e) s diff2)). intro. -set (set' := (not_incident_merge (fst_ext e) s diff2)) in *. -unfold not_incident_merge. -set (h := (fun (y : VertexSet.elt) (s' : EdgeSet.t) => - EdgeSet.remove (snd_ext e, y, Some 0%N) s')) in *. -set (adjsnd := preference_adj (snd_ext e) g) in *. -rewrite VertexSet.fold_1. -set (h' := fun a e => h e a) in *. -induction (VertexSet.elements adjsnd). simpl. assumption. -rewrite MEdgeFacts.fold_left_assoc. -set (tmp' := fold_left h' l set') in *. -unfold h'. unfold h. -apply EdgeSet.remove_2. -intro. elim H0. -destruct (eq_charac _ _ H4); change_rewrite; destruct H5. -left. auto. -right. auto. assumption. - -unfold h'. unfold h. intros. -split; intros. -apply EdgeSet.remove_2. -intro H5. elim (EdgeSet.remove_1 H5 (EdgeSet.remove_3 H4)). -apply EdgeSet.remove_2. -intro H5. elim (EdgeSet.remove_1 H5 H4). -apply (EdgeSet.remove_3 (EdgeSet.remove_3 H4)). - -apply EdgeSet.remove_2. -intro H5. elim (EdgeSet.remove_1 H5 (EdgeSet.remove_3 H4)). -apply EdgeSet.remove_2. -intro H5. elim (EdgeSet.remove_1 H5 H4). -apply (EdgeSet.remove_3 (EdgeSet.remove_3 H4)). - -unfold h'. unfold h. intros. -apply RegRegProps.Equal_remove. auto. - -unfold not_incident_merge. -set (h := (fun (y : VertexSet.elt) (s' : EdgeSet.t) => - EdgeSet.remove (fst_ext e, y, Some 0%N) s')) in *. -rewrite VertexSet.fold_1. -set (h' := fun a e => h e a) in *. -generalize VertexSet.elements_2. intro. -generalize (H3 diff2). clear H3. intro HHH. -induction (VertexSet.elements diff2). simpl. -rewrite H. split. -unfold aff_edge. exists N0. rewrite <-H2. auto. -left. rewrite <-H2. -apply (proj2 (edgemap_to_edgeset_charac _ _ _ (Some N0) (sym_pmap g))). assumption. -rewrite MEdgeFacts.fold_left_assoc. -set (tmp' := fold_left h' l0 s) in *. -unfold h'. unfold h. -apply EdgeSet.remove_2. -intro. -assert (Vertex.eq a a0). -destruct (Vertex.eq_dec a a0). intuition. -assert (eq (fst_ext e, a0, Some N0) b) by auto. generalize H4. clear H3 H4. intro H3. -rewrite <-H3 in H2. destruct (eq_charac _ _ H2); change_rewrite; destruct H4. -auto. -apply Vertex.eq_trans with (y := fst_ext e); auto. -assert (~VertexSet.In a diff2). -intro. unfold diff2 in H5. elim (VertexSet.diff_2 H5). -apply HH. left. intuition. -elim H5. -rewrite H4. apply HHH. left. intuition. -apply IHl0. -intros. apply HHH. right. auto. - -unfold h'. unfold h. intros. -split; intros. -apply EdgeSet.remove_2. -intro H5. elim (EdgeSet.remove_1 H5 (EdgeSet.remove_3 H3)). -apply EdgeSet.remove_2. -intro H5. elim (EdgeSet.remove_1 H5 H3). -apply (EdgeSet.remove_3 (EdgeSet.remove_3 H3)). - -apply EdgeSet.remove_2. -intro H5. elim (EdgeSet.remove_1 H5 (EdgeSet.remove_3 H3)). -apply EdgeSet.remove_2. -intro H5. elim (EdgeSet.remove_1 H5 H3). -apply (EdgeSet.remove_3 (EdgeSet.remove_3 H3)). - -unfold h'. unfold h. intros. -apply RegRegProps.Equal_remove. auto. - -rewrite VertexSet.fold_1. -fold f'. -generalize VertexSet.elements_1. intro HHH. -generalize (HHH diff1 a). clear HHH. intro HHH. -assert (VertexSet.In a diff1). apply VertexSet.diff_3. apply HH. left. intuition. -assumption. -induction (VertexSet.elements diff1). simpl. -generalize (HHH H3). intro H4. inversion H4. -rewrite