cycles.v

(** * Combi.SymGroup.cycles : The Cycle Decomposition of a Permutation *)
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(*      Copyright (C) 2016-2018 Florent Hivert <florent.hivert@lri.fr>        *)
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(** * The Cycle Decomposition of a Permutation

This files deals with decomposition of permutation into cycles. We define the
following notions, where [s] is a permutation over a finite type [T]:

- [psupport s] == the set of non fixed points for [s]
- [porbit_set s] == the set of the supports of the cycles of [s]
- [s \is cyclic] == [s] is a cyclic permutation
- [perm_dec E s] == for [E : {set {set T}}] the set of the restriction of [s]
                    to the elements of [E].
- [cycle_dec s] == the decomposition of [s] in cyclic permutations.
- [disjoint_supports A] == For [A : {set {perm T}}], the element of [A]
                    have pairwise dijoint supports.
- [cycle_dec_spec s A] == [A] is a decomposition of [s] into disjoint cycles.

The main result is Theorem [cycle_decP] which asserts that [cycle_dec s] is
the unique decomposition of [s] into disjoint cycles:

  [ unique (cycle_dec_spec s) (cycle_dec s). ]

**)

Require Import mathcomp.ssreflect.ssreflect.
From mathcomp Require Import ssrfun ssrbool eqtype ssrnat seq fintype.
From mathcomp Require Import tuple path bigop finset div.
From mathcomp Require Import fingroup perm action ssralg.
From mathcomp Require finmodule.

Require Import tools permcomp.

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


Import GroupScope.

#[export] Hint Resolve porbit_id : core.

Section PermCycles.
Variable T : finType.
Implicit Type (s : {perm T}).
Implicit Type (X : {set T}).
Implicit Type (A : {set {perm T}}).

(** ** Support of a permutation *)
Definition psupport s := ~: 'Fix_('P)([set s]).

Lemma in_psupport s x : (x \in psupport s) = (s x != x).
Proof using. by rewrite inE; apply/afix1P; case: eqP. Qed.

Lemma psupport_expg s n : psupport (s ^+ n) \subset psupport s.
Proof using.
by apply/subsetP => x; rewrite !in_psupport; apply contra => /eqP/permX_fix ->.
Qed.

Lemma psupport_perm_on S s : (perm_on S s) = (psupport s \subset S).
Proof using.
apply/subsetP/subsetP => H x.
- rewrite in_psupport; exact: H.
- rewrite inE -in_psupport; exact: H.
Qed.

Lemma psupport1 : psupport (perm_one T) = set0.
Proof using. by apply/setP => x; rewrite in_psupport inE perm1 eq_refl. Qed.

Lemma psupport_eq0 s : (s == 1) = (psupport s == set0).
Proof using.
apply/eqP/eqP => [ -> |]; first exact: psupport1.
move/setP => Heq; rewrite -permP => x.
by move/(_ x): Heq; rewrite in_psupport inE perm1 => /eqP.
Qed.

Lemma psupport_stable s x : (x \in psupport s) = (s x \in psupport s).
Proof using.
rewrite !in_psupport; congr negb; apply/idP/idP => [/eqP -> // | /eqP H].
by rewrite -[s x](permK s) H permK.
Qed.

Lemma card_psupport_noteq1 s : #|psupport s| != 1%N.
Proof using.
apply/cards1P => [[x] Hsupp].
have: s x == x by rewrite -in_set1 -Hsupp -psupport_stable Hsupp inE.
by move => /eqP; apply/eqP; rewrite -in_psupport Hsupp inE.
Qed.

Lemma psupport_card_porbit s x : (#|porbit s x| != 1%N) = (x \in psupport s).
Proof using.
rewrite inE; congr negb; apply/eqP/idP => [H|].
- by apply/afix1P => /=; rewrite -{2}(iter_porbit s x) H.
- rewrite /porbit -afix_cycle_in; last by rewrite inE.
  by move/orbit1P; rewrite /orbit /= => ->; rewrite cards1.
Qed.


(** Complement on porbit *)
Lemma porbit_fix s x : (s x == x) = (porbit s x == [set x]).
Proof using.
rewrite -[LHS]negbK -in_psupport -psupport_card_porbit negbK.
apply/eqP/eqP => [|->]; last by rewrite cards1.
by move/card_orbit1.
Qed.

