# Library GraphTheory.skeleton

Require Import Setoid Morphisms.

From mathcomp Require Import all_ssreflect.
Require Import edone finite_quotient bij preliminaries digraph sgraph treewidth minor checkpoint.
Require Import setoid_bigop structures mgraph mgraph2.

Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.

Local Open Scope quotient_scope.
Set Bullet Behavior "Strict Subproofs".

# Skeletons

Section Skeleton.
Variables Lv Le : Type.
Notation graph := (graph Lv Le).

Definition adjacent (G : graph) (x y : G) :=
[ e, e \in edges x y :|: edges y x].

Proof. movex y. by rewrite /adjacent setUC. Qed.

Lemma adjacentI (G : graph) (x y : G) (e : edge G) : e \in edges x y adjacent x y.
Proof. moveHe. apply/existsP. e. by rewrite inE He. Qed.

Arguments adjacentI [G x y] e _.

Lemma adjacent_edge (G : graph) (e : edge G) : adjacent (source e) (target e).
Proof. apply: (adjacentI e). by rewrite !inE. Qed.

The following lemma avoids the need for without loss of generality reasoning (regarding the direction of the edge) when analyzing adjacent x y
Lemma adjacentP (G : graph) (P : G G Prop) :
Symmetric P
( e : edge G, P (source e) (target e))
( x y : G, adjacent x y P x y).
Proof.
moveS H x y. case/existsPe. rewrite !inE.
case/orP⇒ /andP[/eqP<- /eqP<-] //. exact: S.
Qed.

Definition sk_rel (G : graph) := fun x y : G(x != y) && adjacent x y.

Lemma sk_rel_sym (G : graph) : symmetric (@sk_rel G).
Proof. movex y. by rewrite /sk_rel/= eq_sym adjacent_sym. Qed.

Lemma sk_rel_irrefl (G : graph) : irreflexive (@sk_rel G).
Proof. movex. by rewrite /sk_rel/= eqxx. Qed.

Definition skeleton (G : graph) :=
SGraph (@sk_rel_sym G) (@sk_rel_irrefl G).

Lemma skelP (G : graph) (P : G G Prop) :
Symmetric P
( e : edge G, source e != target e P (source e) (target e))
( x y : G, sk_rel x y P x y).
Proof.
moveS H x y. case/andPxNy. case/existsPe He. move: He xNy. rewrite !inE.
case/orP⇒ /andP[/eqP<- /eqP<-]; last rewrite eq_sym; move⇒ /H//. exact: S.
Qed.

End Skeleton.

Coercion skeleton : graph >-> sgraph.

Section SSkeleton.
Variables (Lv : comMonoid) (Le : elabelType).
Notation graph := (graph Lv Le).
Notation graph2 := (graph2 Lv Le).

Edges of skeletons of quotients
Lemma sk_rel_mergeE (G : graph) (e : equiv_rel G) x y :
@sk_rel _ _ (merge G e) x y
x != y x0 y0 : skeleton G, [/\ \pi x0 = x, \pi y0 = y & x0 -- y0].
Proof.
rewrite /= /sk_rel.
case: (boolP (x != y)) ⇒ //=; last by move_; split ⇒ [|[]].
movexy; split.
- moveA. split ⇒ //. move: xy. pattern x, y. apply adjacentP ⇒ //.
+ move ⇒ {x y A} x y H xy.
case: H ⇒ [|x0 [y0] [? ? /andP [E A]]]; first by rewrite eq_sym.
y0; x0. by rewrite /edge_rel/=/sk_rel eq_sym adjacent_sym A E.
+ movee0. (@endpoint _ _ G false e0). (@endpoint _ _ G true e0).
split ⇒ // ; last apply/andP;split ⇒ //.
× apply: contraNN xy ⇒ /eqP H. by rewrite eqmodE H.
× apply/existsP; e0. by rewrite !inE !eqxx.
- case_ [x0] [y0] [Px Py /andP [Dxy0]]. case/existsP ⇒ [e0 E].
wlog E : x0 y0 x y Dxy0 Px Py {xy E} / e0 \in edges x0 y0.
{ moveW. case/setUP : E; first exact: W.
rewrite adjacent_sym. apply: W ⇒ //. by rewrite eq_sym. }
rewrite inE in E. case/andP : E ⇒ [E1 E2].
apply/existsP; e0. by rewrite !inE /= (eqP E1) (eqP E2) Px Py !eqxx.
Qed.

Lemma hom_skel (G1 G2 : graph) (h: iso G1 G2) :
x y, @edge_rel (skeleton G1) x y @edge_rel (skeleton G2) (h x) (h y).
Proof.
apply skelP.
- movex y. by rewrite sgP.
- movee He. rewrite /edge_rel/=/sk_rel.
rewrite bij_eq ?He //=. apply/existsP; (h.e e).
rewrite !inE !endpoint_iso /=.
case (h.d e)=>/=; by rewrite !eqxx.
Qed.

Instance skel_iso: CProper (iso ==> diso) (@skeleton Lv Le).
Proof.
intros F G h. (iso_v h). apply: hom_skel. apply: (hom_skel (iso_sym h)).
Defined.

Lemma pi_hom (G : graph) (e : equiv_rel G) :
hom_s (pi e : skeleton G skeleton (merge G e)).
Proof.
movex y xy Dxy. apply/sk_rel_mergeE. split ⇒ //. by x; y.
Qed.
Arguments pi_hom [G] e.

