formal/coq/analysis/cardinality.v

(* mathcomp analysis (c) 2017 Inria and AIST. License: CeCILL-C.              *)
From HB Require Import structures.
From mathcomp Require Import all_ssreflect finmap ssralg ssrnum ssrint rat.
From mathcomp Require Import finset.
Require Import boolp mathcomp_extra classical_sets functions.

(******************************************************************************)
(*                              Cardinality                                   *)
(*                                                                            *)
(* This file provides an account of cardinality properties of classical sets. *)
(* This includes standard results of set theory such as the Pigeon Hole       *)
(* principle, the Cantor-Bernstein Theorem, or lemmas about the cardinal of   *)
(* nat, nat * nat, and rat.                                                   *)
(*                                                                            *)
(* Since universe polymorphism is not yet available in our framework, we      *)
(* develop a relational theory of cardinals: there is no type for cardinals   *)
(* only relations A #<= B and A #= B to compare the cardinals of two sets     *)
(* (on two possibly different types).                                         *)
(*                                                                            *)
(*           A #<= B == the cardinal of A is smaller or equal to the one of B *)
(*           A #>= B := B #<= A                                               *)
(*            A #= B == the cardinal of A is equal to the cardinal of B       *)
(*           A #!= B := ~~ (A #= B)                                           *)
(*      finite_set A == the set A is finite                                   *)
(*                   := exists n, A #= `I_n                                   *)
(*                   <-> exists X : {fset T}, A = [set` X]                    *)
(*                   <-> ~ ([set: nat] #<= A)                                 *)
(*    infinite_set A := ~ finite_set A                                        *)
(*       countable A <-> A is countable                                       *)
(*                   := A #<= [set: nat]                                      *)
(*        fset_set A == the finite set corresponding if A : set T is finite,  *)
(*                      set0 otherwise (T : choiceType)                       *)
(*              A.`1 := [fset x.1 | x in A]                                   *)
(*              A.`2 := [fset x.2 | x in A]                                   *)
(* {fimfun aT >-> T} == type of functions with a finite image                 *)
(*                                                                            *)
(******************************************************************************)

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

Reserved Notation "A '#<=' B" (at level 79, format "A  '#<='  B").
Reserved Notation "A '#>=' B" (at level 79, format "A  '#>='  B").
Reserved Notation "A '#=' B" (at level 79, format "A  '#='  B").
Reserved Notation "A '#!=' B" (at level 79, format "A  '#!='  B").

Import Order.Theory GRing.Theory.

Local Open Scope classical_set_scope.
Local Open Scope ring_scope.
Local Open Scope function_scope.

Declare Scope card_scope.
Delimit Scope card_scope with card.
Local Open Scope card_scope.

Definition card_le T U (A : set T) (B : set U) :=
  `[< $|{injfun [set: A] >-> [set: B]}| >].
Notation "A '#<=' B" := (card_le A B) : card_scope.
Notation "A '#>=' B" := (card_le B A) (only parsing) : card_scope.

Definition card_eq T U (A : set T) (B : set U) :=
  `[< $|{bij [set: A] >-> [set: B]}| >].
Notation "A '#=' B" := (card_eq A B) : card_scope.
Notation "A '#!=' B" := (~~ (card_eq A B)) : card_scope.

Definition finite_set {T} (A : set T) := exists n, A #= `I_n.
Notation infinite_set A := (~ finite_set A).

Lemma injPex {T U} {A : set T} :
   $|{inj A >-> U}| <-> exists f : T -> U, set_inj A f.
Proof. by split=> [[f]|[_ /Pinj[f _]]]; first by exists f. Qed.

Lemma surjPex {T U} {A : set T} {B : set U} :
  $|{surj A >-> B}| <-> exists f, set_surj A B f.
Proof. by split=> [[f]|[_ /Psurj[f _]]]; first by exists f. Qed.

Lemma bijPex {T U} {A : set T} {B : set U} :
  $|{bij A >-> B}| <-> exists f, set_bij A B f.
Proof. by split=> [[f]|[_ /Pbij[f _]]]; first by exists f. Qed.

Lemma surjfunPex {T U} {A : set T} {B : set U} :
  $|{surjfun A >-> B}| <-> exists f, B = f @` A.
Proof.
split=> [[f]|[f ->]]; last by squash [fun f in A].
by exists f; apply/seteqP; split=> //; apply: surj.
Qed.

Lemma injfunPex {T U} {A : set T} {B : set U}:
   $|{injfun A >-> B}| <-> exists2 f : T -> U, set_fun A B f & set_inj A f.
Proof. by split=> [[f]|[_ /Pfun[? ->] /funPinj[f]]]; [exists f | squash f]. Qed.

Lemma card_leP {T U} {A : set T} {B : set U} :
  reflect $|{injfun [set: A] >-> [set: B]}| (A #<= B).
Proof. exact: asboolP. Qed.

Lemma inj_card_le {T U} {A : set T} {B : set U} : {injfun A >-> B} -> (A #<= B).
Proof. by move=> f; apply/card_leP; squash (sigLR f). Qed.

Lemma pcard_leP {T} {U : pointedType} {A : set T} {B : set U} :
   reflect $|{injfun A >-> B}| (A #<= B).
Proof.
by apply: (iffP card_leP) => -[f]; [squash (valLR point f) | squash (sigLR f)].
Qed.

Lemma pcard_leTP {T} {U : pointedType} {A : set T} :
  reflect $|{inj A >-> U}| (A #<= [set: U]).
Proof.
by apply: (iffP pcard_leP) => -[f]; [squash f | squash ('totalfun_A f)].
Qed.

Lemma pcard_injP {T} {U : pointedType} {A : set T} :
  reflect (exists f : T -> U, {in A &, injective f}) (A #<= [set: U]).
Proof. by apply: (iffP pcard_leTP); rewrite injPex. Qed.

Lemma ppcard_leP {T U : pointedType} {A : set T} {B : set U} :
   reflect $|{splitinjfun A >-> B}| (A #<= B).
Proof. by apply: (iffP pcard_leP) => -[f]; squash (split f). Qed.

Lemma card_ge0 T U (S : set U) : @set0 T #<= S.
Proof. by apply/card_leP; squash set0fun. Qed.
#[global] Hint Resolve card_ge0 : core.

Lemma card_le0P T U (A : set T) : reflect (A = set0) (A #<= @set0 U).
Proof.
apply: (iffP idP) => [/card_leP[f]|->//].
by rewrite -subset0 => a /mem_set aA; have [x /set_mem] := f (SigSub aA).
Qed.

Lemma card_le0  T U (A : set T) : (A #<= @set0 U) = (A == set0).
Proof. exact/card_le0P/eqP. Qed.

Lemma card_eqP {T U} {A : set T} {B : set U} :
  reflect $|{bij [set: A] >-> [set: B]}| (A #= B).
Proof. exact: asboolP. Qed.

Lemma pcard_eq {T U} {A : set T} {B : set U} : {bij A >-> B} -> A #= B.
Proof. by move=> f; apply/card_eqP; squash (sigLR f). Qed.

Lemma pcard_eqP {T} {U : pointedType} {A : set T} {B : set U} :
   reflect $| {bij A >-> B} | (A #= B).
Proof.
by apply: (iffP card_eqP) => -[f]; [squash (valLR point f) | squash (sigLR f)].
Qed.

Lemma card_bijP {T U} {A : set T} {B : set U} :
   reflect (exists f : A -> B, bijective f) (A #= B).
Proof.
by apply: (iffP card_eqP) => [[f]|[_ /PbijTT[f _]]]; [exists f|squash f].
Qed.

Lemma card_eqVP {T U} {A : set T} {B : set U} :
   reflect $|{splitbij [set: A] >-> [set: B]}| (A #= B).
Proof. by apply: (iffP card_bijP) => [[_ /PbijTT[f _]]//|[f]]; exists f. Qed.

Lemma card_set_bijP {T} {U : pointedType} {A : set T} {B : set U} :
   reflect (exists f, set_bij A B f) (A #= B).
Proof.
by apply: (iffP pcard_eqP) => [[f]|[_ /Pbij[f _]]]; [exists f|squash f].
Qed.

Lemma ppcard_eqP {T U : pointedType} {A : set T} {B : set U} :
   reflect $| {splitbij A >-> B} | (A #= B).
Proof. by apply: (iffP pcard_eqP) => -[f]; [squash (split f)|squash f]. Qed.

Lemma card_eqxx T (A : set T) : A #= A.
Proof. by apply/card_eqP; squash idfun. Qed.
#[global] Hint Resolve card_eqxx : core.