Lemma porbit_mod s x i :
  (s ^+ i)%g x = (s ^+ (i %% #|porbit s x|))%g x.
Proof using.
rewrite {1}(divn_eq i #|porbit s x|) expgD permM; congr aperm.
elim: (i %/ #|porbit s x|) => [| {}i IHi].
- by rewrite mul0n expg0 perm1.
- by rewrite mulSnr expgD permM IHi permX; exact: iter_porbit.
Qed.

Lemma eq_in_porbit s x i j :
  ((s ^+ i)%g x == (s ^+ j)%g x) = (i == j %[mod #|porbit s x|]).
Proof using.
apply/idP/idP.
- rewrite [X in X == _]porbit_mod [X in _ == X]porbit_mod !permX.
  have HC : 0 < #|porbit s x|.
    by rewrite card_gt0; apply/set0Pn; exists x.
  rewrite -!(nth_traject _ (ltn_pmod _ HC)).
  rewrite nth_uniq // ?size_traject ?ltn_pmod //.
  exact: uniq_traject_porbit.
- by move=> /eqP H; apply/eqP; rewrite [LHS]porbit_mod [RHS]porbit_mod H.
Qed.


(** ** porbit_set of a permutation *)
Definition porbit_set s : {set {set T}} := [set x in porbits s | #|x| != 1%N].

Lemma in_porbit_setP s X x:
  reflect (exists2 X, X \in porbit_set s & x \in X) (s x != x).
Proof using.
rewrite -in_psupport; apply (iffP idP) => [Hy | [Y]].
- by exists (porbit s x); first by rewrite inE imset_f ?psupport_card_porbit.
- rewrite inE => /andP [/imsetP [y _ -> {Y}] Hcard].
  by rewrite -psupport_card_porbit -eq_porbit_mem => /eqP ->.
Qed.

Lemma partition_porbits s : partition (porbits s) setT.
Proof using.
rewrite /porbits porbitE.
have /orbit_partition : [acts <[s]>, on [set: T] | 'P].
  by apply/actsP => x _ t; rewrite !inE.
congr partition; apply/setP => x.
by apply/imsetP/imsetP => [] [y _ Hy]; exists y.
Qed.

Lemma partition_psupport s : partition (porbit_set s) (psupport s).
Proof using.
rewrite /porbit_set /porbits porbitE.
have /orbit_partition : [acts <[s]>, on psupport s | 'P].
  apply/actsP => t /cycleP [i -> {t}] x; rewrite !in_psupport /=.
  rewrite -permM -expgSr expgS permM; congr negb.
  by apply/eqP/eqP => [/perm_inj| ->].
congr partition; apply/setP => x.
rewrite !inE; apply/imsetP/andP => [[y Hy -> {x}] | [/imsetP [y _ -> {x}]]].
- split; first exact: imset_f.
  by rewrite -porbitE psupport_card_porbit.
- rewrite -porbitE psupport_card_porbit => Hy.
  by exists y.
Qed.

Lemma porbit_set_eq0 s : (s == perm_one T) = (porbit_set s == set0).
Proof using.
rewrite -subset0 /porbit_set; apply/eqP/subsetP => [-> C| H].
- rewrite !inE => /andP [/imsetP [x _ ->]].
  by rewrite psupport_card_porbit in_psupport permE eq_refl.
- apply/permP => x; rewrite permE; apply/eqP.
  have:= H (porbit s x); rewrite !inE imset_f //= => /contraT.
  by apply contraLR; rewrite -in_psupport psupport_card_porbit.
Qed.

Lemma porbit_set_astabs s X : X \in porbit_set s -> s \in 'N(X | 'P).
Proof using.
rewrite /astabs !inE /= => /andP [/imsetP [x0 _ -> {X} _]].
apply/subsetP => x; rewrite inE apermE.
rewrite -!eq_porbit_mem => /eqP <-.
by rewrite -{2}(expg1 s) porbit_perm.
Qed.

(** ** Cyclic permutations *)
Definition cyclic := [qualify s | #|porbit_set s| == 1%N].