## Strong skeletons

Definition sskeleton (G : graph2) := @sgraph.add_edge (skeleton G) input output.

Lemma sskelP (G : graph2) (P : G G Prop) :
Symmetric P
( e : edge G, source e != target e P (source e) (target e))
(input != output :> G P input output)
( x y : sskeleton G, x -- y P x y).
Proof.
moveS H IO x y. case/orP. exact: skelP.
case/orP ⇒ /= /and3P [A /eqP ? /eqP ?];subst; first exact: IO.
apply: S. exact: IO.
Qed.

Lemma skel_sub (G : graph2) : sgraph.subgraph (skeleton G) (sskeleton G).
Proof. id ⇒ //= x y H _. exact: subrelUl. Qed.

Isomorphim Lemmas

Lemma hom2_sskel (G1 G2 : graph2) (h: iso2 G1 G2) :
x y, @edge_rel (sskeleton G1) x y @edge_rel (sskeleton G2) (h x) (h y).
Proof.
apply sskelP.
- movex y. by rewrite sgP.
- movee He. rewrite /edge_rel /= [_ -- _](_ : _ = true) /= /edge_rel //=.
rewrite /sk_rel /= bij_eq ?He //=. apply/existsP; (h.e e).
rewrite !inE !endpoint_iso.
case (h.d e)=>/=; by rewrite !eqxx.
- moveH. by rewrite /edge_rel /= bij_eq // H iso2_input iso2_output !eqxx.
Qed.

Instance sskel_iso2: CProper (iso2 ==> diso) sskeleton.
Proof.
intros F G h. (iso_v h). exact: hom2_sskel. exact:(hom2_sskel (iso2_sym h)).
Defined.

Lemma decomp_iso2 (G1 G2 : graph2) (T : forest) B1 :
@sdecomp T (sskeleton G1) B1 G1 ≃2 G2
exists2 B2, @sdecomp T (sskeleton G2) B2 & width B2 = width B1.
Proof.
movedec iso. apply: decomp_iso dec _. exact: sskel_iso2.
Qed.

Bridging Lemmas

Lemma adjacent_induced (G : graph) (V : {set G}) (x y : induced V) :
Proof.
apply/existsP/existsP ⇒ -[e]; rewrite ?inEHe.
- (val e). move: He. by rewrite !inE -!val_eqE.
- suff eV : e \in edge_set V by (Sub e eV); rewrite !inE -!val_eqE.
rewrite inE. case/orP: He ⇒ /andP[/eqP→ /eqP->]; apply/andP; split; exact: valP.
Qed.

Lemma edge_set_adj (G : graph2) (x y : G) :
@edge_set _ _ G [set x; y] == set0 ~~ @adjacent _ _ G x y.
Proof.
apply: contraTN ⇒ /existsP[e]. rewrite !inEHe.
apply/set0Pn. e. rewrite !inE.
by case/orP: He ⇒ /andP[->->]; rewrite orbT.
Qed.

Lemma sskeleton_subgraph_for (G : graph2) (V : {set G}) (E : {set edge G})
(con : consistent V E) (i o : sig [eta mem V]) : val i = input val o = output
sgraph.subgraph (sskeleton (point (subgraph_for con) i o)) (sskeleton G).
Proof.
rewrite {2}/sskeleton/= ⇒ <- <-.
val; first exact: val_inj.
rewrite /hom_s. movex y xy _. move: x y xy. apply sskelP.
- movex y. by rewrite sg_sym.
- movee sNt. apply/orP; left. apply/andP. split⇒ //. exact: adjacent_edge.
- move⇒ /= iNo. by rewrite /edge_rel /= iNo !eqxx.
Qed.

Lemma sskeleton_adjacent (G : graph) (i o : G) :
adjacent i o diso G (sskeleton (point G i o)).
Proof.
moveAio.
(bij_id: bij G (sskeleton (point G i o))) ⇒ /=.
by movex y; rewrite {2}/edge_rel/= ⇒ →.
rewrite /is_dhom. apply sskelP.
- movex y. by rewrite sg_sym.
- rewrite /edge_rel/=/sk_rele →. exact: adjacent_edge.
- by rewrite /edge_rel/=/sk_rel ⇒ →.
Qed.

Lemma remove_edge_add (G : graph) (e : edge G) :
remove_edges [set e] [source e, elabel e, target e] G.
Proof.
Iso bij_id (bijD1 e) xpred0. constructor ⇒ //.
- by case ⇒ [f b|[|]].
- by case ⇒ [f|] /=.
Defined.

Lemma remove_loops (G : graph) (E : {set edge G}) :
{in E, e, source e = target e}
diso G (remove_edges E).
Proof.
moveEloops.
have Esame x y : x != y @edges _ _ (remove_edges E) x y = val @^-1: @edges _ _ G x y.
{ movexNy. apply/setPe. by rewrite !inE. }
(bij_id: bij G (remove_edges E)) ⇒ //= x y; rewrite /=/sk_rel ⇒ /andP[xNy].
all: rewrite /edge_rel/=/sk_rel.
all: rewrite xNy /adjacent/= !Esame // 1?eq_sym // -preimsetU.
- case/existsPe He.
suff H : e \notin E by apply/existsP; (Sub e H); rewrite inE.
apply: contraNN xNy ⇒ /Eloops eq_e. move: He. rewrite !inE -eq_e.
by case/orP⇒ /andP[/eqP<- /eqP<-].
- case/existsPe. rewrite inEHe. apply/existsP. by (val e).
Qed.