Lemma card_eq00 T U : @set0 T #= @set0 U.
Proof.
apply/card_eqP/squash; apply: @bijection_of_bijective set0fun _.
by exists set0fun => -[x x0]; have := set_mem x0.
Qed.
#[global] Hint Resolve card_eq00 : core.

Section empty1.
Implicit Types (T : emptyType).
Lemma empty_eq0 T : all_equal_to (set0 : set T).
Proof. by move=> X; apply/setF_eq0/no. Qed.
Lemma card_le_emptyl T U (A : set T) (B : set U) : A #<= B.
Proof. by rewrite empty_eq0. Qed.
Lemma card_le_emptyr T U (A : set T) (B : set U) : (B #<= A) = (B == set0).
Proof. by rewrite empty_eq0; apply/idP/eqP=> [/card_le0P|->//]. Qed.

Definition emptyE_subdef := (empty_eq0, card_le_emptyl, card_le_emptyr, eq_opE).
End empty1.

Theorem Cantor_Bernstein T U (A : set T) (B : set U) :
  A #<= B -> B #<= A -> A #= B.
Proof.
elim/Ppointed: T => T in A *; first by rewrite !emptyE_subdef => _ ->.
elim/Ppointed: U => U in B *; first by rewrite !emptyE_subdef => ->.
suff {A B} card_eq (A B : set U) : B `<=` A -> A #<= B -> A #= B.
  move=> /ppcard_leP[f] /ppcard_leP[g].
  have /(_ _)/ppcard_eqP[|h] := card_eq _ _ (fun_image_sub f).
    by apply/pcard_leP; squash ([fun f in A] \o g).
  by apply/pcard_eqP; squash ((split h)^-1 \o [fun f in A]).
move=> BA /ppcard_leP[u]; have uAB := 'funS_u.
pose C_ := fix C n := if n is n.+1 then u @` C n else A `\` B.
pose C := \bigcup_n C_ n; have CA : C `<=` A.
  by move=> + [] => /[swap]; elim=> [|i IH] y _ []// x /IH/uAB/BA + <-; apply.
have uC: {homo u : x / x \in C}.
  by move=> x; rewrite !inE => -[i _ Cix]; exists i.+1 => //; exists x.
apply/card_set_bijP; exists (fun x => if x \in C then u x else x); split.
- move=> x Ax; case: ifPn; first by move=> _; apply: uAB.
  by move/negP; apply: contra_notP => NBx; rewrite inE; exists 0%N.
- move=> x y xA yA; have := 'inj_u xA yA.
  have [xC|] := boolP (x \in C); have [yC|] := boolP (y \in C) => // + _.
    by move=> /[swap]<-; rewrite uC// xC.
  by move=> /[swap]->; rewrite uC// yC.
- move=> y /[dup] By /BA Ay/=.
  case: (boolP (y \in C)); last by exists y; rewrite // ifN.
  rewrite inE => -[[|i]/= _ []// x Cix <-]; have Cx : C x by exists i.
  by exists x; [exact: CA|rewrite ifT// inE].
Qed.

Lemma card_esym T U (A : set T) (B : set U) : A #= B -> B #= A.
Proof. by move=> /card_eqVP[f]; apply/card_eqP; squash f^-1. Qed.

Lemma card_eq_le T U (A : set T) (B : set U) :
  (A #= B) = (A #<= B) && (B #<= A).
Proof.
apply/idP/andP => [/card_eqVP[f]|[]]; last exact: Cantor_Bernstein.
by split; apply/card_leP; [squash f|squash f^-1].
Qed.

Lemma card_eqPle T U (A : set T) (B : set U) :
  (A #= B) <-> (A #<= B) /\ (B #<= A).
Proof. by rewrite card_eq_le (rwP andP). Qed.

Lemma card_lexx T (A : set T) : A #<= A.
Proof. by apply/card_leP; squash idfun. Qed.
#[global] Hint Resolve card_lexx : core.

Lemma card_leT T (S : set T) : S #<= [set: T].
Proof. by apply/card_leP; squash (to_setT \o inclT _ \o val). Qed.

Lemma subset_card_le T (A B : set T) : A `<=` B -> A #<= B.
Proof. by move=> AB; apply/card_leP; squash (inclT _ \o subfun AB). Qed.

Lemma card_image_le {T U} (f : T -> U) (A : set T) : f @` A #<= A.
Proof.
elim/Ppointed: T => T in A f *; first by rewrite !emptyE_subdef image_set0.
by apply/pcard_leP; squash (pinv A f).
Qed.

Lemma inj_card_eq {T U} {A} {f : T -> U} : {in A &, injective f} -> f @` A #= A.
Proof. by move=> /inj_bij/pcard_eq/card_esym. Qed.
Arguments inj_card_eq {T U A f}.

Lemma card_some {T} {A : set T} : some @` A #= A.
Proof. exact: inj_card_eq. Qed.

Lemma card_image {T U} {A : set T} (f : {inj A >-> U}) : f @` A #= A.
Proof. exact: inj_card_eq. Qed.

Lemma card_imsub {T U} (A : set T) (f : {inj A >-> U}) X : X `<=` A -> f @` X #= X.
Proof. by move=> XA; rewrite (card_image [inj of f \o incl XA]). Qed.

Lemma card_le_trans (T U V : Type) (B : set U) (A : set T) (C : set V) :
  A #<= B -> B #<= C -> A #<= C.
Proof. by move=> /card_leP[f]/card_leP[g]; apply/card_leP; squash (g \o f). Qed.

Lemma card_eq_sym T U (A : set T) (B : set U) : (A #= B) = (B #= A).
Proof. by rewrite !card_eq_le andbC. Qed.

Lemma card_eq_trans T U V (A : set T) (B : set U) (C : set V) :
  A #= B -> B #= C -> A #= C.
Proof. by move=> /card_eqP[f]/card_eqP[g]; apply/card_eqP; squash (g \o f). Qed.

Lemma card_le_eql T T' T'' (A : set T) (B : set T') [C : set T''] :
   A #= B -> (A #<= C) = (B #<= C).
Proof. by move=> /card_eqPle[*]; apply/idP/idP; apply: card_le_trans. Qed.

Lemma card_le_eqr T T' T'' (A : set T) (B : set T') [C : set T''] :
   A #= B -> (C #<= A) = (C #<= B).
Proof. by move=> /card_eqPle[*]; apply/idP/idP => /card_le_trans; apply. Qed.

Lemma card_eql T T' T'' (A : set T) (B : set T') [C : set T''] :
   A #= B -> (A #= C) = (B #= C).
Proof. by move=> e; rewrite !card_eq_le (card_le_eql e) (card_le_eqr e). Qed.

Lemma card_eqr T T' T'' (A : set T) (B : set T') [C : set T''] :
   A #= B -> (C #= A) = (C #= B).
Proof. by move=> e; rewrite !card_eq_le (card_le_eql e) (card_le_eqr e). Qed.

Lemma card_ge_image {T U V} {A : set T} (f : {inj A >-> U}) X (Y : set V) :
  X `<=` A -> (f @` X #<= Y) = (X #<= Y).
Proof. by move=> XA; rewrite (card_le_eql (card_imsub _ _)). Qed.

Lemma card_le_image {T U V} {A : set T} (f : {inj A >-> U}) X (Y : set V) :
  X `<=` A -> (Y #<= f @` X) = (Y #<= X).
Proof. by move=> XA; rewrite (card_le_eqr (card_imsub _ _)). Qed.

Lemma card_le_image2 {T U} (A : set T) (f : {inj A >-> U}) X Y :
   X `<=` A -> Y `<=` A ->
   (f @` X #<= f @` Y) = (X #<= Y).
Proof. by move=> *; rewrite card_ge_image// card_le_image. Qed.

Lemma card_eq_image {T U V} {A : set T} (f : {inj A >-> U}) X (Y : set V) :
  X `<=` A -> (f @` X #= Y) = (X #= Y).
Proof. by move=> XA; rewrite (card_eql (card_imsub _ _)). Qed.

Lemma card_eq_imager {T U V} {A : set T} (f : {inj A >-> U}) X (Y : set V) :
  X `<=` A -> (Y #= f @` X) = (Y #= X).
Proof. by move=> XA; rewrite (card_eqr (card_imsub _ _)). Qed.

Lemma card_eq_image2 {T U} (A : set T) (f : {inj A >-> U}) X Y :
   X `<=` A -> Y `<=` A ->
   (f @` X #= f @` Y) = (X #= Y).
Proof. by move=> *; rewrite card_eq_image// card_eq_imager. Qed.

