Require Import RelationClasses Setoid.
From mathcomp Require Import all_ssreflect.
Require Import edone set_tac finite_quotient preliminaries digraph sgraph treewidth minor equiv.
Require Import structures mgraph ptt mgraph2 skeleton.

Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Set Bullet Behavior "Strict Subproofs".

Section Subalgebra.
Variable L : labels.
Notation graph := (graph L).
Notation graph2 := (graph2 L).

Implicit Types (G H : graph) (U : sgraph) (T : forest).

Open Scope implicit_scope.

# Subalgebra of Tree-Width 2 Graphs

Arguments sdecomp T G B : clear implicits.

Definition compatible (T : forest) (G : graph2) (B : T -> {set G}) :=
exists t, (input \in B t) && (output \in B t).

Lemma sdecomp_sskel (T : forest) (G : graph2) (B : T -> {set G}) :
sdecomp T (sskeleton G) B <-> (sdecomp T G B /\ compatible B).
Proof.
split.
- case => [D1 D2 D3]. split. split => //.
+ move => x y /= xy. apply: D2. by rewrite /edge_rel/= xy.
+ case: (altP (input =P output :> skeleton G)) => E.
* case: (D1 input) => t Ht. exists t. by rewrite -E !Ht.
* suff: (input : sskeleton G) -- output by apply: D2. by rewrite /edge_rel/= E !eqxx.
- move => [[D1 D2 D3] C]. split => //= x y. case/or3P; first exact: D2.
+ case/and3P => ? /eqP E1 /eqP E2. by subst.
+ case/and3P => ? /eqP E1 /eqP E2. subst.
case: C => t. rewrite andbC. by exists t.
Qed.

Lemma hom_eqL (V : finType) (e1 e2 : rel V) (G : sgraph) (h : V -> G)
(e1_sym : symmetric e1) (e1_irrefl : irreflexive e1)
(e2_sym : symmetric e2) (e2_irrefl : irreflexive e2):
e1 =2 e2 ->
@hom_s (SGraph e1_sym e1_irrefl) G h ->
@hom_s (SGraph e2_sym e2_irrefl) G h.
Proof. move => E hom_h x y. rewrite /edge_rel/= -E. exact: hom_h. Qed.

Lemma skel_union_join (G1 G2 : graph) : @sk_rel _ (union G1 G2) =2 @join_rel G1 G2.
Proof.
move => [x|x] [y|y] /=.
- rewrite /edge_rel/= sum_eqE.
case: (boolP (x == y)) => //= E. apply/existsP/existsP.
+ move => [[e|e]]; rewrite !inE //= !sum_eqE. by exists e; rewrite !inE.
+ move => [e] H. exists (inl e). by rewrite !inE /= !sum_eqE in H *.
- apply: contraTF isT => /existsP [[e|e]]; by rewrite !inE //= andbC.
- apply: contraTF isT => /existsP [[e|e]]; by rewrite !inE //= andbC.
- rewrite /edge_rel/= sum_eqE.
case: (boolP (x == y)) => //= E. apply/existsP/existsP.
+ move => [[e|e]]; rewrite !inE //= !sum_eqE. by exists e; rewrite !inE.
+ move => [e] H. exists (inr e). by rewrite !inE /= !sum_eqE in H *.
Qed.

Section Quotients.
Variables (G1 G2 : graph2).

Let P : {set union G1 G2} := [set unl input; unl output; unr input; unr output].

Definition admissible (eqv : rel (union G1 G2)) :=
forall x y, eqv x y -> x = y \/ [set x;y] \subset P.

Lemma admissible_eqv_clot (l: pairs (union G1 G2)):
(forall p, p \in l -> p.1 \in P /\ p.2 \in P) -> admissible (eqv_clot l).
Proof.
move => H x y. rewrite eqv_clotE. case/connectP => p.
have relE u v : rel_of_pairs l u v -> ((u \in P) * (v \in P)) %type.
{ by move /H => /= [-> ->]. }
elim: p x => /= [x _ -> //|z p IH x /andP [E pth_p] lst_p]; first by left.
right. rewrite subUset !sub1set. case: (IH _ pth_p lst_p) => [Z|S]; subst.
- rewrite -Z. case/orP : E => E; by rewrite !(relE _ _ E).
- case/orP: E => E; rewrite (relE _ _ E) /=; by abstract (set_tac).
Qed.