Lemma remove_edges_connected (G : graph) (E : {set edge G}) :
{in E, e : edge G, connect (@sk_rel _ _ (remove_edges E)) (source e) (target e)}
connected [set: skeleton G] connected [set: skeleton (remove_edges E)].
Proof.
moveE_conn G_conn. apply: connectedTIx y. have := connectedTE G_conn x y.
apply: connect_sub x y. move⇒ /=. apply skelP.
- by movex y; rewrite connect_symI //; exact: sk_rel_sym (remove_edges E).
- movee sNt. case: (boolP (e \in E)) ⇒ [/E_conn//|He].
apply: connect1. rewrite /sk_rel/= sNt /=.
by apply/existsP; (Sub e He); rewrite !inE !eqxx.
Qed.

Lemma remove_edges_cross (G : graph) (V : {set G}) (E : {set edge G}) (x y : G) :
E \subset edge_set V sk_rel x y y \notin V @sk_rel _ _ (remove_edges E) x y.
Proof.
moveE_subV xy yNV.
have {yNV} : (x \notin V) || (y \notin V) by rewrite yNV.
pattern x, y. revert x y xy.
apply skelP ; first by movex y; rewrite orbC sk_rel_sym.
movee sNt stNV. rewrite /sk_rel sNt /=.
case: (boolP (e \in E)) ⇒ He; last first.
- apply/existsP; (Sub e He); by rewrite !inE !eqxx.
- move: He stNV ⇒ /(subsetP E_subV). rewrite inE. by case/andP⇒ → →.
Qed.

Lemma remove_edges_restrict (G : graph) (V : {set G}) (E : {set edge G}) (x y : G) :
E \subset edge_set V connect (restrict (~: V) (@sk_rel _ _ G)) x y
connect (@sk_rel _ _ (remove_edges E)) x y.
Proof.
moveE_subV. apply: connect_mono x yx y /=. rewrite !inE -andbA.
case/and3PxNV yNV xy. exact: remove_edges_cross xy yNV.
Qed.

Lemma sskeleton_remove_io (G : graph2) (E : {set edge G}) :
E \subset @edge_set _ _ G IO
diso (sskeleton (point (remove_edges E) input output)) (sskeleton G).
Proof.
moveE_subIO.
(bij_id: bij (sskeleton (point (remove_edges E) input output)) (sskeleton G)) ⇒ //.
all: rewrite /is_dhom; apply sskelP.
- movex y. by rewrite sg_sym.
- movee sNt. apply/orP; left. rewrite /edge_rel/=/sk_rel sNt /=. exact: adjacent_edge.
- rewrite /edge_rel /= ⇒ →. by rewrite !eqxx.
- movex y. by rewrite sg_sym.
- movee sNt /=. case: (boolP (e \in E)) ⇒ Ee; last first.
+ apply/orP; left. rewrite /edge_rel/=/sk_rel sNt /=. apply/existsP; (Sub e Ee).
by rewrite !inE !eqxx.
+ move: Ee sNt ⇒ /(subsetP E_subIO). rewrite !inE /edge_rel/=.
by case/andP⇒ /orP[]/eqP→ /orP[]/eqP->; rewrite !eqxx // eq_sym ⇒ →.
- rewrite /edge_rel/= ⇒ →. by rewrite !eqxx.
Qed.

## Interval and Bag Graphs

NOTE: for interval graphs, we need to remove self loops on the ends, because these edges will be edges of the bag graph for the end nodes.
Definition interval_edges (G : graph) (x y : G) :=
edge_set (@interval G x y) :\: (edges x x :|: edges y y).

Lemma interval_edges_sym (G : graph) (x y : G) :
interval_edges x y = interval_edges y x.
Proof. by rewrite /interval_edges interval_sym setUC. Qed.

Lemma igraph_proof (G : graph) (x y : skeleton G) :
consistent (interval x y) (interval_edges x y).
Proof.
movee. rewrite inE =>[b]. rewrite inE =>/andP [_ H].
rewrite /edge_set in_set in H. move: H=>/andP[? ?]. by case b.
Qed.

Definition igraph (G : graph) (x y : skeleton G) :=
@point (subgraph_for (@igraph_proof G x y))
(Sub x (intervalL x y))
(Sub y (intervalR x y)).

Definition bgraph (G : graph) (U : {set G}) (x:G) :=
@point (induced (@bag G U x))
(Sub x (@bag_id G U x))
(Sub x (@bag_id G U x)).

Lemma bgraph_eq_io (G : graph) (U : {set G}) (x : G) : input = output :> @bgraph G U x.
Proof. by []. Qed.