Lemma card_ge_some {T T'} {A : set T} {B : set T'} :
  (some @` A #<= B) = (A #<= B).
Proof. by rewrite (card_le_eql card_some). Qed.

Lemma card_le_some {T T'} {A : set T} {B : set T'} :
  (A #<= some @` B) = (A #<= B).
Proof. by rewrite (card_le_eqr card_some). Qed.

Lemma card_le_some2 {T T'} {A : set T} {B : set T'} :
  (some @` A #<= some @` B) = (A #<= B).
Proof. by rewrite card_ge_some card_le_some. Qed.

Lemma card_eq_somel {T T'} {A : set T} {B : set T'} :
  (some @` A #= B) = (A #= B).
Proof. by rewrite (card_eql card_some). Qed.

Lemma card_eq_somer {T T'} {A : set T} {B : set T'} :
  (A #= some @` B) = (A #= B).
Proof. by rewrite (card_eqr card_some). Qed.

Lemma card_eq_some2 {T T'} {A : set T} {B : set T'} :
  (some @` A #= some @` B) = (A #= B).
Proof. by rewrite card_eq_somel card_eq_somer. Qed.

Lemma card_eq0 {T U} {A : set T} : (A #= @set0 U) = (A == set0).
Proof. by rewrite card_eq_le card_le0 card_ge0 andbT. Qed.

Lemma card_eq_emptyr (T : emptyType) U (A : set T) (B : set U) :
  (B #= A) = (B == set0).
Proof. by rewrite empty_eq0; exact: card_eq0. Qed.

Lemma card_eq_emptyl (T : emptyType) U (A : set T) (B : set U) :
  (A #= B) = (B == set0).
Proof. by rewrite card_eq_sym card_eq_emptyr. Qed.

Definition emptyE := (emptyE_subdef, card_eq_emptyr, card_eq_emptyl).

Lemma card_setT T (A : set T) : [set: A] #= A.
Proof. by apply/card_esym/card_eqP; squash to_setT. Qed.
#[global] Hint Resolve card_setT : core.

Lemma card_setT_sym T (A : set T) : A #= [set: A].
Proof. exact/card_esym/card_setT. Qed.
#[global] Hint Resolve card_setT : core.

Lemma surj_card_ge {T U} {A : set T} {B : set U} : {surj B >-> A} -> A #<= B.
Proof.
by move=> g; rewrite (card_le_trans (subset_card_le 'surj_g)) ?card_image_le.
Qed.
Arguments surj_card_ge {T U A B} g.

Lemma pcard_surjP {T : pointedType} {U} {A : set T} {B : set U} :
  reflect (exists g, set_surj B A g) (A #<= B).
Proof.
apply: (iffP idP) => [|[_ /Psurj[g _]]]; last exact: surj_card_ge.
elim/Ppointed: U => U in B *; first by rewrite ?emptyE => ->; exists any.
by move=> /pcard_leP[f]; exists (pinv A f); apply: subl_surj surj.
Qed.

Lemma pcard_geP {T : pointedType} {U} {A : set T} {B : set U} :
  reflect $|{surj B >-> A}| (A #<= B).
Proof. by apply: (iffP pcard_surjP); rewrite surjPex. Qed.

Lemma ocard_geP {T U} {A : set T} {B : set U} :
  reflect $|{surj B >-> some @` A}| (A #<= B).
Proof.
by elim/Pchoice: T => T in A *; rewrite -card_ge_some; apply: pcard_geP.
Qed.

Lemma pfcard_geP {T U} {A : set T} {B : set U} :
  reflect (A = set0 \/ $|{surjfun B >-> A}|) (A #<= B).
Proof.
apply: (iffP idP); last by move=> [->//|[f]]; apply: surj_card_ge; exact: f.
elim/Ppointed: T => T in A *; first by rewrite !emptyE; left.
elim/Ppointed: U => U in B *; first by rewrite !emptyE => ->; right; squash any.
move=> /pcard_geP[f]; case: (eqVneq A set0); first by left.
move=> /set0P[x Ax]; right; apply/surjfunPex.
exists (fun y => if f y \in A then f y else x).
apply/seteqP; split.
  by move=> x' /[dup] /= /'surj_f [y By <-] Afy; exists y; rewrite ?ifT// inE.
by apply/image_subP => y By; case: ifPn; rewrite (inE, notin_set).
Qed.

Lemma card_le_II n m : (`I_n #<= `I_m) = (n <= m)%N.
Proof.
apply/idP/idP=> [/card_leP[f]|?];
  last by apply/subset_card_le => k /leq_trans; apply.
by have /leq_card := in2TT 'inj_(IIord \o f \o IIord^-1); rewrite !card_ord.
Qed.

Lemma ocard_eqP {T U} {A : set T} {B : set U} :
  reflect $|{bij A >-> some @` B}| (A #= B).
Proof.
elim/Pchoice: U => U in B *.
by rewrite -(card_eqr card_some); exact: (iffP pcard_eqP).
Qed.

Lemma oocard_eqP {T U} {A : set T} {B : set U} :
  reflect $|{splitbij some @` A >-> some @` B}| (A #= B).
Proof.
elim/Pchoice: U => U in B *; elim/Pchoice: T => T in A *.
rewrite -(card_eql card_some) -(card_eqr card_some).
exact: (iffP ppcard_eqP).
Qed.

Lemma card_eq_II {n m} : reflect (n = m) (`I_n #= `I_m).
Proof. by rewrite card_eq_le !card_le_II -eqn_leq; apply: eqP. Qed.

Lemma sub_setP  {T} {A : set T} (X : set A) : set_val @` X `<=` A.
Proof. by move=> x [/= a Xa <-]; apply: set_valP. Qed.
Arguments sub_setP {T A}.
Arguments image_subset {aT rT} f [A B].

Lemma card_subP T U (A : set T) (B : set U) :
  reflect (exists2 C, C #= A & C `<=` B) (A #<= B).
Proof.
apply: (iffP idP) => [/card_leP[f]|[C CA CB]]; last first.
  by rewrite -(card_le_eql CA); apply/card_leP; squash (inclT _ \o subfun CB).
exists (set_val @` range f); last exact: (subset_trans (sub_setP _)).
by rewrite ?(card_eql (inj_card_eq _))//; apply: in2W; apply: in2TT; apply: inj.
Qed.

(* remove *)
Lemma pigeonhole m n (f : nat -> nat) : {in `I_m &, injective f} ->
  f @` `I_m `<=` `I_n -> (m <= n)%N.
Proof.
move=> /Pinj[{}f->] /subset_card_le.
by rewrite (card_le_eql (inj_card_eq _))// card_le_II.
Qed.

Definition countable T (A : set T) := A #<= @setT nat.

Lemma eq_countable T U (A : set T) (B : set U) :
  A #= B -> countable A = countable B.
Proof. by move=> /card_le_eql leA; rewrite /countable leA. Qed.

Lemma countable_setT_countMixin (T : Type) :
  countable (@setT T) -> Countable.mixin_of T.
Proof.
by move=> /pcard_leP/unsquash f; exists f 'oinv_f; apply: in1TT 'funoK_f.
Qed.

Lemma countableP (T : countType) (A : set T) : countable A.
Proof. by apply/card_leP; squash (to_setT \o choice.pickle). Qed.
#[global] Hint Resolve countableP : core.

Lemma countable0 T : countable (@set0 T). Proof. exact: card_ge0. Qed.
#[global] Hint Resolve countable0 : core.

Lemma countable_injP T (A : set T) :
  reflect (exists f : T -> nat, {in A &, injective f}) (countable A).
Proof. exact: pcard_injP. Qed.

Lemma sub_countable T U (A : set T) (B : set U) : A #<= B ->
  countable B -> countable A.
Proof. exact: card_le_trans. Qed.

Lemma finite_setP T (A : set T) : finite_set A <-> exists n, A #= `I_n.
Proof. by []. Qed.

Lemma finite_II n : finite_set `I_n. Proof. by apply/finite_setP; exists n. Qed.
#[global] Hint Resolve finite_II : core.

Lemma card_II {n} : `I_n #= [set: 'I_n].
Proof. by apply/card_esym/pcard_eqP/bijPex; exists val; split. Qed.

Lemma finite_fsetP {T : choiceType} {A : set T} :
  finite_set A <-> exists X : {fset T}, A = [set` X].
Proof.
rewrite finite_setP; split=> [[n]|[X {A}->]]; last first.
  exists #|{: X}|; rewrite (card_eqr card_II).
  by apply/card_eqP; squash (to_setT \o enum_rank \o val_finset).
rewrite (card_eqr card_II) => /card_esym/card_eqVP[f]; pose g := f \o to_setT.
exists [fset val (g i) | i in 'I_n]%fset.
apply/seteqP; split=> [x /mem_set Ax|_ /imfsetP[i _ ->]]; last exact: set_valP.
by apply/imfsetP; exists (g^-1 (SigSub Ax)); rewrite ?[g _]invK//= inE.
Qed.