Lemma cyclicP c :
  reflect (exists2 x, x \in psupport c & psupport c = porbit c x)
          (c \is cyclic).
Proof using.
apply (iffP cards1P) => [[sc Hsc] | [x Hx Hsc]].
- have:= partition_psupport c; rewrite Hsc => /cover_partition.
  rewrite /cover big_set1 => Hsupp; subst sc.
  have : psupport c != set0.
    rewrite -psupport_eq0 porbit_set_eq0 Hsc.
    apply/negP => /eqP Habs.
    by have:= set11 (psupport c); rewrite Habs in_set0.
  move=> /set0Pn [x Hx]; exists x; first by [].
  have : porbit c x \in porbit_set c.
    by rewrite inE imset_f //= psupport_card_porbit.
  by rewrite Hsc in_set1 => /eqP ->.
- exists (porbit c x); apply triv_part.
  + by rewrite inE imset_f //= psupport_card_porbit.
  + by rewrite -Hsc; apply: partition_psupport.
Qed.

Lemma cycle_cyclic t :
  t \is cyclic -> cycle t = [set t ^+ i | i : 'I_#|psupport t|].
Proof using.
move/cyclicP => [x Hx Hsupp]; rewrite Hsupp.
apply/setP => C; apply/cycleP/imsetP => [[i -> {C}] | [i Hi -> {C}]].
- have /(ltn_pmod i) Hmod : 0 < #|porbit t x|.
    by rewrite card_gt0; apply/set0Pn; exists x.
  exists (Ordinal Hmod) => //=; apply/permP => y /=.
  case: (boolP (y \in porbit t x)).
  + by rewrite -eq_porbit_mem => /eqP <-; exact: porbit_mod.
  + rewrite -Hsupp in_psupport negbK => /eqP Ht.
    by rewrite !permX_fix.
- by exists i.
Qed.

Lemma order_cyclic t : t \is cyclic -> #[t] = #|psupport t|.
Proof using.
rewrite /order => Hcy.
rewrite (cycle_cyclic Hcy) card_imset ?card_ord //.
move: Hcy => /cyclicP [x Hx Hsupp].
move=> [i Hi] [j Hj] /= /(congr1 (fun s => s x)) Hij.
apply val_inj => /=; apply/eqP.
rewrite -(nth_uniq x _ _ (uniq_traject_porbit t x)) ?size_traject -?Hsupp //.
by rewrite !nth_traject // -!permX Hij.
Qed.



(** Complement about restr_perm *)
Lemma psupport_restr_perm_incl X s :
  psupport (restr_perm X s) \subset X.
Proof using.
apply/subsetP => x; rewrite in_psupport.
by apply contraR => /out_perm -> //; apply: restr_perm_on.
Qed.

Lemma restr_perm_neq X s x :
  restr_perm X s x != x -> restr_perm X s x = s x.
Proof using.
move=> Hx; rewrite restr_permE //; move: Hx; apply contraR.
- by move=> /triv_restr_perm ->; rewrite perm1.
- by move/(out_perm (restr_perm_on X s)) ->.
Qed.

Lemma psupport_restr_perm X s :
  X \in porbit_set s -> psupport (restr_perm X s) = X.
Proof using.
move => HX; apply/eqP; rewrite eqEsubset psupport_restr_perm_incl /=.
apply/subsetP => y Hin; rewrite in_psupport.
rewrite restr_permE ?porbit_set_astabs // -in_psupport.
rewrite -(cover_partition (partition_psupport s)).
by apply/bigcupP; exists X.
Qed.