Lemma interval_bag_edges_disj (G : graph) (U : {set G}) (x y : G) :
connected [set: skeleton G] x \in @CP (skeleton G) U y \in @CP (skeleton G) U
[disjoint edge_set (@bag G U x) & @interval_edges G x y].
Proof.
moveG_conn x_cp y_cp. rewrite -setI_eq0 -subset0. apply/subsetPe.
rewrite 3!inE ⇒ /and3P[]/andP[src_p tgt_p].
rewrite 4!inE ![_ \in @interval G _ _]inE (lock sinterval) !inE -lock.
rewrite negb_or !negb_and -!orbA ⇒ /andP[eNx eNy].
case/andP⇒ /or3P[src_x|src_y|src_sI] /or3P[tgt_x|tgt_y|tgt_sI].
all: move: eNx eNy; rewrite ?src_x ?src_y ?tgt_x ?tgt_y ?orbF //= ⇒ eNx eNy.
- move: eNx. by rewrite (eqP tgt_y) -(@bag_cp G _ U) // -(eqP tgt_y) tgt_p.
- by case: (disjointE (@interval_bag_disj G U x y y_cp) tgt_p tgt_sI).
- move: eNx. by rewrite (eqP src_y) -(@bag_cp G _ U) // -(eqP src_y) src_p.
- by case: (disjointE (@interval_bag_disj G U x y y_cp) tgt_p tgt_sI).
- by case: (disjointE (@interval_bag_disj G U x y y_cp) src_p src_sI).
- by case: (disjointE (@interval_bag_disj G U x y y_cp) src_p src_sI).
- by case: (disjointE (@interval_bag_disj G U x y y_cp) src_p src_sI).
Qed.

Lemma bag_edges_disj (G : graph) (U : {set G}) (x y : G) :
connected [set: skeleton G] x \in @CP G U y \in @CP G U x != y
[disjoint edge_set (@bag G U x) & edge_set (@bag G U y)].
Proof.
moveG_conn x_cp y_cp xNy. rewrite -setI_eq0 -subset0. apply/subsetPe.
rewrite 3!inE. case/andP⇒ /andP[src_x tgt_x] /andP[src_y tgt_y].
by case: (disjointE (@bag_disj G _ U x y _ _ _) src_x src_y).
Qed.

Lemma interval_edges_disj_cp (G : graph) (x y z : G) :
z \in @cp G x y [disjoint interval_edges x z & interval_edges z y].
Proof.
movez_cpxy. rewrite -setI_eq0 -subset0. apply/subsetPe.
rewrite 6!inE negb_or !negb_and. case/andP⇒ /and3P[_ src_Ixz tgt_Ixz].
rewrite 5!inE negb_or !negb_and. case/and3P⇒ /andP[e_z _] src_Izy tgt_Izy.
move: e_z.
have : source e \in [set z] by rewrite -(intervalI_cp z_cpxy) inE src_Ixz src_Izy.
rewrite inE =>->/=.
have : target e \in [set z] by rewrite -(intervalI_cp z_cpxy) inE tgt_Ixz tgt_Izy.
by rewrite inE =>->.
Qed.

Lemma interval_bag_edge_cover (G : graph) (x y : G) :
connected [set: skeleton G] x != y
[set: edge G] = edge_set (@bag G [set x; y] x) :|:
@interval_edges G x y :|:
edge_set (@bag G [set x; y] y).
Proof.
moveG_conn xNy. apply/eqP. rewrite eq_sym -subTset. apply/subsetPe _.
move: (sinterval_bag_cover G_conn xNy) ⇒ compU.
have : target e \in [set: G] by []. have : source e \in [set: G] by [].
rewrite !{}compU !in_setU -!orbA.
case: (boolP (source e \in @sinterval G x y)) ⇒ [Hsrc _|Nsrc /= Hsrc].
+ case: (boolP (target e \in @sinterval G x y)) ⇒ [Htgt _|Ntgt /= Htgt].
{ rewrite ![e \in _]inE ![_ \in @interval G _ _]inE Hsrc Htgt.
rewrite !orbT negb_or !negb_and.
suff [-> ->] : source e != x source e != y by [].
by split; apply: contraTneq Hsrc ⇒ ->; rewrite (@sinterval_bounds G). }
wlog Htgt : x y xNy Hsrc Ntgt {Htgt} / target e \in @bag G [set x; y] x.
{ moveHyp. case/orP: Htgt; first exact: Hyp.
rewrite setUC orbC orbAC orbC interval_edges_sym.
apply: Hyp; rewrite 1?sinterval_sym //. by rewrite eq_sym. }
have Nsrc : source e \notin @bag G [set x; y] x.
{ rewrite (disjointFl (@interval_bag_disj G _ x y _) Hsrc) //.
apply: CP_extensive. by rewrite !inE eqxx. }
have src_tgt : @sedge G (target e) (source e).
by apply: contraNneq Ntgt =>->. }
have : target e = x := bag_exit_edge G_conn _ Htgt Nsrc src_tgt.
move/(_ (CP_extensive _)). rewrite 3!inE eqxx ⇒ /(_ _)/Wrap[]// tgt_x.
rewrite ![e \in _]inE tgt_x (@intervalL G) [_ \in @interval G _ _]inE Hsrc.
rewrite orbT !andbT negb_or !negb_and.
suff [->->] : source e != x source e != y by [].
by split; apply: contraTneq Hsrc =>->; rewrite (@sinterval_bounds G).
+ wlog Hsrc : x y xNy {Nsrc Hsrc} / source e \in @bag G [set x; y] x.
{ moveHyp. case/orP: Hsrc; first exact: Hyp.
rewrite setUC sinterval_sym interval_edges_sym orbC orbAC orbC.
rewrite [(e \in _) || _]orbC orbAC [_ || (e \in _)]orbC.
by apply: Hyp; rewrite eq_sym. }
case: (boolP (target e \in @bag G [set x; y] x)) ⇒ Ntgt /=.
by rewrite [e \in _]inE Hsrc Ntgt.
have src_tgt : @sedge G (source e) (target e).
{ rewrite /=/sk_rel adjacent_edge andbT. by apply: contraNneq Ntgt =><-. }
have : source e = x := bag_exit_edge G_conn _ Hsrc Ntgt src_tgt.
move/(_ (CP_extensive _)). rewrite 3!inE eqxx ⇒ /(_ _)/Wrap[]// src_x.
case/orPHtgt.
× rewrite ![e \in _]inE src_x (@intervalL G) (negbTE xNy) orbF eqxx/=.
have → : target e != x.
{ apply: contraTneq Htgt =>->. by rewrite (@sinterval_bounds G). }
by have → : target e \in @interval G x y by rewrite inE Htgt.
× have Nsrc : source e \notin @bag G [set x; y] y.
{ have yNx : y != x by rewrite eq_sym.
rewrite (disjointFl (bag_disj G_conn _ _ yNx) Hsrc) //;
by apply: CP_extensive; rewrite !inE eqxx. }
have := bag_exit_edge G_conn (CP_extensive _) Htgt Nsrc _.
rewrite sg_sym 3!inE eqxx ⇒ /(_ _ src_tgt)/Wrap[]// tgt_y.
rewrite ![e \in _]inE src_x tgt_y (@intervalL G) (@intervalR G).
by rewrite !eqxx eq_sym (negbTE xNy).
Qed.