Lemma finite_subfset {T : choiceType} (X : {fset T}) {A : set T} :
  A `<=` [set` X] -> finite_set A.
Proof.
move=> AX; apply/finite_fsetP; exists [fset x in X | x \in A]%fset.
apply/seteqP; split=> x; rewrite /= ?inE; last by move=> /andP[_ /set_mem].
by move=> Ax; rewrite mem_set ?andbT//; apply: AX.
Qed.
Arguments finite_subfset {T} X {A}.

Lemma finite_set0 T : finite_set (set0 : set T).
Proof. by apply/finite_setP; exists 0%N; rewrite II0. Qed.
#[global] Hint Resolve finite_set0 : core.

Lemma finite_seqP {T : eqType} A :
   finite_set A <-> exists s : seq T, A = [set` s].
Proof.
elim/eqPchoice: T => T in A *; rewrite finite_fsetP.
split=> [[X ->]|[s ->]]; first by exists X.
by exists [fset x | x in s]%fset; apply/seteqP; split=> x /=; rewrite inE.
Qed.

Lemma finite_seq {T : eqType} (s : seq T) : finite_set [set` s].
Proof. by apply/finite_seqP; exists s. Qed.
#[global] Hint Resolve finite_seq : core.

Lemma finite_fset {T : choiceType} (X : {fset T}) : finite_set [set` X].
Proof. by apply/finite_fsetP; exists X. Qed.
#[global] Hint Resolve finite_fset : core.

Lemma finite_finpred {T : finType} {pT : predType T} (P : pT) :
  finite_set [set` P].
Proof.
rewrite finite_seqP; exists (enum P).
by apply/seteqP; split=> x/=; rewrite mem_enum.
Qed.
#[global]
Hint Extern 0 (finite_set [set` _]) => solve [apply: finite_finpred] : core.

Lemma finite_finset {T : finType} {X : set T} : finite_set X.
Proof.
by have -> : X = [set` mem X] by apply/seteqP; split=> x /=; rewrite ?inE.
Qed.
#[global] Hint Resolve finite_finset : core.

Lemma finite_set_countable T (A : set T) : finite_set A -> countable A.
Proof. by move=> /finite_setP[n /eq_countable->]. Qed.

Lemma infiniteP T (A : set T) : infinite_set A <-> [set: nat] #<= A.
Proof.
elim/Ppointed: T => T in A *.
  by rewrite !emptyE; split=> // /(congr1 (@^~ 0%N))/=; rewrite propeqE => -[].
split=> [Ainfinite| + /finite_setP[n eqAI]]; last first.
  rewrite (card_le_eqr eqAI) => le_nat_n.
  suff: `I_n.+1 #<= `I_n by rewrite card_le_II ltnn.
  exact: card_le_trans (subset_card_le _) le_nat_n.
have /all_sig2[f Af fX] : forall X : {fset T}, {x | x \in A & x \notin X}.
  move=> X; apply/sig2W; apply: contra_notP Ainfinite => nAX; apply/finite_fsetP.
  exists [fset x in X | x \in A]%fset; rewrite eqEsubset; split; last first.
    by move=> x/=; rewrite !inE => /andP[_]; rewrite inE.
  move=> x Ax /=; rewrite !inE/=; apply/andP; split; rewrite ?inE//.
  by apply: contra_notT nAX => xNX; exists x; rewrite ?inE.
do [under [forall x : {fset _}, _]eq_forall do rewrite inE] in Af *.
suff [g gE] : exists g : nat -> T,
    forall n, g n = f [fset g k | k in iota 0 n]%fset.
  have /Pinj[h hE] : {in setT &, injective g}.
    move=> i j _ _; apply: contra_eq; wlog lt_ij : i j / (i < j)%N => [hwlog|_].
    by case: ltngtP => // ij _; [|rewrite eq_sym];
       apply: hwlog=> //; rewrite lt_eqF//.
    rewrite [g j]gE; set X := (X in f X); have := fX X.
    by apply: contraNneq => <-; apply/imfsetP; exists i => //=; rewrite mem_iota.
  have/injPfun[i _] : {homo h : x / setT x >-> A x} by move=> i; rewrite -hE gE.
  by apply/pcard_leP; squash i.
pose g := fix g n k := if n isn't n'.+1 then f fset0
                       else f [fset g n' i | i in iota 0 k]%fset.
exists (fun n => g n n) => n.
suff {n} gn n k : (k <= n)%N -> g n k = f [fset g k k | k in iota 0 k]%fset.
  by rewrite gn//; congr f; apply/fsetP => k.
have [m] := ubnP n; elim: m n k => //= m IHm [|n] k /=.
  rewrite leqn0 => _ /eqP->/=.
  congr f; apply/fsetP => x; rewrite !inE; symmetry.
  by apply/imfsetP => /= -[].
rewrite ltnS => ltmn lekSn /=; congr f; apply/fsetP => i.
by apply/imfsetP/imfsetP => /= -[j]; rewrite mem_iota/= => jk ->;
   exists j; rewrite ?mem_iota//= ?add0n ?IHm//;
   by [rewrite (leq_trans jk)// (leq_trans lekSn)|rewrite -ltnS (leq_trans jk)].
Qed.

Lemma finite_setPn T (A : set T) : finite_set A <-> ~ ([set: nat] #<= A).
Proof. by rewrite -infiniteP notK. Qed.

Lemma card_le_finite T U (A : set T) (B : set U) :
  A #<= B -> finite_set B -> finite_set A.
Proof.
by move=> ?; rewrite !finite_setPn; apply: contra_not => /card_le_trans; apply.
Qed.

Lemma sub_finite_set T (A B : set T) : A `<=` B ->
  finite_set B -> finite_set A.
Proof. by move=> ?; apply/card_le_finite/subset_card_le. Qed.

Lemma finite_set_leP T (A : set T) : finite_set A <-> exists n, A #<= `I_n.
Proof.
split=> [[n /card_eqPle[]]|[n leAn]]; first by exists n.
by apply: card_le_finite leAn _; exists n.
Qed.

Lemma card_ge_preimage {T U} (B : set U) (f : T -> U) :
  {in f @^-1` B &, injective f} -> f @^-1` B #<= B.
Proof.
move=> /Pinj[g eqg]; rewrite -(card_le_eql (card_image g)) -eqg.
by apply: subset_card_le; apply: image_preimage_subset.
Qed.

Corollary finite_preimage {T U} (B : set U) (f : T -> U) :
  {in f @^-1` B &, injective f} -> finite_set B -> finite_set (f @^-1` B).
Proof. by move=> /card_ge_preimage fB; apply: card_le_finite. Qed.

Lemma eq_finite_set T U (A : set T) (B : set U) :
  A #= B -> finite_set A = finite_set B.
Proof.
move=> eqAB; apply/propeqP.
by split=> -[n Xn]; exists n; move: Xn; rewrite (card_eql eqAB).
Qed.

Lemma card_le_setD T (A B : set T) : A `\` B #<= A.
Proof. by apply: subset_card_le; rewrite setDE; apply: subIset; left. Qed.

Lemma finite_image T T' A (f : T -> T') : finite_set A -> finite_set (f @` A).
Proof. exact/card_le_finite/card_image_le. Qed.

Lemma finite_set1 T (x : T) : finite_set [set x].
Proof.
elim/Pchoice: T => T in x *.
by apply/finite_fsetP; exists (fset1 x); rewrite set_fset1.
Qed.
#[global] Hint Resolve finite_set1 : core.

Lemma finite_setD T (A B : set T) : finite_set A -> finite_set (A `\` B).
Proof. exact/card_le_finite/card_le_setD. Qed.

Lemma finite_setU T (A B : set T) :
  finite_set (A `|` B) = (finite_set A /\ finite_set B).
Proof.
pose fP := @finite_fsetP [choiceType of {classic T}]; rewrite propeqE; split.
  by move=> finAUB; split; apply: sub_finite_set finAUB.
by case=> /fP[X->]/fP[Y->]; apply/fP; exists (X `|` Y)%fset; rewrite set_fsetU.
Qed.

Lemma finite_set2 T (x y : T) : finite_set [set x; y].
Proof. by rewrite !finite_setU; split; apply: finite_set1. Qed.
#[global] Hint Resolve finite_set2 : core.