Lemma restr_perm_psupportE X s :
  restr_perm (psupport (restr_perm X s)) s = restr_perm X s.
Proof using.
case: (boolP (s \in 'N(X | 'P))) => [Hs|]; first last.
  move/triv_restr_perm ->; rewrite psupport1; apply/permP => x.
  by rewrite perm1 (out_perm (restr_perm_on _ _)) ?inE.
apply/permP => x.
case: (altP (restr_perm X s x =P x)) => Hx.
- rewrite Hx; move: Hx => /eqP.
  rewrite -[_ == _]negbK -in_psupport => /out_perm -> //.
  exact: restr_perm_on.
- rewrite (restr_perm_neq Hx).
  move: Hx; rewrite -in_psupport => Hxsupp.
  rewrite restr_permE //.
  apply/astabsP => y /=; rewrite apermE.
  case: (boolP (y \in psupport (restr_perm X s))) => [|Hy].
  + rewrite !in_psupport => Hy; have:= Hy => /restr_perm_neq => Hy2.
    move: Hy; apply contra => /eqP; rewrite -{2}Hy2 => /perm_inj.
    by rewrite Hy2 => ->.
  + apply negbTE; move: Hy; apply contra => Hsy.
    have bla i :
        (s ^+ i.+1) y \in psupport (restr_perm (T:=T) X s) /\
        (restr_perm (T:=T) X s ^+ i) (s y) = (s ^+ i.+1) y.
      elim: i => [|i [IHi1 IHi2]]; first by rewrite expg0 expg1 perm1.
      have := IHi1; rewrite in_psupport => /restr_perm_neq.
      rewrite -[RHS]permM -expgSr => <-.
      split; first by rewrite -psupport_stable.
      by rewrite !expgSr !permM IHi2 expgSr permM.
    have {bla} [] := bla #[s].-1 => [H _].
    move: H; rewrite prednK; last exact: order_gt0.
    by rewrite expg_order perm1.
Qed.

Lemma porbit_restr_perm s x :
  porbit (restr_perm (porbit s x) s) x = porbit s x.
Proof using.
case: (boolP (porbit s x \in porbit_set s)) => [Hx|].
- have Hiter (i : nat) : (restr_perm (porbit s x) s ^+ i) x = (s^+i) x.
    elim: i => [|n Hn]; first by rewrite !expg0 !perm1.
    rewrite !expgSr !permM {}Hn (restr_permE (porbit_set_astabs _)) //.
    by rewrite -eq_porbit_mem porbit_perm eq_porbit_mem.
  apply/setP => z; apply/porbitP/porbitP => [] [n].
  + by rewrite Hiter => ->; exists n.
  + by rewrite -Hiter => ->; exists n.
- rewrite inE imset_f //= psupport_card_porbit in_psupport negbK => H.
  apply/eqP; have:= H; rewrite porbit_fix => /eqP ->; rewrite -porbit_fix.
  by rewrite restr_permE ?inE //=; apply/subsetP => y; rewrite !inE => /eqP ->.
Qed.

Lemma porbit_set_restr s X :
  X \in porbit_set s -> porbit_set (restr_perm X s) = [set X].
Proof using.
move=> H; have:= H; rewrite inE => /andP [/imsetP [x _ Hx] HX]; subst X.
apply/setP => Y; rewrite [RHS]inE.
apply/idP/idP => [HY | /eqP -> {Y}].
- have HYX : Y \subset (porbit s x).
    rewrite -(psupport_restr_perm H).
    rewrite -(cover_partition (partition_psupport _)).
    by apply (bigcup_max _ HY).
  rewrite eqEsubset; apply/andP; split => //.
  move: HYX => /subsetP HYX.
  move: HY; rewrite inE => /andP [/imsetP [y _ Hy] _].
  have:= porbit_id (restr_perm (T:=T) (porbit s x) s) y.
  rewrite -Hy => /HYX; rewrite -eq_porbit_mem => /eqP Heq.
  by rewrite Hy -Heq porbit_restr_perm.
- rewrite inE HX andbT.
  apply/imsetP; exists x => //.
  by apply esym; apply porbit_restr_perm.
Qed.


(** * Decomposition of a permutation by restriction to disjoint stable subsets *)
Definition perm_dec (S : {set {set T}}) s : {set {perm T}} :=
  [set restr_perm X s | X in S].
Definition cycle_dec s : {set {perm T}} := perm_dec (porbit_set s) s.

Lemma cyclic_dec s : {in (cycle_dec s), forall C, C \is cyclic}.
Proof using.
move => C /imsetP [X HX ->].
by rewrite unfold_in porbit_set_restr ?cards1.
Qed.