Lemma decomp_quot (T1 T2 : forest) D1 D2 (e : equiv_rel (union G1 G2)):
sdecomp T1 (sskeleton G1) D1 -> sdecomp T2 (sskeleton G2) D2 ->
width D1 <= 3 -> width D2 <= 3 ->
admissible e -> #|[set pi e x | x in P]| <= 3 ->
exists T D, [/\ sdecomp T (skeleton (merge _ e)) D,
width D <= 3 & exists t,
D t = [set \pi x | x in P]].
Proof using.
move => /sdecomp_sskel [dec1 comp1] /sdecomp_sskel [dec2 comp2] W1 W2 adm_e collapse_P.
move: comp1 comp2 => [t1 /andP[t1_in t1_out]] [t2 /andP[t2_in t2_out]].
pose T := tjoin T1 T2.
pose D : tjoin T1 T2 -> {set sjoin G1 G2}:= decompU D1 D2.
have dec_D: sdecomp T (sjoin G1 G2) D by exact: join_decomp.
have dis_t1_t2 : ~~ connect (@sedge T)(inl t1) (inr t2) by rewrite join_disc.
have dis_T12 : {in [set inl t1;inr t2] &, forall x y, x != y -> ~~ connect (@sedge T) x y}.
{ move => [?|?] [?|?] /setUP[] /set1P-> /setUP[]/set1P ->.
all: by rewrite ?eqxx // => _; rewrite sconnect_sym. }
pose T' := @tlink T _ dis_T12.
pose D' := decompL D P : _ -> {set sjoin G1 G2}.
have dec_D' : sdecomp T' (sjoin G1 G2) D'.
apply/subsetP => x.
rewrite !inE -!orbA => /or4P [] /eqP->.
all: apply/bigcupP; solve
[ by exists (inl t1) => //; rewrite ?inE ?mem_imset ?eqxx
| by exists (inr t2) => //; rewrite ?inE ?mem_imset ?eqxx]. }
pose h := pi e : skeleton (union G1 G2) -> skeleton (merge _ e).
exists T'. exists (rename D' h); split; last by exists None.
- apply: rename_decomp => //.
+ apply: hom_eqL (pi_hom (e := e)). exact: skel_union_join.
+ exact: pi_surj.
+ move => x y. move/sk_rel_mergeE => [? [x0] [y0] /= [? ? ?]].
exists x0;exists y0. split => //. by rewrite /edge_rel/= -skel_union_join.
+ move => x y. move/eqquotP => /=. case/adm_e => [<-|A].
* case: (sbag_cover dec_D' x) => t Ht. left. exists t. by rewrite Ht.
* left. exists None. by rewrite /= -!sub1set -subUset.
- rewrite /width (bigop.bigID (pred1 None)).
rewrite bigop.big_pred1_eq. rewrite geq_max. apply/andP;split => //.
+ rewrite (bigop.reindex Some) /=.
* apply: (@leq_trans (maxn (width D1) (width D2))); last by rewrite geq_max W1 W2.
apply: leq_trans (join_width _ _).
apply: max_mono => t. exact: leq_imset_card.
* apply: subon_bij; last by (apply bij_on_codom => //; exact: (inl t1)).
by move => [x|]; rewrite !in_simpl // codom_f.
Qed.

Lemma decomp_par2 (T1 T2 : forest) D1 D2 :
sdecomp T1 (sskeleton G1) D1 -> sdecomp T2 (sskeleton G2) D2 ->
width D1 <= 3 -> width D2 <= 3 ->
exists T D, [/\ sdecomp T (sskeleton (G1 G2)) D & width D <= 3].
Proof using.
move => dec1 dec2 W1 W2.
case: (decomp_quot (e:=eqv_clot [:: (unl input, unr input); (unl output, unr output)])
dec1 dec2 W1 W2 _ _).
- apply admissible_eqv_clot. case => u v.
rewrite !inE /= !xpair_eqE => /orP [] /andP [/eqP -> /eqP->]; by rewrite !eqxx.
- pose P' : {set union G1 G2} := [set unl input; unl output].
apply: (@leq_trans #|P'|); last by rewrite cards2; by case (_ != _).
apply: (@leq_trans #|[set (\pi x : G1 G2) | x in P']|); last exact: leq_imset_card.
apply: subset_leq_card.
apply/subsetP => ? /imsetP [x H1 ->]. case/setUP : H1 => H1; first case/setUP : H1 => H1.
* by rewrite mem_imset.
* move/set1P : H1 => ->. apply/imsetP. exists (unl input); first by rewrite !inE eqxx ?orbT.
apply/eqquotP. eqv.
* move/set1P : H1 => ->. apply/imsetP. exists (unl output); first by rewrite !inE eqxx ?orbT.
apply/eqquotP. eqv.
- move => T [D] [A B [t C]]. exists T. exists D. split => //.
apply/sdecomp_sskel. split => //.
exists t. by rewrite C !mem_imset // !inE ?eqxx ?orbT.
Qed.