MEdgeFacts.fold_left_assoc. -set (tmp' := fold_left f' l0 set) in *. -unfold f'. unfold f. -destruct (Vertex.eq_dec a a0). -apply EdgeSet.add_1. -apply eq_ordered_eq. constructor; simpl; try split; intuition. -apply OptionN_as_OT.eq_refl. -apply EdgeSet.add_2. apply IHl0. intro. -generalize (HHH H4). intro H5. inversion H5; subst. -elim (n0 H7). -assumption. - -unfold f'. unfold f. intros. -apply RegRegProps.add_add. - -unfold f'. unfold f. intros. -apply RegRegProps.Dec.F.add_m. apply eq_refl. auto. - -apply IHl. -assumption. -intros. apply HH. right. auto. - -unfold f'. unfold f. intros. -apply RegRegProps.add_add. - -unfold f'. unfold f. intros. -apply RegRegProps.Dec.F.add_m. apply eq_refl. auto. -Qed. - -Lemma AE_aux_3 : forall e g s p q, -(forall a, EdgeSet.In a s <-> aff_edge a /\ In_graph_edge a g) -> -(forall b, EdgeSet.In b (AE (merge e g p q)) - <-> - (~incident b (snd_ext e) /\ - ((EdgeSet.In b s /\ ~Interfere (fst_ext b) (snd_ext b) (merge e g p q)) \/ - EdgeSet.In b (VertexSet.fold - (fun y s' => EdgeSet.add (fst_ext e, y, Some N0) s') - (preference_adj (fst_ext e) (merge e g p q)) - EdgeSet.empty)))). - -Proof. -intros e g s p q. generalize I. intro H. -split; intros. -generalize (proj1 (In_graph_aff_edge_in_AE _ _) H1). intro H3. -destruct H3 as [H3 H4]. -assert (~incident b (snd_ext e)) as Hinc. -intro. destruct H2; elim (merge_1 e g p q); auto. -unfold In_graph. rewrite H2. apply (proj1 (In_graph_edge_in_ext _ _ H4)). -unfold In_graph. rewrite H2. apply (proj2 (In_graph_edge_in_ext _ _ H4)). - -split. -assumption. - -assert (get_weight b = Some N0) as Hw. -apply AE_weights with (g := merge e g p q). assumption. - -clear H1. -generalize (merge_4 e _ g p q H4). intro H1. -destruct H1. destruct H1 as [H1 H2]. destruct H2 as [H2 HH2]. -assert (aff_edge x) as Haffb. -unfold same_type in HH2. destruct HH2; destruct H5;[auto|congruence]. - -unfold redirect in H2. -destruct (OTFacts.eq_dec (fst_ext x) (snd_ext e)). -right. -unfold weak_eq in H2. change_rewrite. destruct H2; destruct H2. - -set (f := (fun (y : VertexSet.elt) (s' : EdgeSet.t) => - EdgeSet.add (fst_ext e, y, Some 0%N) s')) in *. -set (set := preference_adj (fst_ext e) (merge e g p q)) in *. -rewrite VertexSet.fold_1. set (f' := fun a e => f e a) in *. -generalize VertexSet.elements_1. intro HH. -generalize (HH set (snd_ext x)). clear HH. intro HH. -induction (VertexSet.elements set). simpl. -assert (InA Vertex.eq (snd_ext x) nil). -apply HH. -unfold set. rewrite <-H5. rewrite (compat_preference_adj _ _ _ (Vertex.eq_sym H2)). -apply (proj1 (edgemap_to_edgeset_charac _ _ _ (Some N0) (sym_pmap_merge_map e g q p))). -cut (eq b (fst_ext b, snd_ext b, Some N0)). intro Heq. -rewrite <-Heq. destruct H4. assumption. -generalize (IE_weights _ _ H4). congruence. -rewrite (edge_eq b); change_rewrite. -assert (get_weight b = Some N0). -apply AE_weights with (g:= merge e g p q). rewrite In_graph_aff_edge_in_AE. -split; auto. -rewrite H6. apply eq_refl. -inversion H6. - -rewrite MEdgeFacts.fold_left_assoc. -set (tmp := fold_left f' l EdgeSet.empty) in *. -unfold f'. unfold f. -destruct (Vertex.eq_dec (snd_ext x) a). -apply EdgeSet.add_1. -apply