Lemma interval_cp_edge_cover (G : graph) (x y z : G) :
connected [set: skeleton G] z \in @cp G x y :\: [set x; y]
@interval_edges G x y = @interval_edges G x z :|:
edge_set (@bag G [set x; y] z) :|:
@interval_edges G z y.
Proof.
moveG_conn zPcpxy.
have z_cpxy : z \in @cp G x y by move: z zPcpxy; apply/subsetP; exact: subsetDl.
have z_cp : z \in @CP G [set x; y].
{ apply/bigcupP; (x, y); by [rewrite in_setX !in_set2 !eqxx |]. }
have [x_cp y_cp] : x \in @CP G [set x; y] y \in @CP G [set x; y]
by split; apply: CP_extensive; rewrite !inE eqxx.
apply/eqP. rewrite eqEsubset andbC !subUset [(_ \subset _) && _ && _]andbAC.
apply/andP; split; first (apply/andP; split).
- wlog suff : x y z zPcpxy {z_cpxy x_cp y_cp z_cp} /
interval_edges x z \subset interval_edges x y.
{ moveHyp. rewrite (Hyp x y z zPcpxy) /=.
by rewrite interval_edges_sym (interval_edges_sym x) Hyp 1?cp_sym 1?setUC. }
move: zPcpxy. rewrite in_setD in_set2 negb_or ⇒ /andP[]/andP[zNx zNy] z_cpxy.
apply/subsetPe. rewrite ![e \in _]inE !negb_or.
case/and3P⇒ /andP[eNx eNz] Hsrc Htgt. rewrite eNx /=.
have → : source e \in @interval G x y.
{ move: Hsrc. rewrite !in_setU !in_set1 -orbA.
case/or3P⇒ [->//|/eqP->|/(subsetP (sinterval_sub G_conn z_cpxy))->//].
move: z_cpxy ⇒ /(subsetP (cp_sub_interval G_conn _ _)).
by rewrite !in_setU !in_set1. }
have → : target e \in @interval G x y.
{ move: Htgt. rewrite !in_setU !in_set1 -orbA.
case/or3P⇒ [->//|/eqP->|/(subsetP (sinterval_sub G_conn z_cpxy))->//].
move: z_cpxy ⇒ /(subsetP (cp_sub_interval G_conn _ _)).
by rewrite !in_setU !in_set1. }
rewrite /= andbT.
suff : y \notin @interval G x z by apply: contraNN ⇒ /andP[/eqP<-].
rewrite in_setU in_set2 [_ == z]eq_sym (negbTE zNy) orbF negb_or.
apply/andP; split; first by apply: contraTneq z_cpxy =>->; rewrite cpxx inE.
by rewrite inE negb_and !negbK (@cp_sym G y x) z_cpxy.
- apply/subsetPe. rewrite ![e \in _]inE negb_or.
have /subsetP PsubI := bag_sub_sinterval G_conn x_cp y_cp zPcpxy.
case/andP⇒ /PsubI Hsrc /PsubI Htgt.
rewrite ![_ \in @interval G x y]inE Hsrc Htgt !orbT /= andbT.
apply/andP; split; apply: contraTN Hsrc ⇒ /andP[/eqP<-];
by rewrite sinterval_bounds.
- apply/subsetPe.
rewrite in_setD andbC !in_setU inE (interval_cp_cover G_conn zPcpxy).
rewrite negb_or 2!in_setU 2!(in_setU (target e)) ⇒ /andP[]/andP[].
rewrite [(_ \in _) || _]orbC -2!orbA. case/orPHsrc Htgt.
+ wlog Htgt : x y z zPcpxy Hsrc x_cp y_cp z_cp {z_cpxy Htgt} /
(target e \in @bag G [set x; y] z) || (target e \in x |: @sinterval G x z).
{ case/or3P: Htgt ⇒ /= Htgt Hyp.
× apply: Hyp ⇒ //; by rewrite Htgt.
× apply: Hyp ⇒ //; by rewrite Htgt.
× rewrite andbC orbC orbCA orbC setUC (interval_edges_sym x)interval_edges_sym.
by apply: Hyp; rewrite setUC 1?cp_sym // sinterval_sym Htgt. }
case/orP: HtgtHtgt _; first by rewrite ![e \in _]inE Hsrc Htgt.
case: (altP (source e =P target e)) ⇒ He.
by rewrite ![e \in _]inE -He Hsrc.
have src_tgt : @sedge G (source e) (target e).
{ by rewrite /=/sk_rel He adjacent_edge. }
have zNx : z != x by move: zPcpxy; rewrite !inE negb_or -andbA ⇒ /and3P[].
have Ntgt : target e \notin @bag G [set x; y] z.
{ move: Htgt; rewrite sinterval_sym in_setU1.