Lemma finite_set3 T (x y z : T) : finite_set [set x; y; z].
Proof. by rewrite !finite_setU; do !split; apply: finite_set1. Qed.
#[global] Hint Resolve finite_set3 : core.

Lemma finite_set4 T (x y z t : T) : finite_set [set x; y; z; t].
Proof. by rewrite !finite_setU; do !split; apply: finite_set1. Qed.
#[global] Hint Resolve finite_set4 : core.

Lemma finite_set5 T (x y z t u : T) : finite_set [set x; y; z; t; u].
Proof. by rewrite !finite_setU; do !split; apply: finite_set1. Qed.
#[global] Hint Resolve finite_set5 : core.

Lemma finite_set6 T (x y z t u v : T) : finite_set [set x; y; z; t; u; v].
Proof. by rewrite !finite_setU; do !split; apply: finite_set1. Qed.
#[global] Hint Resolve finite_set6 : core.

Lemma finite_set7 T (x y z t u v w : T) : finite_set [set x; y; z; t; u; v; w].
Proof. by rewrite !finite_setU; do !split; apply: finite_set1. Qed.
#[global] Hint Resolve finite_set7 : core.

Lemma finite_setI T (A B : set T) :
  (finite_set A \/ finite_set B) -> finite_set (A `&` B).
Proof.
by case; apply: contraPP; rewrite !infiniteP => /card_le_trans; apply;
   apply: subset_card_le.
Qed.

Lemma finite_setIl T (A B : set T) : finite_set A -> finite_set (A `&` B).
Proof. by move=> ?; apply: finite_setI; left. Qed.

Lemma finite_setIr T (A B : set T) : finite_set B -> finite_set (A `&` B).
Proof. by move=> ?; apply: finite_setI; right. Qed.

Lemma finite_setM T T' (A : set T) (B : set T') :
  finite_set A -> finite_set B -> finite_set (A `*` B).
Proof.
elim/Pchoice: T => T in A *; elim/Pchoice: T' => T' in B *.
move=> /finite_fsetP[{}A ->] /finite_fsetP[{}B ->].
apply/finite_fsetP; exists (A `*` B)%fset; apply/predeqP => x.
by split; rewrite /= inE => /andP.
Qed.

Lemma finite_image2 [aT bT rT : Type] [A : set aT] [B : set bT] (f : aT -> bT -> rT) :
  finite_set A -> finite_set B -> finite_set [set f x y | x in A & y in B]%classic.
Proof. by move=> fA fB; rewrite image2E; apply/finite_image/finite_setM. Qed.

Lemma finite_image11 [xT aT bT rT : Type] [X : set xT]
    (g : aT -> bT -> rT) (fa : xT -> aT) (fb : xT -> bT) :
    finite_set (fa @` X) -> finite_set (fb @` X) ->
  finite_set [set g (fa x) (fb x) | x in X]%classic.
Proof.
move=> /(finite_image2 g) /[apply]; apply: sub_finite_set; rewrite image2E.
by move=> r/= [x Xx <-]; exists (fa x, fb x) => //; split; exists x.
Qed.

Definition fset_set (T : choiceType) (A : set T) :=
  if pselect (finite_set A) is left Afin
  then projT1 (cid (finite_fsetP.1 Afin)) else fset0.

Lemma fset_setK (T : choiceType) (A : set T) : finite_set A ->
  [set` fset_set A] = A.
Proof. by rewrite /fset_set; case: pselect => // Afin _; case: cid. Qed.

Lemma in_fset_set (T : choiceType) (A : set T) : finite_set A ->
  fset_set A =i A.
Proof.
by move=> fA x; rewrite -[A in RHS]fset_setK//; apply/idP/idP; rewrite ?inE.
Qed.

Lemma fset_set_sub (T : choiceType) (A B : set T) :
  finite_set A -> finite_set B -> A `<=` B = (fset_set A `<=` fset_set B)%fset.
Proof.
move=> finA finB; apply/propext; split=> [AB|/fsubsetP AB t].
  by apply/fsubsetP => t; rewrite in_fset_set// in_fset_set// 2!inE => /AB.
by have := AB t; rewrite !in_fset_set// !inE.
Qed.

Lemma fset_set_set0 (T : choiceType) (A : set T) : finite_set A ->
  fset_set A = fset0 -> A = set0.
Proof.
move=> finA; rewrite /fset_set; case: pselect => // {}finA.
by case: cid => _/= -> ->; rewrite set_fset0.
Qed.

Lemma fset_set0 {T : choiceType} : fset_set (set0 : set T) = fset0.
Proof.
by apply/fsetP=> x; rewrite in_fset_set ?inE//; apply/negP; rewrite inE.
Qed.

Lemma fset_set1 {T : choiceType} (x : T) : fset_set [set x] = [fset x]%fset.
Proof.
apply/fsetP=> y; rewrite in_fset_set ?inE//.
by apply/idP/idP; rewrite inE => /eqP.
Qed.

Lemma fset_setU {T : choiceType} (A B : set T) :
  finite_set A -> finite_set B ->
  fset_set (A `|` B) = (fset_set A `|` fset_set B)%fset.
Proof.
move=> fA fB; apply/fsetP=> x.
rewrite ?(inE, in_fset_set)//; last by rewrite finite_setU.
by apply/idP/orP; rewrite ?inE.
Qed.

Lemma fset_setI {T : choiceType} (A B : set T) :
  finite_set A -> finite_set B ->
  fset_set (A `&` B) = (fset_set A `&` fset_set B)%fset.
Proof.
move=> fA fB; apply/fsetP=> x.
rewrite ?(inE, in_fset_set)//; last by apply: finite_setI; left.
by apply/idP/andP; rewrite ?inE.
Qed.

Lemma fset_setU1 {T : choiceType} (x : T) (A : set T) :
  finite_set A -> fset_set (x |` A) = (x |` fset_set A)%fset.
Proof. by move=> fA; rewrite fset_setU// fset_set1. Qed.

Lemma fset_setD {T : choiceType} (A B : set T) :
  finite_set A -> finite_set B ->
  fset_set (A `\` B) = (fset_set A `\` fset_set B)%fset.
Proof.
move=> fA fB; apply/fsetP=> x.
rewrite ?(inE, in_fset_set)//; last exact: finite_setD.
by apply/idP/andP; rewrite ?inE => -[]; rewrite ?notin_set.
Qed.

Lemma fset_setD1 {T : choiceType} (x : T) (A : set T) :
  finite_set A -> fset_set (A `\ x) = (fset_set A `\ x)%fset.
Proof. by move=> fA; rewrite fset_setD// fset_set1. Qed.

Lemma fset_setM {T1 T2 : choiceType} (A : set T1) (B : set T2) :
    finite_set A -> finite_set B ->
  fset_set (A `*` B) = (fset_set A `*` fset_set B)%fset.
Proof.
move=> Afin Bfin; have ABfin : finite_set (A `*` B) by apply: finite_setM.
apply/fsetP => i; apply/idP/idP; rewrite !(inE, in_fset_set)//=.
  by move=> [/mem_set-> /mem_set->].
by move=> /andP[]; rewrite !inE.
Qed.

Definition fst_fset (T1 T2 : choiceType) (A : {fset (T1 * T2)}) : {fset T1} :=
  [fset x.1 | x in A]%fset.
Definition snd_fset (T1 T2 : choiceType) (A : {fset (T1 * T2)}) : {fset T2} :=
  [fset x.2 | x in A]%fset.
Notation "A .`1" := (fst_fset A) : fset_scope.
Notation "A .`2" := (snd_fset A) : fset_scope.

Lemma finite_set_fst (T1 T2 : choiceType) (A : set (T1 * T2)) :
  finite_set A -> finite_set A.`1.
Proof.
move=> /finite_fsetP[B A_B]; apply/finite_fsetP; exists (B.`1)%fset.
by apply/seteqP; split=> [x/= [y]|_/= /imfsetP[[x1 x2]/= +] ->]; rewrite A_B;
  [move=> xyB; apply/imfsetP; exists (x, y)|move=> ?; exists x2].
Qed.

Lemma finite_set_snd (T1 T2 : choiceType) (A : set (T1 * T2)) :
  finite_set A -> finite_set A.`2.
Proof.
move=> /finite_fsetP[B A_B]; apply/finite_fsetP; exists (B.`2)%fset.
apply/seteqP; split=> [y/= [x]|_/= /imfsetP[[x1 x2]/= +] ->]; rewrite A_B;
  by [move=> xyB; apply/imfsetP; exists (x, y)|move=> ?; exists x1].
Qed.