eq_ordered_eq. -rewrite (edge_eq b). -constructor; simpl. split; intuition. -rewrite <-e0. intuition. -assert (get_weight b = Some N0). -apply AE_weights with (g:= merge e g p q). rewrite In_graph_aff_edge_in_AE. -split; intuition. -rewrite H6. apply OptionN_as_OT.eq_refl. -apply EdgeSet.add_2. -apply IHl. intro. generalize (HH H6). intro H7. -inversion H7; subst. -elim (n H9). -auto. - -unfold f'. unfold f. intros. -apply RegRegProps.add_add. - -unfold f'. unfold f. intros. -apply RegRegProps.Dec.F.add_m. apply eq_refl. auto. - -set (f := (fun (y : VertexSet.elt) (s' : EdgeSet.t) => - EdgeSet.add (fst_ext e, y, Some 0%N) s')) in *. -set (set := preference_adj (fst_ext e) (merge e g p q)) in *. -rewrite VertexSet.fold_1. set (f' := fun a e => f e a) in *. -generalize VertexSet.elements_1. intro HH. -generalize (HH set (snd_ext x)). clear HH. intro HH. -induction (VertexSet.elements set). simpl. -assert (InA Vertex.eq (snd_ext x) nil). -apply HH. -unfold set. rewrite <-H2. rewrite (compat_preference_adj _ _ _ (Vertex.eq_sym H5)). -apply (proj1 (edgemap_to_edgeset_charac _ _ _ (Some N0) (sym_pmap_merge_map e g q p))). -cut (eq b (snd_ext b, fst_ext b, Some N0)). intro Heq. -rewrite <-Heq. destruct H4. assumption. -generalize (IE_weights _ _ H4). congruence. -rewrite (edge_eq b); change_rewrite. -assert (get_weight b = Some N0). -apply AE_weights with (g:= merge e g p q). rewrite In_graph_aff_edge_in_AE. -split; auto. -rewrite H6. apply edge_comm. -inversion H6. - -rewrite MEdgeFacts.fold_left_assoc. -set (tmp := fold_left f' l EdgeSet.empty) in *. -unfold f'. unfold f. -destruct (Vertex.eq_dec (snd_ext x) a). -apply EdgeSet.add_1. -fold (eq (fst_ext e, a, Some N0) b). -rewrite edge_comm. apply eq_ordered_eq. -rewrite (edge_eq b). -constructor; simpl. split; intuition. -rewrite <-e0. intuition. -assert (get_weight b = Some N0). -apply AE_weights with (g:= merge e g p q). rewrite In_graph_aff_edge_in_AE. -split; auto. -rewrite H6. apply OptionN_as_OT.eq_refl. -apply EdgeSet.add_2. -apply IHl. intro. generalize (HH H6). intro H7. -inversion H7; subst. -elim (n H9). -auto. - -unfold f'. unfold f. intros. -apply RegRegProps.add_add. - -unfold f'. unfold f. intros. -apply RegRegProps.Dec.F.add_m. apply eq_refl. auto. - -destruct (OTFacts.eq_dec (snd_ext x) (snd_ext e)). - -right. -unfold weak_eq in H2. change_rewrite. destruct H2; destruct H2. -assert (get_weight x = Some N0). -apply AE_weights with (g:=g). apply (proj2 (In_graph_aff_edge_in_AE _ _)). -split; auto. - -set (f := (fun (y : VertexSet.elt) (s' : EdgeSet.t) => - EdgeSet.add (fst_ext e, y, Some 0%N) s')) in *. -set (set := preference_adj (fst_ext e) (merge e g p q)) in *. -rewrite VertexSet.fold_1. set (f' := fun a e => f e a) in *. -generalize VertexSet.elements_1. intro HH. -generalize (HH set (fst_ext x)). clear HH. intro HH. -induction (VertexSet.elements set). simpl. -assert (InA Vertex.eq (fst_ext x) nil). -apply HH. -unfold set. rewrite <-H2. rewrite (compat_preference_adj _ _ _ (Vertex.eq_sym H5)). -apply (proj1 (edgemap_to_edgeset_charac _ _ _ (Some N0) (sym_pmap_merge_map e g q p))). -cut (eq b (snd_ext b, fst_ext b, Some N0)). intro