case/orP⇒ [/eqP->|Htgt]; last first.
by rewrite (disjointFl (interval_bag_disj _ x_cp) Htgt).
apply/bagPn. x; by [|rewrite cpxx inE]. }
move/cpP/(_ (edgep src_tgt)): (bag_exit G_conn z_cp Hsrc Ntgt).
rewrite mem_edgep orbC. have /negbTE->/= : z != target e
by apply: contraNneq Ntgt =><-; rewrite (@bag_id G).
move⇒ /eqP Esrc. rewrite ![e \in _]inE -Esrc eqxx (negbTE zNx) /=.
rewrite {1}Esrc eq_sym He (@intervalR G) in_setU in_set2.
by move: Htgt; rewrite in_setU1; case/orP=>->.
+ wlog Hsrc : x y z zPcpxy x_cp y_cp z_cp Htgt z_cpxy {Hsrc} /
source e \in x |: @sinterval G x z.
{ case/orP: Hsrc ⇒ /= Hsrc; first by [apply]; moveHyp.
rewrite andbC orbC orbCA orbC setUC (interval_edges_sym x) interval_edges_sym.
apply: Hyp; rewrite 1?[_ |: set1 _]setUC 1?cp_sym //.
× by rewrite orbC orbAC orbC (@sinterval_sym G y) sinterval_sym.
× by rewrite sinterval_sym. }
case/or3P: HtgtHtgt.
× case/andPeNx _. rewrite 2!inE negb_or eNx 2!inE negb_and.
have ->/= : source e != z.
{ apply: contraTneq Hsrc ⇒ →.
rewrite in_setU1 (@sinterval_bounds G) orbF.
move: zPcpxy; rewrite in_setD in_set2 negb_or; by case/andP⇒ /andP[]. }
have → : source e \in @interval G x z.
{ by move: Hsrc; rewrite !in_setU !in_set1 ⇒ /orP[]->. }
suff → : target e \in @interval G x z by [].
by move: Htgt; rewrite !in_setU !in_set1 ⇒ /orP[]->.
× case: (altP (source e =P target e)) ⇒ He _.
by rewrite ![e \in _]inE He Htgt.
have tgt_src : @sedge G (target e) (source e).
have zNx : z != x by move: zPcpxy; rewrite !inE negb_or -andbA ⇒ /and3P[].
have Nsrc : source e \notin @bag G [set x; y] z.
{ move: Hsrc; rewrite sinterval_sym in_setU1.
case/orP⇒ [/eqP->|Hsrc]; last first.
by rewrite (disjointFl (interval_bag_disj _ x_cp) Hsrc).
apply/bagPn. x; by [|rewrite cpxx inE]. }
move/cpP/(_ (edgep tgt_src)): (bag_exit G_conn z_cp Htgt Nsrc).
rewrite mem_edgep orbC. have /negbTE->/= : z != source e
by apply: contraNneq Nsrc =><-; rewrite (@bag_id G).
move⇒ /eqP Etgt. rewrite ![e \in _]inE -Etgt eqxx (negbTE zNx) andbF andbT /=.
rewrite {1}Etgt He (@intervalR G) in_setU in_set2.
by move: Hsrc; rewrite in_setU1; case/orP=>->.
× case: (altP (source e =P target e)) Hsrc Htgt ⇒ [<-|He Hsrc Htgt].
{ rewrite !in_setU1; case/orP⇒ [/eqP->|Hsrc].
- rewrite inE z_cpxy /= orbF ⇒ /eqP Exy.
move: zPcpxy. by rewrite -Exy cpxx setUid setDv inE.
- rewrite (disjointFr (sinterval_disj_cp z_cpxy) Hsrc) orbF ⇒ /eqP Htgt.
move: Hsrc. by rewrite Htgt inE (@cp_sym G y x) z_cpxy andbF. }
have {He} He : @sedge G (source e) (target e).
{ by rewrite /=/sk_rel He adjacent_edge. }
have [/negbTE srcNz /negbTE tgtNz] : source e != z target e != z.
{ split; [apply: contraTneq Hsrc|apply: contraTneq Htgt]=>->.
all: rewrite in_setU1 negb_or inE mem_cpl ?andbF andbT.
all: move: zPcpxy; by rewrite !inE negb_or -andbA =>/and3P[]. }
have {Hsrc} Hsrc : source e \in @interval G x z.
{ move: Hsrc; rewrite /interval !in_setU !in_set1 ⇒ /orP[/eqP->|->//].
by rewrite eqxx. }
have {Htgt} Htgt : target e \in @interval G z y.
{ move: Htgt; rewrite /interval !in_setU !in_set1 ⇒ /orP[/eqP->|->//].
by rewrite eqxx. }
have := interval_edge_cp z_cpxy He Hsrc Htgt. by rewrite srcNz tgtNz.
Qed.