Lemma psupport_cycle_dec s :
  [set psupport C | C in cycle_dec s] = porbit_set s.
Proof using.
apply/setP => X; apply/imsetP/idP.
- move => [x /imsetP[x0 Hx0 ->] ->].
  by rewrite psupport_restr_perm.
- rewrite inE => /andP [HX1 HX2].
  have HX: X \in porbit_set s by rewrite inE HX1 HX2.
  exists (restr_perm X s); last by rewrite psupport_restr_perm.
  by apply/imsetP; exists X.
Qed.

(** ** Disjoint psupport and commutation *)
Definition disjoint_psupports A :=
  trivIset [set psupport C| C in A] /\ {in A &, injective psupport}.

Lemma disjoint_psupport_subset (S1 S2 : {set {perm T}}) :
  S1 \subset S2 -> disjoint_psupports S2 -> disjoint_psupports S1.
Proof using.
rewrite /disjoint_psupports => Hsubs [Htriv Hinj].
split.
- exact: (trivIsetS (imsetS _ Hsubs) Htriv).
- move/subsetP in Hsubs.
  by move=> s t /Hsubs Hs /Hsubs; exact: Hinj.
Qed.

Lemma disjoint_perm_dec S s :
  trivIset S -> disjoint_psupports (perm_dec S s).
Proof using.
move=> Htriv.
have Hinj : {in perm_dec S s &, injective psupport}.
  move => C1 C2 /imsetP [c1 Hc1 ->] /imsetP [c2 HC2 ->] H.
  by rewrite -(restr_perm_psupportE c1 s) H restr_perm_psupportE.
split; last exact: Hinj.
apply/trivIsetP.
move=> C1 C2 /imsetP [c1 /imsetP [S1 HS1 -> ->]].
move=>       /imsetP [c2 /imsetP [S2 HS2 -> ->]] Hs12.
rewrite -setI_eq0 -subset0; apply/subsetP => x.
rewrite inE in_set0 => /andP [].
move=> /(subsetP (psupport_restr_perm_incl _ _)) => HxS1.
move=> /(subsetP (psupport_restr_perm_incl _ _)) => HxS2.
move: Htriv => /trivIsetP/(_ S1 S2 HS1 HS2) H.
have {}/H : S1 != S2 by move: Hs12; apply contra => /eqP ->.
rewrite -setI_eq0 => /eqP/setP/(_ x).
by rewrite !inE HxS1 HxS2.
Qed.

Lemma disjoint_cycle_dec s :
  disjoint_psupports (cycle_dec s).
Proof using.
apply: disjoint_perm_dec.
by have /and3P [] := partition_psupport s.
Qed.

Lemma psupport_disjointC s t :
  [disjoint psupport s & psupport t] -> commute s t.
Proof using.
move=> Hdisj; apply/permP => x; rewrite !permM.
case: (boolP (x \in psupport s)) => [Hs |].
- have:= Hdisj; rewrite disjoints_subset => /subsetP H.
  have:= H x Hs; rewrite inE in_psupport negbK => /eqP ->.
  move: Hs; rewrite psupport_stable => /H.
  by rewrite inE in_psupport negbK => /eqP ->.
- rewrite in_psupport negbK => /eqP Hs; rewrite Hs.
  case: (boolP (x \in psupport t)) => [Ht |].
  + move: Ht; rewrite psupport_stable.
    move: Hdisj; rewrite -setI_eq0 setIC setI_eq0 disjoints_subset => /subsetP.
    by move=> H{}/H; rewrite inE in_psupport negbK => /eqP ->.
  + by rewrite in_psupport negbK => /eqP ->; rewrite Hs.
Qed.

Lemma abelian_disjoint_psupports A : disjoint_psupports A -> abelian <<A>>.
Proof using.
move=> [] /trivIsetP Htriv Hinj.
rewrite abelian_gen abelianE; apply/subsetP => C HC.
apply/centP => D HD.
case: (altP (C =P D)) => [-> // | HCD].
apply psupport_disjointC.
apply Htriv; try exact: imset_f.
move: HCD; apply contra => /eqP Hsupp; apply/eqP.
exact: Hinj.
Qed.

Lemma abelian_perm_dec S s : trivIset S -> abelian <<perm_dec S s>>.
Proof using.
by move=> /disjoint_perm_dec Htriv; apply abelian_disjoint_psupports.
Qed.