Heq. -rewrite <-Heq. destruct H4. assumption. -generalize (IE_weights _ _ H4). congruence. -rewrite (edge_eq b); change_rewrite. -assert (get_weight b = Some N0). -apply AE_weights with (g:= merge e g p q). rewrite In_graph_aff_edge_in_AE. -split; auto. -rewrite H7. apply edge_comm. -inversion H7. - -rewrite MEdgeFacts.fold_left_assoc. -set (tmp := fold_left f' l EdgeSet.empty) in *. -unfold f'. unfold f. -destruct (Vertex.eq_dec (fst_ext x) a). -apply EdgeSet.add_1. -fold (eq (fst_ext e, a, Some N0) b). -rewrite edge_comm. apply eq_ordered_eq. -rewrite (edge_eq b). -constructor; simpl. split; intuition. -rewrite <-e0. intuition. -rewrite Hw. auto. apply OptionN_as_OT.eq_refl. -apply EdgeSet.add_2. -apply IHl. intro. generalize (HH H7). intro H8. -inversion H8; subst. -elim (n0 H10). -auto. - -unfold f'. unfold f. intros. -apply RegRegProps.add_add. - -unfold f'. unfold f. intros. -apply RegRegProps.Dec.F.add_m. apply eq_refl. auto. - -set (f := (fun (y : VertexSet.elt) (s' : EdgeSet.t) => - EdgeSet.add (fst_ext e, y, Some 0%N) s')) in *. -set (set := preference_adj (fst_ext e) (merge e g p q)) in *. -rewrite VertexSet.fold_1. set (f' := fun a e => f e a) in *. -generalize VertexSet.elements_1. intro HH. -generalize (HH set (fst_ext x)). clear HH. intro HH. -induction (VertexSet.elements set). simpl. -assert (InA Vertex.eq (fst_ext x) nil). -apply HH. -unfold set. rewrite <-H5. rewrite (compat_preference_adj _ _ _ (Vertex.eq_sym H2)). -apply (proj1 (edgemap_to_edgeset_charac _ _ _ (Some N0) (sym_pmap_merge_map e g q p))). -cut (eq b (fst_ext b, snd_ext b, Some N0)). intro Heq. -rewrite <-Heq. destruct H4. assumption. -generalize (IE_weights _ _ H4). congruence. -rewrite (edge_eq b); change_rewrite. -assert (get_weight b = Some N0). -apply AE_weights with (g:= merge e g p q). rewrite In_graph_aff_edge_in_AE. -split; auto. -rewrite H6. apply eq_refl. -inversion H6. - -rewrite MEdgeFacts.fold_left_assoc. -set (tmp := fold_left f' l EdgeSet.empty) in *. -unfold f'. unfold f. -destruct (Vertex.eq_dec (fst_ext x) a). -apply EdgeSet.add_1. -apply eq_ordered_eq. -rewrite (edge_eq b). -constructor; simpl. split; intuition. -rewrite H5. intuition. -rewrite Hw. apply OptionN_as_OT.eq_refl. -apply EdgeSet.add_2. -apply IHl. intro. generalize (HH H6). intro H7. -inversion H7; subst. -elim (n0 H9). -auto. - -unfold f'. unfold f. intros. -apply RegRegProps.add_add. - -unfold f'. unfold f. intros. -apply RegRegProps.Dec.F.add_m. apply eq_refl. auto. - -assert (eq b x) as Heq. -unfold weak_eq in H2. destruct H2; destruct H2. -apply eq_ordered_eq. constructor; simpl. -inversion H2; auto. -fold (get_weight b). fold (get_weight x). -rewrite Hw. -assert (get_weight x = Some N0). -apply AE_weights with (g:=g). -rewrite In_graph_aff_edge_in_AE. -split; auto. -rewrite H6. apply OptionN_as_OT.eq_refl. -rewrite (edge_eq b). rewrite (edge_eq x). -rewrite edge_comm. apply eq_ordered_eq. constructor; simpl. -inversion H2; simpl in *; intuition. -rewrite H6. rewrite H7. auto. -rewrite H6. rewrite H7. auto. -assert (get_weight x = Some N0). -apply AE_weights with (g:=g). -rewrite In_graph_aff_edge_in_AE. -split; intuition. -rewrite