Lemma bigcup_finite {I T} (D : set I) (F : I -> set T) :
    finite_set D -> (forall i, D i -> finite_set (F i)) ->
  finite_set (\bigcup_(i in D) F i).
Proof.
elim/Pchoice: I => I in D F *.
elim/Ppointed: T => T in F *; first by rewrite emptyE.
move=> Dfin Ffin; pose G (i : fset_set D) := fset_set (F (val i)).
suff: (\bigcup_(i in D) F i #<= [set: {i & G i}])%card.
  by move=> /card_le_finite; apply; apply: finite_finset.
apply/pcard_geP/surjPex.
exists (fun (k : {i : fset_set D & G i}) => val (projT2 k)).
move=> y [i Di Fky]/=.
have Dk : i \in fset_set D by rewrite in_fset_set// inE.
pose k : fset_set D := [` Dk]%fset.
have Gy : y \in G k by rewrite in_fset_set ?inE//; apply: Ffin.
by exists (Tagged G [` Gy]%fset).
Qed.

Lemma trivIset_sum_card (T : choiceType) (F : nat -> set T) n :
  (forall n, finite_set (F n)) -> trivIset [set: nat] F ->
  (\sum_(i < n) #|` fset_set (F i)| =
   #|` fset_set (\big[setU/set0]_(k < n) F k)|)%N.
Proof.
move=> finF tF; elim: n => [|n ih]; first by rewrite !big_ord0 fset_set0.
rewrite big_ord_recr//= ih big_ord_recr/= fset_setU//; last first.
  by rewrite -bigcup_mkord; exact: bigcup_finite.
rewrite cardfsU [X in (_ - X)%N](_ : _  = O) ?subn0// ?EFinD ?natrD//.
apply/eqP; rewrite cardfs_eq0 -fset_setI//; last first.
  by rewrite -bigcup_mkord; exact: bigcup_finite.
rewrite (@trivIset_bigsetUI _ xpredT)// ?fset_set0//.
by rewrite [X in trivIset X F](_ : _ = [set: nat])//; exact/seteqP.
Qed.

Lemma finite_setMR (T T' : choiceType) (A : set T) (B : T -> set T') :
  finite_set A -> (forall x, A x -> finite_set (B x)) -> finite_set (A `*`` B).
Proof.
move=> Afin Bfin; rewrite -bigcupM1l.
by apply: bigcup_finite => // i Ai; apply/finite_setM/Bfin.
Qed.

Lemma finite_setML (T T' : choiceType) (A : T' -> set T) (B : set T') :
  (forall x, B x -> finite_set (A x)) -> finite_set B -> finite_set (A ``*` B).
Proof.
move=> Afin Bfin; rewrite -bigcupM1r.
by apply: bigcup_finite => // i Ai; apply/finite_setM=> //; apply: Afin.
Qed.

Lemma fset_set_II n : fset_set `I_n = [fset val i | i in 'I_n]%fset.
Proof.
apply/fsetP => i; rewrite /= ?inE in_fset_set//.
apply/idP/imfsetP; rewrite ?inE/=.
  by move=> lt_in; exists (Ordinal lt_in).
by move=> [j _ ->].
Qed.

Lemma set_fsetK (T : choiceType) (A : {fset T}) : fset_set [set` A] = A.
Proof.
apply/fsetP => x; rewrite in_fset_set//=.
by apply/idP/idP; rewrite ?inE.
Qed.

Lemma fset_set_image {T U : choiceType} (f : T -> U) (A : set T) :
  finite_set A -> fset_set (f @` A) = (f @` fset_set A)%fset.
Proof.
move=> Afset; apply/fsetP=> i.
rewrite !in_fset_set; last exact: finite_image.
apply/idP/imfsetP; rewrite !inE/=.
  by move=> [x Ax <-]; exists x; rewrite ?in_fset_set ?inE.
by move=> [x + ->]; rewrite in_fset_set// inE; exists x.
Qed.

Lemma fset_set_inj {T : choiceType} (A B : set T) :
  finite_set A -> finite_set B -> fset_set A = fset_set B -> A = B.
Proof. by move=> Afin Bfin /(congr1 pred_set); rewrite !fset_setK. Qed.

Lemma bigsetU_fset_set T (I : choiceType) (A : set I) (F : I -> set T) :
  finite_set A -> \big[setU/set0]_(i <- fset_set A) F i =\bigcup_(i in A) F i.
Proof.
move=> finA; rewrite -bigcup_fset /fset_set; case: pselect => [{}finA|//].
apply/seteqP; split=> [x [i /=]|x [i Ai Fix]].
  by case: cid => /= B -> iB Fix; exists i.
by exists i => //; case: cid => // B AB /=; move: Ai; rewrite AB.
Qed.

#[deprecated(note="Use -bigsetU_fset_set instead")]
Lemma bigcup_fset_set T (I : choiceType) (A : set I) (F : I -> set T) :
  finite_set A -> \bigcup_(i in A) F i = \big[setU/set0]_(i <- fset_set A) F i.
Proof. by move=> /bigsetU_fset_set->. Qed.

Lemma bigsetU_fset_set_cond T (I : choiceType) (A : set I) (F : I -> set T)
    (P : pred I) : finite_set A ->
  \big[setU/set0]_(i <- fset_set A | P i) F i = \bigcup_(i in A `&` P) F i.
Proof.
by move=> *; rewrite bigcup_mkcondr big_mkcond -bigcup_fset_set ?mem_setE.
Qed.

#[deprecated(note="Use -bigsetU_fset_set_cond instead")]
Lemma bigcup_fset_set_cond T (I : choiceType) (A : set I) (F : I -> set T)
    (P : pred I) : finite_set A ->
  \bigcup_(i in A `&` P) F i = \big[setU/set0]_(i <- fset_set A | P i) F i.
Proof. by move=> /bigsetU_fset_set_cond->. Qed.

Lemma bigsetI_fset_set T (I : choiceType) (A : set I) (F : I -> set T) :
  finite_set A -> \big[setI/setT]_(i <- fset_set A) F i =\bigcap_(i in A) F i.
Proof.
by move=> *; apply: setC_inj; rewrite setC_bigcap setC_bigsetI bigsetU_fset_set.
Qed.

#[deprecated(note="Use -bigsetI_fset_set instead")]
Lemma bigcap_fset_set T (I : choiceType) (A : set I) (F : I -> set T) :
  finite_set A -> \bigcap_(i in A) F i = \big[setI/setT]_(i <- fset_set A) F i.
Proof. by move=> /bigsetI_fset_set->. Qed.

Lemma bigsetI_fset_set_cond T (I : choiceType) (A : set I) (F : I -> set T)
    (P : pred I) : finite_set A ->
  \big[setI/setT]_(i <- fset_set A | P i) F i = \bigcap_(i in A `&` P) F i.
Proof.
by move=> *; rewrite bigcap_mkcondr big_mkcond -bigcap_fset_set ?mem_setE.
Qed.

Lemma super_bij T U (X A : set T) (Y B : set U) (f : {bij X >-> Y}) :
  X `<=` A -> Y `<=` B -> A `\` X #= B `\` Y ->
  exists g : {bij A >-> B}, {in X, g =1 f}.
Proof.
elim/Ppointed: U => U in Y B f *.
  rewrite !emptyE in f * => XA _; rewrite setD_eq0 => AX.
  by suff /seteqP->// : A `<=>` X by exists f.
move=> XA YB /pcard_eqP[g].
rewrite -(joinIB X A) -(joinIB Y B) !meetEset.
have /disj_set2P AX : (A `&` X) `&` (A `\` X) = set0 by apply: meetIB.
have /disj_set2P BY : (B `&` Y) `&` (B `\` Y) = set0 by apply: meetIB.
rewrite !(setIidr XA) !(setIidr YB) in AX BY *.
by exists [bij of glue AX BY f g] => x /= xX; rewrite glue1.
Qed.

Lemma card_eq_fsetP {T : choiceType} {A : {fset T}} {n} :
  reflect (#|` A| = n) ([set` A] #= `I_n).
Proof.
elim/choicePpointed: T => T in A *.
  rewrite -{1}[A]set_fsetK !emptyE fset_set0 cardfs0.
  by apply: (iffP eqP) => [/IIn_eq0->//|<-]; rewrite II0.
rewrite (card_eqr card_II) card_eq_sym.
apply: (iffP pcard_eqP) => [[f]|]; last first.
  rewrite cardfE => eqAn.
  by squash (set_val \o finset_val \o enum_val \o cast_ord (esym eqAn)).
suff -> : A = [fset f i | i in 'I_n]%fset by rewrite card_imfset ?size_enum_ord.
apply/fsetP => x; apply/idP/imfsetP => /= [xA|[i _ ->]].
  by have [i _ <-] := 'surj_f xA; exists i.
by have /(_ i I) := 'funS_f.
Qed.