Lemma has_edge (G : graph) (x y : G) :
connected [set: skeleton G] x != y 0 < #|edge G|.
Proof.
move/connectedTE/(_ x y). case/connect_irredPp _ xy.
case: (splitL p xy) ⇒ x' [/= xx'] _. apply/card_gt0P.
case/andP: xx'_. case/existsPe _. by e.
Qed.

Lemma card_val (T : finType) (P : pred T) (s : subFinType P) (A : pred s) :
#|val @: A| = #|A|.
Proof. rewrite card_imset //. exact: val_inj. Qed.

Lemma connected_skeleton' (G : graph) V E (con : @consistent _ _ G V E) :
U : {set subgraph_for con}, @connected (skeleton G) (val @: U)
( e, e \in edge_set (val @: U) source e != target e
let x := source e in let y := target e in
exists2 e, e \in E & (e \in edges x y) || (e \in edges y x))
@connected (skeleton (subgraph_for con)) U.
Proof.
moveU. set U' := val @: Uconn_U all_edges x y x_U y_U.
case/connectP: (conn_U _ _ (mem_imset val x_U) (mem_imset val y_U)) ⇒ p.
elim: p x x_U ⇒ [|a p IH] x x_U; first by move_ /val_inj <-; exact: connect0.
rewrite (lock edge_rel) /= -!andbA -lock. case/and4P_ a_U' xa.
have Ha : a \in V by case/imsetP: a_U'b _ ->; exact: valP.
set b : subgraph_for con := Sub a Ha. rewrite -[a]/(val b) ⇒ p_path p_last.
have b_U : b \in U by rewrite -(inj_imset _ _ val_inj).
apply: connect_trans (IH b b_U p_path p_last). apply: connect1 ⇒ /=.
apply/andP; split; first by apply/andP; split. move: xa.

rewrite -[a]/(val b).
move: b x b_U x_U {p_last p_path a_U' IH p} ⇒ {a Ha} b x b_U x_U.
rewrite /edge_rel/=/sk_rel -val_eqE. case/andPxNb. rewrite xNb /= ⇒ adjv_xb.
wlog [e0] : b x b_U x_U xNb {adjv_xb} / e, e \in edges (val x) (val b).
{ moveHyp. case/existsP: adjv_xbe. rewrite inE ⇒ /orP[]He.
- by apply: Hyp ⇒ //; e.
- rewrite adjacent_sym. apply: Hyp ⇒ //; by [rewrite eq_sym | e]. }
rewrite inE. case/andP⇒ /eqP Hsrc /eqP Htgt.
have Ne0 : source e0 != target e0 by rewrite Hsrc Htgt.
have : e0 \in edge_set U' by rewrite inE Hsrc Htgt !mem_imset.
case/all_edges⇒ [//|] e' Ee'. rewrite Hsrc HtgtHe'.
set e : edge (subgraph_for con) := Sub e' Ee'.
have He : e \in edges x b :|: edges b x.
{ move: He'. by rewrite !inE -[e']/(val e) -!(inj_eq val_inj). }
apply/existsP. by e.
Qed.

Lemma connected_skeleton (G : graph) V E (con : @consistent _ _ G V E) :
U : {set subgraph_for con}, @connected (skeleton G) (val @: U)
( e, e \in edge_set (val @: U) source e != target e e \in E)
@connected (skeleton (subgraph_for con)) U.
Proof.
moveU conn_U all_edges. apply: connected_skeleton' conn_U _.
movee E1 E2; e; rewrite ?inE ?eqxx //. exact: all_edges.
Qed.

Lemma connected_bgraph (G : graph2) (U : {set G}) (x : G) :
connected [set: skeleton G] x \in @CP (skeleton G) U
connected [set: skeleton (bgraph U x)].
Proof.
moveconn_G cp_x.
apply: connected_skeleton; rewrite imset_valT memKset //. exact: connected_bag.
Qed.

Lemma connected_igraph (G : graph2) (x y: G) :
connected [set: skeleton G]
connected [set: skeleton (igraph x y)].
Proof.
moveconn_G. apply: connected_skeleton; rewrite imset_valT memKset.
+ exact: connected_interval.
+ movee E1 E2. rewrite 4!inE E1 andbT negb_or. apply/andP;split.
× apply: contraNN E2 ⇒ /andP [/eqP → /eqP ->]; by rewrite eqxx.
× apply: contraNN E2 ⇒ /andP [/eqP → /eqP ->]; by rewrite eqxx.
Qed.

Lemma iso_pointxx (G : graph) (x : G) :
diso (sskeleton (point _ x x)) (skeleton G).
Proof.
(bij_id : bij (sskeleton (point G x x)) (skeleton G)) ⇒ u v /=.
- case/or3P ⇒ // /and3P[uNv /eqP eq1 /eqP eq2];
move: uNv; by rewrite eq1 eq2 eqxx.
- by rewrite {2}/edge_rel/= ⇒ →.
Qed.

Lemma sub_pointxx (G : graph) (x:G) :
sgraph.subgraph (sskeleton (point _ x x))
(skeleton G).
Proof. apply: iso_subgraph; exact: iso_pointxx. Qed.

Lemma sub_sub (G H : graph) :
subgraph G H sgraph.subgraph G H.
Proof.
casehv [he] [hd] [hom] [inj_hv inj_he].
hv ⇒ // x y xy _. move: x y xy.
apply skelP; first by movex y; rewrite sg_sym.
movee sNt. rewrite /edge_rel/=/sk_rel (inj_eq inj_hv) sNt.
generalize (endpoint_ihom (is_ihom:=hom) e false).
generalize (endpoint_ihom (is_ihom:=hom) e true).
case (hd e)=>/= <- <-.
Qed.