Lemma abelian_cycle_dec s : abelian <<cycle_dec s>>.
Proof using.
by apply abelian_disjoint_psupports; apply disjoint_cycle_dec.
Qed.

Lemma restr_perm_inj s :
  {in porbit_set s &, injective ((restr_perm (T:=T))^~ s)}.
Proof using.
by move=> C D /psupport_restr_perm {2}<- /psupport_restr_perm {2}<- ->.
Qed.

Lemma out_perm_prod A x :
  {in A, forall C, x \notin psupport C} -> (\prod_(C in A) C) x = x.
Proof using.
move=> H.
rewrite -big_filter; have [/= S _ [_ mem_S] _] := big_enumP (mem A).
have {mem_S}H : {in S, forall C, x \notin psupport C}.
  by move=> C; rewrite mem_S; apply: H.
elim: S H => [| a S IHS HaS]; first by rewrite big_nil perm1.
rewrite big_cons permM.
have:= HaS _ (mem_head a S); rewrite in_psupport negbK => /eqP ->.
rewrite {}IHS // => C HC.
by apply: HaS; rewrite in_cons HC orbT.
Qed.

Import finmodule.FiniteModule morphism.

Lemma prod_of_disjoint A C x:
  disjoint_psupports A -> C \in A ->
  x \in psupport C -> (\prod_(C0 in A) C0) x = C x.
Proof using.
move=> Hdisj HC Hx.
have {Hx} Hnotin : {in A :\ C, forall C0 : {perm T}, x \notin psupport C0}.
  move: Hdisj => [/trivIsetP Hdisj Hinj] C0.
  rewrite 2!inE => /andP [HC0 HC0A].
  move/(_ _ _ (imset_f _ HC) (imset_f _ HC0A)): Hdisj.
  have Hdiff: psupport C != psupport C0.
    by move: HC0; apply contra => /eqP/Hinj ->.
  move=> /(_ Hdiff) /disjoint_setI0 /setP /(_ x).
  rewrite inE in_set0 => /nandP [] //.
  by move => /negbTE; rewrite Hx.
have Hin : forall i : {perm T}, (i \in A) && (i != C) -> i \in <<A>>.
  by move => c /andP [Hi _]; apply: mem_gen.
have abel := abelian_disjoint_psupports Hdisj.
(* We tranfert the computation in the Z-module associated [abel] *)
(* Product in the group become commutative sum in the Z-module   *)
rewrite [X in fun_of_perm X](_ :
           _ = val (\sum_(C0 in A) fmod abel C0)%R); first last.
  rewrite -morph_prod /=; last by move=> i; apply: mem_gen.
  rewrite -[LHS](fmodK abel) //.
  by apply group_prod => i; apply: mem_gen.
rewrite (bigD1 C) //= GRing.addrC -morph_prod //=.
rewrite -fmodM /=; [ | exact: group_prod | exact: mem_gen].
rewrite fmodK; last by apply groupM; [exact: group_prod |  exact: mem_gen].
(* Back in symmetric group computation *)
rewrite {abel Hin Hdisj HC} permM; congr fun_of_perm.
apply big_rec; first by rewrite perm1.
move=> i S Hi {2}<-; rewrite permM; congr fun_of_perm.
apply/eqP; rewrite -[_ == _]negbK -in_psupport.
by apply Hnotin; rewrite !inE andbC.
Qed.

Lemma expg_prod_of_disjoint A C x i:
  disjoint_psupports A -> C \in A ->
  x \in psupport C -> ((\prod_(C0 in A) C0) ^+ i) x = (C ^+ i) x.
Proof using.
move => Hdisj HC Hx.
have Hin j : (C ^+ j) x \in psupport C.
  elim j => [|k Hk]; first by rewrite expg0 perm1.
  by rewrite expgSr permM -psupport_stable.
elim: i => [|j Hj].
- by rewrite !expg0 perm1.
- by rewrite !expgSr !permM Hj (prod_of_disjoint Hdisj HC (Hin j)).
Qed.