Lemma card_fset_set {T : choiceType} (A : set T) n :
  A #= `I_n -> #|`fset_set A| = n.
Proof.
move=> An; apply/card_eq_fsetP; rewrite fset_setK//.
by apply/finite_setP; exists n.
Qed.

Lemma geq_card_fset_set {T : choiceType} (A : set T) n :
  A #<= `I_n -> (#|`fset_set A| <= n)%N.
Proof.
move=> An; have /finite_setP[m Am] : finite_set A
  by apply/finite_set_leP; exists n.
by rewrite (card_fset_set Am) -card_le_II -(card_le_eql Am).
Qed.

Lemma leq_card_fset_set {T : choiceType} (A : set T) n :
  finite_set A -> A #>= `I_n -> (#|`fset_set A| >= n)%N.
Proof.
move=> /finite_setP[m Am]; rewrite (card_fset_set Am).
by rewrite (card_le_eqr Am) card_le_II.
Qed.

Lemma infinite_set_fset {T : choiceType} (A : set T) (n : nat) :
  infinite_set A ->
    exists2 B : {fset T}, [set` B] `<=` A & (#|` B| >= n)%N.
Proof.
elim/choicePpointed: T => T in A *; first by rewrite emptyE.
move=> /infiniteP/ppcard_leP[f]; exists (fset_set [set f i | i in `I_n]).
  rewrite fset_setK//; last exact: finite_image.
  by apply: subset_trans (fun_image_sub f); apply: image_subset.
rewrite fset_set_image// card_imfset//= fset_set_II/=.
by rewrite card_imfset//= ?size_enum_ord//; apply: val_inj.
Qed.

Lemma infinite_set_fsetP {T : choiceType} (A : set T) :
  infinite_set A <->
   forall n, exists2 B : {fset T}, [set` B] `<=` A & (#|` B| >= n)%N.
Proof.
split; first by move=> ? ?; apply: infinite_set_fset.
elim/choicePpointed: T => T in A *.
  move=> /(_ 1%N)[B _]; rewrite cardfs_gt0 => /fset0Pn[x xB].
  by have: [set` B] x by []; rewrite emptyE.
move=> Bge /finite_setP[n An]; have [B BA] := Bge n.+1.
apply/negP; rewrite -leqNgt -(card_fset_set An) fsubset_leq_card//.
apply/fsubsetP => x /BA; rewrite in_fset_set ?inE//.
by apply/finite_setP; exists n.
Qed.

Lemma fcard_eq {T T' : choiceType} (A : set T) (B : set T') :
    finite_set A -> finite_set B ->
  reflect (#|`fset_set A| = #|`fset_set B|) (A #= B).
Proof.
move=> /finite_setP/cid[n An] /finite_setP/cid[m Bm].
rewrite (card_fset_set An) (card_fset_set Bm).
by rewrite (card_eql An) (card_eqr Bm); apply: card_eq_II.
Qed.

Lemma card_IID {n k} : `I_n `\` `I_k #= `I_(n - k)%N.
Proof.
apply/fcard_eq => //; first exact: finite_setD.
rewrite fset_setD//= cardfsD/= -fset_setI// setI_II.
rewrite !fset_set_II !card_imfset// /= !size_enum_ord.
by case: leqP; rewrite // subnn => /eqP->.
Qed.

Lemma finite_set_bij T (A : set T) n S : A != set0 ->
    A #= `I_n -> S `<=` A ->
  exists (f : {bij `I_n >-> A}) k, (k <= n)%N /\ `I_n `&` (f @^-1` S) = `I_k.
Proof.
elim/Ppointed: T => T in A S *; first by rewrite !emptyE eqxx.
move=> AN0 An SA; have [k kn Sk] : exists2 k, (k <= n)%N & S #= `I_k.
  have /finite_setP[k Sk]: finite_set S by apply: sub_finite_set SA _; exists n.
  exists k => //; rewrite -card_le_II.
  by rewrite -(card_le_eqr An) -(card_le_eql Sk); apply: subset_card_le.
have /card_esym/ppcard_eqP[f] := Sk.
have eqAS : A `\` S #= `I_n `\` `I_k.
  have An' := An; have Sk' := Sk.
  do [have /finite_fsetP[{An'}A ->] : finite_set A by exists n] in An AN0 SA *.
  do [have /finite_fsetP[{Sk'}S ->] : finite_set S by exists k] in Sk f SA *.
  have [/card_eq_fsetP {}An /card_eq_fsetP {}Sk] := (An, Sk).
  rewrite -set_fsetD (card_eqr card_IID); apply/card_eq_fsetP.
  by rewrite cardfsD (fsetIidPr _) ?An ?Sk //; apply/fsubsetP.
case: (super_bij [bij of f^-1] SA _ eqAS) => [x /= /leq_trans->// | g].
have [{}g ->] := pPbij 'bij_g => /= gE.
exists [bij of g^-1], k; split=> //=; rewrite -inv_sub_image //= invV.
by under eq_imagel do rewrite /= gE ?inE//; rewrite image_eq.
Qed.

#[deprecated(note="use countable0 instead")]
Notation countable_set0 := countable0.

Lemma countable1 T (x : T) : countable [set x].
Proof. exact: finite_set_countable. Qed.
#[global] Hint Resolve countable1 : core.

Lemma countable_fset (T : choiceType) (X : {fset T}) : countable [set` X].
Proof. exact: finite_set_countable. Qed.
#[global] Hint Resolve countable_fset : core.

Lemma countable_finpred (T : finType) (pT : predType T) (P : pT) : countable [set` P].
Proof. exact: finite_set_countable. Qed.
#[global] Hint Extern 0 (is_true (countable [set` _])) => solve [apply: countable_finpred] : core.

Lemma eq_card_nat T (A : set T):
  countable A -> ~ finite_set A -> A #= [set: nat].
Proof. by move=> Acnt /infiniteP leNA; rewrite card_eq_le leNA andbT. Qed.

Lemma infinite_nat : ~ finite_set [set: nat].
Proof. exact/infiniteP/card_lexx. Qed.

Lemma infinite_prod_nat : infinite_set [set: nat * nat].
Proof.
by apply/infiniteP/pcard_leTP/injPex; exists (pair 0%N) => // m n _ _ [].
Qed.

Lemma card_nat2 : [set: nat * nat] #= [set: nat].
Proof. exact/eq_card_nat/infinite_prod_nat/countableP. Qed.

Canonical rat_pointedType := PointedType rat 0.

Lemma infinite_rat : infinite_set [set: rat].
Proof.
apply/infiniteP/pcard_leTP/injPex; exists (GRing.natmul 1) => // m n _ _.
exact/Num.Theory.mulrIn/oner_neq0.
Qed.

Lemma card_rat : [set: rat] #= [set: nat].
Proof. exact/eq_card_nat/infinite_rat/countableP. Qed.

Lemma choicePcountable {T : choiceType} : countable [set: T] ->
  {T' : countType | T = T' :> Type}.
Proof.
move=> /pcard_leP/unsquash f.
by exists (CountType T (CountMixin (in1TT 'funoK_f))).
Qed.

Lemma eqPcountable {T : eqType} : countable [set: T] ->
  {T' : countType | T = T' :> Type}.
Proof. by elim/eqPchoice: T => T /choicePcountable. Qed.

Lemma Pcountable {T : Type} : countable [set: T] ->
  {T' : countType | T = T' :> Type}.
Proof. by elim/Pchoice: T => T /choicePcountable. Qed.

Lemma bigcup_countable {I T} (D : set I) (F : I -> set T) :
    countable D -> (forall i, D i -> countable (F i)) ->
  countable (\bigcup_(i in D) F i).
Proof.
elim/Ppointed: T => T in F *; first by rewrite emptyE.
rewrite -(eq_countable (card_setT _)) => cD cF; rewrite bigcup_set_type.
set G := (fun i : D => F (val i)).
have {cF}cG i : countable (G i) by apply: cF; apply: set_valP.
move: (D : Type) cD G cG => {F I}_ /Pcountable[{}D ->] G cG.
suff: (\bigcup_i G i #<= [set: {i & G i}])%card.
  have cGT i : countable [set: G i] by rewrite (eq_countable (card_setT _)).
  have /all_sig[H GE] := fun i => Pcountable (cGT i).
  by move=> /sub_countable->//; rewrite (eq_fun GE).
apply/pcard_geP/surjPex; exists (fun (k : {i & G i}) => val (projT2 k)).
by move=> x [i _] Gix/=; exists (Tagged G (SigSub (mem_set Gix))).
Qed.