H6. rewrite Hw. apply OptionN_as_OT.eq_refl. -left. split. -rewrite Heq. -rewrite (H0 x). split. -exists N0. rewrite <-Heq. auto. auto. -intro. elim (simple_graph (merge e g p q) (fst_ext b) (snd_ext b)). -split. -unfold Interfere in H5. -apply (proj1 (edgemap_to_edgeset_charac _ _ _ None (sym_imap_merge_map e g q p))). -rewrite edge_comm. -destruct H5. -generalize (AE_weights _ _ H5). simpl. congruence. -assumption. -apply (proj1 (edgemap_to_edgeset_charac _ _ _ (Some N0) (sym_pmap_merge_map e g q p))). -rewrite edge_comm. -cut (eq (fst_ext b, snd_ext b, Some N0) b). intro Heq2. -rewrite Heq2. -destruct H4. -assumption. -generalize (IE_weights _ _ H4). congruence. -rewrite (edge_eq b). change_rewrite. rewrite Hw. apply eq_refl. - -(* second part !!! *) - -destruct H1. -cut (get_weight b = Some N0). intro Hw. -assert (eq b (fst_ext b, snd_ext b, Some N0)). -rewrite (edge_eq b); change_rewrite. -apply eq_ordered_eq. constructor; simpl. -split; intuition. -rewrite Hw. apply OptionN_as_OT.eq_refl. - -destruct H2. -rewrite H3. -rewrite In_graph_aff_edge_in_AE. split. -exists N0; simpl; auto. -cut (eq b (redirect (snd_ext e) (fst_ext e) b)). intro Hcut. -rewrite <-H3. rewrite Hcut. rewrite (edge_eq _). -cut (get_weight (redirect (snd_ext e) (fst_ext e) b) = Some N0). intro Hw'. -cut (Prefere (fst_ext (redirect (snd_ext e) (fst_ext e) b)) - (snd_ext (redirect (snd_ext e) (fst_ext e) b)) - (merge e g p q)). intro. -unfold Prefere in H4. destruct H4. -assert (get_weight ((fst_ext (redirect (snd_ext e) (fst_ext e) b), - snd_ext (redirect (snd_ext e) (fst_ext e) b), Some x)) = Some N0). -apply (AE_weights) with (g:=merge e g p q). -rewrite In_graph_aff_edge_in_AE. split. -exists x. auto. auto. -rewrite Hw'. rewrite <-H5. simpl. auto. -destruct H2. apply merge_5. -apply (proj1 (H0 _) H2). -intro. elim H1. -destruct (eq_charac _ _ H5). right; intuition. left; intuition. -exists N0; simpl; auto. -intro. unfold Interfere in H5. -cut (eq (fst_ext (redirect (snd_ext e) (fst_ext e) b), - snd_ext (redirect (snd_ext e) (fst_ext e) b), - None) - (fst_ext b, snd_ext b, None)). intro H6. -rewrite H6 in H5. -elim H4. unfold Interfere. assumption. -unfold redirect; change_rewrite. -destruct (OTFacts.eq_dec (fst_ext b) (snd_ext e)). -elim H1. left. auto. -destruct (OTFacts.eq_dec (snd_ext b) (snd_ext e)). -elim H1. right. auto. -apply eq_refl. -rewrite <-Hcut. assumption. -unfold redirect; change_rewrite. -destruct (OTFacts.eq_dec (fst_ext b) (snd_ext e)). -elim H1. left. auto. -destruct (OTFacts.eq_dec (snd_ext b) (snd_ext e)). -elim H1. right. auto. -apply eq_refl. - -rewrite H3. -apply (proj2 (edgemap_to_edgeset_charac _ _ _ (Some N0) (sym_pmap_merge_map e g q p))). -set (f := (fun (y : VertexSet.elt) (s' : EdgeSet.t) => - EdgeSet.add (fst_ext e, y, Some 0%N) s')) in *. -set (s' := preference_adj (fst_ext e) (merge e g p q)) in *. -rewrite VertexSet.fold_1 in H2. -generalize VertexSet.elements_2. intro HH. -generalize (HH s'). clear HH. intro HH. -induction (VertexSet.elements s'). simpl in H2. -elim (EdgeSet.empty_1 H2). -set (f' := fun a e => f e a) in *. -rewrite MEdgeFacts.fold_left_assoc in H2. -set (tmp := fold_left f' l