Lemma psupport_of_disjoint A :
  disjoint_psupports A ->
  psupport (\prod_(C0 in A) C0) = \bigcup_(C0 in A) psupport C0.
Proof using.
move=> Hdisj; apply/setP => x.
rewrite in_psupport; apply/idP/bigcupP => [| [C] HC Hx].
- move=> H; apply/ exists_inP; move: H.
  apply contraR; rewrite negb_exists_in => /forallP Hsupp.
  apply/eqP; apply out_perm_prod => C.
  by move: (Hsupp C)=> /implyP.
- by rewrite (prod_of_disjoint Hdisj HC Hx) -in_psupport.
Qed.

Lemma porbit_set_of_disjoint A :
  disjoint_psupports A ->
  porbit_set (\prod_(C0 in A) C0) = \bigcup_(C0 in A) porbit_set C0.
Proof using.
move=> Hdisj; apply/setP => X; rewrite inE.
apply/andP/bigcupP => [[]/imsetP [x _ ->] Hcard|[C] HC].
- have:= Hcard; rewrite psupport_card_porbit psupport_of_disjoint //.
  move=> /bigcupP => [][C HC Hx]; exists C => //.
  rewrite inE; apply/andP; split => //.
  apply/imsetP; exists x => //.
  apply/setP => y.
  by apply/porbitP/porbitP =>[][i ->];
    exists i; rewrite (expg_prod_of_disjoint _ _ HC).
- rewrite inE => /andP [/imsetP[x _ ->] Hcard].
  split =>//.
  apply/imsetP; exists x => //.
  apply/setP=> y; rewrite psupport_card_porbit in Hcard.
  by apply/porbitP/porbitP =>[][i ->];
    exists i; rewrite (expg_prod_of_disjoint _ _ HC).
Qed.

Lemma perm_decE S s :
  trivIset S -> psupport s \subset cover S ->
  s \in 'C(S | ('P)^* ) ->
  \prod_(C in perm_dec S s) C = s.
Proof using.
move=> Htriv /subsetP Hcover Hact.
apply/permP => x.
case: (boolP (x \in psupport s)) => Hx; first last.
  rewrite out_perm_prod.
  + by move: Hx; rewrite in_psupport negbK => /eqP ->.
  + move=> Ctmp /imsetP [C] HC -> {Ctmp}.
    move: Hx; apply contra; rewrite !in_psupport => H.
    by have:= H; rewrite (restr_perm_neq H).
have:= Hcover x Hx => /bigcupP => [[C HC HxC]].
have Hrestr : (restr_perm (T:=T) C s) x = s x.
  rewrite restr_permE // -astab1_set.
  move: Hact => /astab_act/(_ HC) Hact.
  by apply/astabP => D; rewrite inE => /eqP ->{D}.
rewrite (prod_of_disjoint (C := restr_perm C s)).
- exact: Hrestr.
- exact: disjoint_perm_dec.
- exact: imset_f.
- by move: Hx; rewrite !in_psupport Hrestr.
Qed.

Lemma cycle_decE s : \prod_(C in cycle_dec s) C = s.
Proof using.
have /and3P [/eqP Hcov Htriv _] := partition_psupport s.
apply perm_decE => //; first by rewrite Hcov.
apply/astabP => C /porbit_set_astabs.
rewrite -astab1_set => /astabP; apply.
by rewrite inE.
Qed.

Lemma disjoint_psupports_of_decomp A B :
  disjoint_psupports A -> disjoint_psupports B ->
    \prod_(C in A) C = \prod_(C in B) C ->
    {in A & B, forall c1 c2, psupport c1 = psupport c2 -> c1 = c2}.
Proof using.
move=> HdisjA HdisjB /permP Heq c1 c2 Hc1 Hc2 /setP Hsupp.
apply/permP => x.
case: (boolP (x \in psupport c1)) => H0;
  have:= H0; rewrite {}Hsupp; move: H0.
- move => Hx1 Hx2; move/(_ x): Heq.
  by rewrite (prod_of_disjoint _ Hc1) ?(prod_of_disjoint _ Hc2).
- by rewrite !in_psupport !negbK => /eqP -> /eqP ->.
Qed.