Definition flesh_out_graph (G : sgraph) sym0 tt (z : G) : graph2 :=
{| graph_of :=
{| vertex := G ; edge := [finType of { p : G × G | p.1 -- p.2 }];
endpoint b e := if b then snd (val e) else (val e).1;
vlabel _ := tt;
elabel := fun _sym0 |} ;
input := z; output := z |}.

Lemma flesh_out (G : sgraph) (sym0 : Le) (tt: Lv) (z : G) :
{ G' & (diso (sskeleton G') G × diso (skeleton G') G)%type}.
Proof.
pose G' := flesh_out_graph sym0 tt z. G'.
suff iso : diso (skeleton G') G.
{ split⇒ //. rewrite <-iso. exact: iso_pointxx. }
(bij_id : bij (skeleton G') G) ⇒ x y; rewrite /sk_rel.
- case/andP_ /existsP[][/=][u v /=] uv. rewrite !inE /=.
case/orP⇒ /andP[/eqP<- /eqP<-] //; by rewrite sg_sym.
- movexy. rewrite /edge_rel/=/sk_rel sg_edgeNeq //=.
apply/existsP; (Sub (x, y) xy). by rewrite !inE !eqxx.
Qed.

## Disjoint Union of a connected component and the remainder of the graph

Lemma edge_component (G : graph) (e : edge G) :
@component_of G (source e) = @component_of G (target e).
Proof.
case: (altP (source e =P target e)) ⇒ [->//|sDt].
apply: same_pblock ⇒ //. apply/components_pblockP.
have st : (source e : skeleton G) -- (target e) .
{ by rewrite /edge_rel/=/sk_rel sDt adjacent_edge. }
by (prev (edgep st)).
Qed.
Opaque merge.

Arguments unl [Lv Le G H] x : simpl never.
Arguments unr [Lv Le G H] x : simpl never.

Lemma edge_set_component (G : graph) (C : {set skeleton G}) (e : edge G) :
C \in @components G [set: G] (source e \in C) = (target e \in C).
Proof.
movecomp_C. apply/idP/idP.
- moveinC. rewrite (mem_component comp_C inC).
by rewrite edge_component (@in_component_of G).
- moveinC. rewrite (mem_component comp_C inC).
by rewrite -edge_component (@in_component_of G).
Qed.
Arguments edge_set_component [G C e].

Lemma edge_setN (G : graph) (C : {set G}) (e : edge G):
e \in edge_set C e \notin edge_set (~: C).
Proof. rewrite !inE. by case/andP ⇒ → →. Qed.

Lemma edge_comp (G : graph) (C : {set G}) (e : edge G):
C \in components [set: skeleton G]
(e \in edge_set C) + (e \in edge_set (~: C))%type.
Proof.
movecomp_C. case: (boolP (source e \in C)) ⇒ p; [left|right].
- by rewrite inE -(edge_set_component comp_C) p.
- by rewrite !inE -(edge_set_component comp_C) p.
Qed.

Section IsoComponents.
Variables (G : graph) (C : {set G}).
Hypothesis comp_C : C \in components [set: skeleton G].
Let G1 := induced C.
Let G2 := induced (~: C).

Lemma decC z : (z \in C) + (z \in ~: C)%type.
Proof. case: (boolP (z \in C)) ⇒ p; [by left|right]. by rewrite inE p. Qed.

Lemma decE (e : edge G) : (e \in edge_set C) + (e \in edge_set (~: C)).
Proof. exact: edge_comp. Qed.

Definition component_v (x : union G1 G2) := match x with inl xval x | inr xval x end.
Definition component_v' (z : G) : union G1 G2 :=
match decC z with inl pinl (Sub z p) | inr pinr (Sub z p) end.

Definition component_e (e : edge (union G1 G2)) := match e with inl eval e | inr eval e end.
Definition component_e' (e : edge G) : edge (union G1 G2) :=
match decE e with inl pinl (Sub e p) | inr pinr (Sub e p) end.

Lemma component_can_v : cancel component_v component_v'.
Proof.
casea; rewrite /component_v/component_v'; case: (decC _) ⇒ p.
- congr inl. exact: val_inj.
- exfalso. move: (valP a) ⇒ /= X. by rewrite inE X in p.
- exfalso. move: (valP a) ⇒ /=. by rewrite inE p.
- congr inr. exact: val_inj.
Qed.

Lemma component_can_v' : cancel component_v' component_v.
Proof. by movex; rewrite /component_v/component_v'; case: (decC x). Qed.

Lemma component_can_e : cancel component_e component_e'.
Proof.
casee; rewrite /component_e/component_e'; case: (decE _) ⇒ p.
- congr inl. exact: val_inj.
- exfalso. move: (valP e) ⇒ /= /edge_setN X. by contrab.
- exfalso. move: (valP e) ⇒ /= X. move/edge_setN in p. by rewrite X in p.
- congr inr. exact: val_inj.
Qed.

Lemma component_can_e' : cancel component_e' component_e.
Proof. movex; rewrite /component_e/component_e'; by case (decE _). Qed.

Lemma component_hom : is_ihom component_v component_e xpred0.
Proof. repeat split; by case. Qed.

Definition iso_component : iso (union (induced C) (induced (~: C))) G :=
Eval hnf in Iso' component_can_v component_can_v' component_can_e component_can_e' component_hom.

End IsoComponents.

End SSkeleton.