EdgeSet.empty) in *. -unfold f' in H2. unfold f in H2. -destruct (proj1 (RegRegProps.Dec.F.add_iff _ _ _) H2). -fold (eq (fst_ext e, a, Some N0) b) in H4. -rewrite (edge_eq b) in H4. -destruct (eq_charac _ _ H4); change_rewrite. -destruct H5. -assert (VertexSet.In (snd_ext b) s'). -apply HH. left. intuition. -unfold s' in H7. -unfold adj_set. rewrite (MapFacts.find_o _ (Vertex.eq_sym H5)). -assumption. -destruct H5. -assert (VertexSet.In (fst_ext b) s'). -apply HH. left. intuition. -unfold s' in H7. -unfold preference_adj in H7. unfold adj_set in H7. rewrite (MapFacts.find_o _ H5) in H7. -apply (sym_pmap_merge_map e g q p). assumption. -apply IHl. -assumption. -intros. apply HH. right. auto. - -unfold f'. unfold f. intros. -apply RegRegProps.add_add. - -unfold f'. unfold f. intros. -apply RegRegProps.Dec.F.add_m. apply eq_refl. auto. - -destruct H2. -rewrite AE_weights with (g:=g). auto. -rewrite (In_graph_aff_edge_in_AE). rewrite <-H0. intuition. -set (f := (fun (y : VertexSet.elt) (s' : EdgeSet.t) => - EdgeSet.add (fst_ext e, y, Some 0%N) s')) in *. -set (s' := (preference_adj (fst_ext e) (merge e g p q))) in *. -rewrite VertexSet.fold_1 in H2. -set (f' := fun a e => f e a) in *. -induction (VertexSet.elements s'). simpl in H2. -elim (EdgeSet.empty_1 H2). -rewrite MEdgeFacts.fold_left_assoc in H2. -set (tmp := fold_left f' l EdgeSet.empty) in *. -unfold f' in H2. unfold f in H2. -destruct (proj1 (RegRegProps.Dec.F.add_iff _ _ _) H2). -fold (eq (fst_ext e, a, Some N0) b) in H3. rewrite <-H3. -auto. -apply IHl. assumption. - -unfold f'. unfold f. intros. -apply RegRegProps.add_add. - -unfold f'. unfold f. intros. -apply RegRegProps.Dec.F.add_m. apply eq_refl. auto. -Qed. - -Lemma AE_merge_wl_aux : forall e g s p q, -(forall a, EdgeSet.In a s <-> aff_edge a /\ In_graph_edge a g) -> -(EdgeSet.Equal (AE_merge_up (preference_adj (fst_ext e) (merge e g p q)) - (preference_adj (fst_ext e) g) - (preference_adj (snd_ext e) g) - e s) - (AE (merge e g p q))). - -Proof. -intros. -unfold EdgeSet.Equal. intro. rewrite (AE_aux_2 e g s p q H). - rewrite (AE_aux_3 e g s p q H). -reflexivity. -Qed. - -Lemma AE_merge_wl : forall e g s p q, -(forall a, EdgeSet.In a s <-> aff_edge a /\ In_graph_edge a g) -> -(forall b, EdgeSet.In b (AE_merge_up (preference_adj (fst_ext e) (merge e g p q)) - (preference_adj (fst_ext e) g) - (preference_adj (snd_ext e) g) - e s) - <-> - aff_edge b /\ In_graph_edge b (merge e g p q)). - -Proof. -intros. -rewrite <-(In_graph_aff_edge_in_AE). -apply AE_merge_wl_aux; assumption. -Qed. - -(* It implements the interface *) - -(* Specification of the interference neighborhood *) -Lemma in_interf : forall x y g, -VertexSet.In x (interference_adj y g) <-> In_graph_edge (x,y,None) g. - -Proof. -split; intros. -apply in_interf_interf. auto. -apply interf_in_interf. auto. -Qed. - -(* Specification of the preference neighborhood *) -Lemma in_pref : forall x y g, -VertexSet.In x (preference_adj y g) <-> exists w, In_graph_edge (x,y,Some w) g. - -Proof. -split; intros. -apply in_pref_pref. auto. -destruct H. apply pref_in_pref with (w:=x0). auto. -Qed.
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