diff options
Diffstat (limited to 'backend/InterfGraphMapImp.v')
| -rwxr-xr-x | backend/InterfGraphMapImp.v | 9401 |
1 files changed, 9401 insertions, 0 deletions
diff --git a/backend/InterfGraphMapImp.v b/backend/InterfGraphMapImp.v new file mode 100755 index 00000000..cdd83278 --- /dev/null +++ b/backend/InterfGraphMapImp.v @@ -0,0 +1,9401 @@ +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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