Lemma countableMR T T' (A : set T) (B : T -> set T') :
  countable A -> (forall i, A i -> countable (B i)) -> countable (A `*`` B).
Proof.
elim/Ppointed: T => T in A B *; first by rewrite emptyE -bigcupM1l bigcup_set0.
elim/Ppointed: T' => T' in B *.
  by rewrite -bigcupM1l bigcup0// => i; rewrite emptyE setM0.
move=> Ac Bc; rewrite -bigcupM1l bigcup_countable// => i Ai.
have /ppcard_leP[f] := Bc i Ai; apply/pcard_geP/surjPex.
exists (fun k => (i, f^-1%FUN k)) => -[_ j]/= [-> dj].
by exists (f j) => //=; rewrite funK ?inE.
Qed.

Lemma countableM T1 T2 (D1 : set T1) (D2 : set T2) :
  countable D1 -> countable D2 -> countable (D1 `*` D2).
Proof. by move=> D1c D2c; apply: countableMR (fun _ _ => D2c). Qed.

Lemma countableML T T' (A : T' -> set T) (B : set T') :
  countable B -> (forall i, B i -> countable (A i)) -> countable (A ``*` B).
Proof.
move=> Bc Ac; rewrite -bigcupM1r; apply: bigcup_countable => // i Bi.
by apply: countableM => //; apply: Ac.
Qed.

Lemma infiniteMRl T T' (A : set T) (B : T -> set T') :
  infinite_set A -> (forall i, B i !=set0) -> infinite_set (A `*`` B).
Proof.
move=> /infiniteP/pcard_geP[f] /(_ _)/cid-/all_sig[b Bb].
apply/infiniteP/pcard_geP/surjPex; exists (fun x => f x.1).
by move=> i iT; have [a Aa fa] := 'oinvP_f iT; exists (a, b a) => /=.
Qed.

Lemma cardMR_eq_nat T T' (A : set T) (B : T -> set T') :
    (A #= [set: nat] -> (forall i, countable (B i) /\ B i !=set0) ->
   A `*`` B #= [set: nat])%card.
Proof.
rewrite !card_eq_le => /andP[Acnt /infiniteP Ainfty] /all_and2[Bcnt Bn0].
by rewrite [(_ #<= _)%card]countableMR//=; apply/infiniteP/infiniteMRl.
Qed.

HB.mixin Record FiniteImage aT rT (f : aT -> rT) := {
  fimfunP : finite_set (range f)
}.
HB.structure Definition FImFun aT rT := {f of @FiniteImage aT rT f}.

Reserved Notation "{ 'fimfun' aT >-> T }"
  (at level 0, format "{ 'fimfun'  aT  >->  T }").
Reserved Notation "[ 'fimfun' 'of' f ]"
  (at level 0, format "[ 'fimfun'  'of'  f ]").
Notation "{ 'fimfun' aT >-> T }" := (@FImFun.type aT T) : form_scope.
Notation "[ 'fimfun' 'of' f ]" := [the {fimfun _ >-> _} of f] : form_scope.
#[global] Hint Resolve fimfunP : core.

Lemma fimfun_inP {aT rT} (f : {fimfun aT >-> rT}) (D : set aT) :
  finite_set (f @` D).
Proof. by apply: (@sub_finite_set _ _ (range f)) => // y [x]; exists x. Qed.

#[global] Hint Resolve fimfun_inP : core.

Section fimfun_pred.
Context {aT rT : Type}.
Definition fimfun : {pred aT -> rT} := mem [set f | finite_set (range f)].
Definition fimfun_key : pred_key fimfun.
Proof. exact. Qed.
Canonical fimfun_keyed := KeyedPred fimfun_key.
End fimfun_pred.

Section fimfun.
Context {aT rT : Type}.
Notation T := {fimfun aT >-> rT}.
Notation fimfun := (@fimfun aT rT).
Section Sub.
Context (f : aT -> rT) (fP : f \in fimfun).
Definition fimfun_Sub_subproof := @FiniteImage.Build aT rT f (set_mem fP).
#[local] HB.instance Definition _ := fimfun_Sub_subproof.
Definition fimfun_Sub := [fimfun of f].
End Sub.

Lemma fimfun_rect (K : T -> Type) :
  (forall f (Pf : f \in fimfun), K (fimfun_Sub Pf)) -> forall u : T, K u.
Proof.
move=> Ksub [f [[Pf]]]/=.
by suff -> : Pf = (set_mem (@mem_set _ [set f | _] f Pf)) by apply: Ksub.
Qed.

Lemma fimfun_valP f (Pf : f \in fimfun) : fimfun_Sub Pf = f :> (_ -> _).
Proof. by []. Qed.

Canonical fimfun_subType := SubType T _ _ fimfun_rect fimfun_valP.
End fimfun.

Lemma fimfuneqP aT rT (f g : {fimfun aT >-> rT}) :
  f = g <-> f =1 g.
Proof. by split=> [->//|fg]; apply/val_inj/funext. Qed.

Definition fimfuneqMixin aT (rT : eqType) :=
  [eqMixin of {fimfun aT >-> rT} by <:].
Canonical fimfuneqType aT (rT : eqType) :=
  EqType {fimfun aT >-> rT} (fimfuneqMixin aT rT).
Definition fimfunchoiceMixin aT (rT : choiceType) :=
  [choiceMixin of {fimfun aT >-> rT} by <:].
Canonical fimfunchoiceType aT (rT : choiceType) :=
  ChoiceType {fimfun aT >-> rT} (fimfunchoiceMixin aT rT).

Lemma finite_image_cst {aT rT : Type} (x : rT) :
  finite_set (range (cst x : aT -> _)).
Proof.
elim/Ppointed: aT => aT; rewrite ?emptyE ?image_set0//.
suff -> : cst x @` [set: aT] = [set x] by apply: finite_set1.
by apply/predeqP => y; split=> [[t' _ <-]//|->//] /=; exists point.
Qed.

Lemma cst_fimfun_subproof aT rT x : @FiniteImage aT rT (cst x).
Proof. by split; exact: finite_image_cst. Qed.
HB.instance Definition _ aT rT x := @cst_fimfun_subproof aT rT x.
Definition cst_fimfun {aT rT} x := [the {fimfun aT >-> rT} of cst x].

Lemma fimfun_cst aT rT x : @cst_fimfun aT rT x =1 cst x. Proof. by []. Qed.

Lemma comp_fimfun_subproof aT rT sT
   (f : {fimfun aT >-> rT}) (g : rT -> sT) : @FiniteImage aT sT (g \o f).
Proof. by split; rewrite -(image_comp f g); apply: finite_image. Qed.
HB.instance Definition _ aT rT sT f g := @comp_fimfun_subproof aT rT sT f g.

Section zmod.
Context (aT : Type) (rT : zmodType).
Lemma fimfun_zmod_closed : zmod_closed (@fimfun aT rT).
Proof.
split=> [|f g]; rewrite !inE/=; first exact: finite_image_cst.
by move=> fA gA; apply: (finite_image11 (fun x y => x - y)).
Qed.
Canonical fimfun_add := AddrPred fimfun_zmod_closed.
Canonical fimfun_zmod := ZmodPred fimfun_zmod_closed.
Definition fimfun_zmodMixin := [zmodMixin of {fimfun aT >-> rT} by <:].
Canonical fimfun_zmodType := ZmodType {fimfun aT >-> rT} fimfun_zmodMixin.

Implicit Types (f g : {fimfun aT >-> rT}).

Lemma fimfunD f g : f + g = f \+ g :> (_ -> _). Proof. by []. Qed.
Lemma fimfunN f : - f = \- f :> (_ -> _). Proof. by []. Qed.
Lemma fimfunB f g : f - g = f \- g :> (_ -> _). Proof. by []. Qed.
Lemma fimfun0 : (0 : {fimfun aT >-> rT}) = cst 0 :> (_ -> _). Proof. by []. Qed.
Lemma fimfun_sum I r (P : {pred I}) (f : I -> {fimfun aT >-> rT}) (x : aT) :
  (\sum_(i <- r | P i) f i) x = \sum_(i <- r | P i) f i x.
Proof. by elim/big_rec2: _ => //= i y ? Pi <-. Qed.

HB.instance Definition _ f g := FImFun.copy (f \+ g) (f + g).
HB.instance Definition _ f g := FImFun.copy (\- f) (- f).
HB.instance Definition _ f g := FImFun.copy (f \- g) (f - g).
End zmod.