Lemma disjoint_psupports_cycles A :
  {in A, forall C, C \is cyclic} ->
  disjoint_psupports A ->
  {in A, forall C, psupport C \in porbits (\prod_(C in A) C)}.
Proof using.
move=> Hcycle Hdisj C HC; move/(_ C HC): Hcycle.
rewrite /cyclic => /cards1P [X] Hpsupp.
have:= eq_refl X; rewrite -in_set1 -Hpsupp inE => /andP [/imsetP [x _] Hx].
subst X; rewrite porbitE=> Hcard.
have:= cover_partition (partition_psupport C); rewrite Hpsupp.
rewrite /cover big_set1 => <-; apply/imsetP; exists x => //.
have Hx : x \in psupport C.
  rewrite in_psupport; apply/eqP; rewrite -apermE => /afix1P.
  rewrite -afix_cycle => /orbit1P Hcontra.
  by move: Hcard; rewrite Hcontra cards1.
by apply/setP => y; apply/porbitP/porbitP => [] [i] ->;
  exists i; apply esym; rewrite (expg_prod_of_disjoint i Hdisj HC).
Qed.


Lemma disjoint_psupports_porbits A :
  {in A, forall C, C \is cyclic} ->
  disjoint_psupports A ->
  {in porbit_set (\prod_(C in A) C),
    forall X, exists2 C, C \in A & psupport C = X}.
Proof using.
move => Hcycle Hdisj X; rewrite inE => /andP [/imsetP [x] _ -> {X} Hcard].
case: (boolP (x \in psupport (\prod_(C in A) C))) => [Hin|].
- have: exists2 C0, (C0 \in A) & (x \in psupport C0).
    apply/exists_inP.
    move: Hin; apply contraLR; rewrite negb_exists => /forallP Hnotin.
    rewrite in_psupport negbK; apply/eqP; apply out_perm_prod => C HC.
    by have:= Hnotin C; rewrite HC andTb.
  move => [C0] HC0 Hx.
  exists C0 => //; move: Hx.
  have: psupport C0 \in (porbits (\prod_(C in A) C)).
    exact: disjoint_psupports_cycles.
  move => /imsetP [y] _ ->.
  by rewrite -eq_porbit_mem => /eqP.
- rewrite in_psupport negbK porbitE -apermE => /eqP /afix1P.
  rewrite -afix_cycle => /orbit1P.
  rewrite -porbitE => Hcontra; move: Hcard.
  by rewrite Hcontra cards1.
Qed.

(** ** Cycle decomposition of a permutation *)
Variant cycle_dec_spec s A : Prop :=
  CycleDecSpec of
    {in A, forall C, C \is cyclic} &
    disjoint_psupports A &
    \prod_(C in A) C = s : cycle_dec_spec s A.

Theorem cycle_decP s : unique (cycle_dec_spec s) (cycle_dec s).
Proof using.
split; first by constructor;
    [ exact: cyclic_dec |  exact: disjoint_cycle_dec | exact: cycle_decE].
move=> A [Hcy Hdisj Hprod].
apply/setP => /= C; apply/imsetP/idP=> /= [| HC].
- rewrite -{1}Hprod => [] [X HX1] ->.
  have:= disjoint_psupports_porbits Hcy Hdisj HX1 => [] [x Hx].
  rewrite -{1}(psupport_restr_perm HX1).
  move=> /(disjoint_psupports_of_decomp Hdisj (disjoint_cycle_dec s)).
  rewrite Hprod cycle_decE => /(_ erefl Hx) <- //.
  by apply/imsetP; exists X; rewrite -Hprod.
- have:= HC => /disjoint_psupports_cycles /= /(_ Hcy Hdisj) /imsetP [x _].
  rewrite Hprod => Hsupp.
  have Hx: porbit s x \in porbit_set s.
    rewrite inE; apply/andP; split.
    + by apply/imsetP; exists x.
    + by rewrite -Hsupp; rewrite card_psupport_noteq1.
  exists (porbit s x) => //.
  have:= disjoint_psupports_of_decomp Hdisj (disjoint_cycle_dec s).
  rewrite Hprod cycle_decE => /(_ erefl C (restr_perm (porbit s x) s) HC).
  apply; last by rewrite psupport_restr_perm.
  by apply/imsetP; exists (porbit s x).
Qed.

End PermCycles.

Arguments cyclic {T}.