Lemma decomp_dot2 (T1 T2 : forest) D1 D2 :
sdecomp T1 (sskeleton G1) D1 -> sdecomp T2 (sskeleton G2) D2 ->
width D1 <= 3 -> width D2 <= 3 ->
exists T D, [/\ sdecomp T (sskeleton (G1 · G2)) D & width D <= 3].
Proof using.
move => dec1 dec2 W1 W2.
case: (decomp_quot (e:=eqv_clot [:: (unl output, unr input)]) dec1 dec2 W1 W2 _ _).
- apply admissible_eqv_clot. case => u v.
rewrite !inE /= !xpair_eqE => /andP [/eqP -> /eqP->]; by rewrite !eqxx.
- pose P' : {set union G1 G2} := [set unl input; unl output; unr output].
apply: (@leq_trans #|P'|); last apply cards3.
apply: (@leq_trans #|[set (\pi x : G1 · G2) | x in P']|); last exact: leq_imset_card.
apply: subset_leq_card.
apply/subsetP => ? /imsetP [x H1 ->]. move: H1.
rewrite /P -!setUA [[set _;_]]setUC !setUA. case/setUP => H1.
* by rewrite mem_imset.
* move/set1P : H1 => ->. apply/imsetP. exists (unl output); first by rewrite !inE eqxx ?orbT.
apply/eqquotP. eqv.
- move => T [D] [A B [t C]]. exists T. exists D. split => //.
apply/sdecomp_sskel. split => //.
exists t. by rewrite C !mem_imset // !inE ?eqxx ?orbT.
Qed.

End Quotients.

Lemma decomp_cnv (G : graph2) T D :
sdecomp T (sskeleton G) D -> sdecomp T (sskeleton (G°)) D.
Proof.
move/sdecomp_sskel => [dec cmp]. apply/sdecomp_sskel; split => //.
move: cmp => [t] H. exists t => /=. by rewrite andbC.
Qed.

Lemma decomp_dom (G : graph2) T D :
sdecomp T (sskeleton G) D -> sdecomp T (sskeleton (dom G)) D.
Proof.
move/sdecomp_sskel => [dec cmp]. apply/sdecomp_sskel; split => //.
move: cmp => [t] /andP [H _]. exists t => /=. by rewrite !H.
Qed.

Theorem eval_TW2 A (f: A -> graph2):
(forall a, exists T D, [/\ sdecomp T (sskeleton (f a)) D & width D <= 3]) ->
forall u, exists T D, [/\ sdecomp T (sskeleton (eval f u)) D & width D <= 3].
Proof.
move => Hf. elim => [u IHu v IHv | u IHu v IHv | u IHu | u IHu | | | a].
- move: IHu IHv => [T1] [D1] [? ?] [T2] [D2] [? ?].
exact: (decomp_dot2 (D1 := D1) (D2 := D2)).
- move: IHu IHv => [T1] [D1] [? ?] [T2] [D2] [? ?].
exact: (decomp_par2 (D1 := D1) (D2 := D2)).
- move: IHu => [T] [D] [D1 D2]. exists T. exists D. split => //.
exact: decomp_cnv.
- move: IHu => [T] [D] [D1 D2]. exists T. exists D. split => //.
exact: decomp_dom.
- apply: decomp_small. by rewrite card_unit.
- apply: decomp_small. by rewrite card_sum card_unit.
- apply Hf.
Qed.

End Subalgebra.

Section s.
Variable A: Type.
Let graph_of_term: term A -> graph2 (flat_labels A) := eval (@g2_var (flat_labels A)).
Theorem graph_of_TW2 (u : term A) :
exists T D, [/\ @sdecomp T (sskeleton (graph_of_term u)) D & width D <= 3].
Proof.
apply eval_TW2 => a.
apply: decomp_small. by rewrite card_sum card_unit.
Qed.

Lemma sskel_K4_free (u : term A) : K4_free (sskeleton (graph_of_term u)).
Proof.
case: (graph_of_TW2 u) => T [B] [B1 B2].
exact: TW2_K4_free B1 B2.
Qed.

Lemma skel_K4_free (u : term A) : K4_free (skeleton (graph_of_term u)).
Proof.
apply: minor_K4_free (@sskel_K4_free u).
exact: sub_minor (skel_sub _).
Qed.

(* TODO: define the subalgebra as a ptt_algebra? (with sigma types) *)

End s.