# Theory of Sets EII-6 Equivalence relations

Copyright INRIA (2009-2013) Apics; Marelle Team (Jose Grimm). Part of this code comes from Carlos Simpson

Require Import ssreflect ssrfun ssrbool eqtype ssrnat.
Require Export sset3.

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

## EII-6-1 Definition of an equivalence relation

Module Relation.

the substrate of a relation are elements that are relate

Definition substrate r := (domain r) \cup (range r).

Lemma pr1_sr r y:
inc y r -> inc (P y) (substrate r).
Proof. move => yr; apply: setU2_1; apply/funI_P; ex_tac. Qed.

Lemma pr2_sr r y:
inc y r -> inc (Q y) (substrate r).
Proof. move => yr; apply: setU2_2; apply/funI_P; ex_tac. Qed.

Lemma arg1_sr r x y:
related r x y -> inc x (substrate r).
Proof. by move=> rel; rewrite -(pr1_pair x y); apply: pr1_sr. Qed.

Lemma arg2_sr r x y:
related r x y -> inc y (substrate r).
Proof. by move=> rel;rewrite -(pr2_pair x y); apply: pr2_sr. Qed.

Ltac substr_tac :=
match goal with
| H:inc ?x ?r |- inc (P ?x) (substrate ?r)
=> apply: (pr1_sr H)
| H:inc ?x ?r |- inc (Q ?x) (substrate ?r)
=> apply: (pr2_sr H)
| H:related ?r ?x _ |- inc ?x (substrate ?r)
=> apply: (arg1_sr H)
| H:related ?r _ ?y |- inc ?y (substrate ?r)
=> apply: (arg2_sr H)
| H:inc(J ?x _ ) ?r|- inc ?x (substrate ?r)
=> apply: (arg1_sr H)
| H: inc (J _ ?y) ?r |- inc ?y (substrate ?r)
=> apply: (arg2_sr H)
end.

Lemma substrate_smallest r s:
(forall y, inc y r -> inc (P y) s) ->
(forall y, inc y r -> inc (Q y) s) ->
sub (substrate r) s.
Proof. move=> h1 h2 x; case /setU2_P; move/funI_P => [z zr ->]; fprops. Qed.

Lemma substrate_P r: sgraph r -> forall x,
inc x (substrate r) <->
((exists y, inc (J x y) r) \/ (exists y, inc (J y x) r)).
Proof.
move => gr x; split; last by case => [] [y yr]; substr_tac.
by case /setU2_P => /funI_P [z zr ->]; [left | right]; ex_tac; rewrite gr.
Qed.

Lemma substr_i r x: inc (J x x) r -> inc x (substrate r).
Proof. move => h; substr_tac. Qed.

Reflexivity etc. properties for a relation

Section Definitions.
Implicit Type (r: relation).

Definition reflexive_re r E :=
forall x, inc x E <-> r x x.

Definition reflexive_rr r :=
forall x y, r x y -> (r x x /\ r y y).

Definition equivalence_r r :=
symmetric_r r /\ transitive_r r.

Definition equivalence_re r E :=
equivalence_r r /\ reflexive_re r E.

Definition order_r r :=
[/\ transitive_r r, antisymmetric_r r & reflexive_rr r].

Definition preorder_r r :=
transitive_r r /\ reflexive_rr r.

Definition order_re (r: relation) x :=
order_r r /\ reflexive_re r x.

End Definitions.

Same definitions for a graph

Definition reflexivep r := forall y, inc y (substrate r) -> related r y y.
Definition symmetricp r := symmetric_r (related r).
Definition antisymmetricp r := antisymmetric_r (related r).
Definition transitivep r := transitive_r (related r).

Definition equivalence r :=
[/\ sgraph r, reflexivep r, transitivep r & symmetricp r].

Definition order r :=
[/\ sgraph r, reflexivep r, transitivep r & antisymmetricp r].

Definition preorder r :=
[/\ sgraph r, reflexivep r & transitivep r].

Definition order_on r E := order r /\ substrate r = E.

Basic properties

Lemma equivalence_sgraph r: equivalence r -> sgraph r.
Proof. by move=> []. Qed.

Lemma order_sgraph r: order r -> sgraph r.
Proof. by move => []. Qed.

Lemma preorder_sgraph r: preorder r -> sgraph r.
Proof. by move => []. Qed.

Hint Resolve order_sgraph equivalence_sgraph : fprops.

Lemma reflexive_domain g: reflexivep g -> domain g = substrate g.
Proof.
move=> pb; set_extens u; first by apply: setU2_1.
case /setU2_P => // ug; apply/funI_P; exists (J u u);aw.
exact:(pb _ (setU2_2 (domain g) ug)).
Qed.

Lemma domain_sr g: equivalence g -> domain g = substrate g.
Proof. move=> [_ pb _ _]; apply: (reflexive_domain pb). Qed.

Lemma domain_sr1 r: order r -> domain r = substrate r.
Proof. move=> [_ pb _ _]; apply: (reflexive_domain pb). Qed.

Lemma substrate_sub: {compat substrate : x y / sub x y}.
Proof.
move => r s rs; move: (domain_S rs) (range_S rs) => pa pb t.
case /setU2_P => h; apply/setU2_P; fprops.
Qed.

Lemma symmetric_transitive_equivalence r:
sgraph r -> symmetricp r -> transitivep r -> equivalence r.
Proof.
move=> gr sr tr; red; split => //.
move => y /(substrate_P gr) [][z pr].
apply: (tr _ _ _ pr (sr _ _ pr)).
apply: (tr _ _ _ (sr _ _ pr) pr).
Qed.

Lemma equivalence_relation_pr1 g:
sgraph g -> equivalence_r (related g) -> equivalence g.
Proof.
move=> gg [sr tr]; apply: symmetric_transitive_equivalence => //.
Qed.

Lemma reflexivity_e r u:
equivalence r -> inc u (substrate r) -> related r u u.
Proof. by move=> [_ rr _ _] us; apply:rr. Qed.

Lemma symmetricity_e r u v:
equivalence r -> related r u v -> related r v u.
Proof. by move=> [_ _ _]; apply. Qed.

Lemma transitivity_e r v u w:
equivalence r -> related r u v -> related r v w -> related r u w.
Proof.
move=> [_ _ tr _] r1 r2; apply: (tr _ _ _ r1 r2).
Qed.

Ltac equiv_tac:=
match goal with
| H: equivalence ?r, H1: inc ?u (substrate ?r) |- related ?r ?u ?u
=> apply: (reflexivity_e H H1)
| H: equivalence ?r |- inc (J ?u ?u) ?r
=> apply: reflexivity_e
| H:equivalence ?r, H1:related ?r ?u ?v |- related ?r ?v ?u
=> apply: (symmetricity_e H H1)
| H:equivalence ?r, H1: inc (J ?u ?v) ?r |- inc (J ?v ?u) ?r
=> apply: (symmetricity_e H H1)
| H:equivalence ?r, H1:related ?r ?u ?v, H2: related ?r ?v ?w
|- related ?r ?u ?w
=> apply: (transitivity_e H H1 H2)
| H:equivalence ?r, H1:related ?r ?v ?u, H2: related ?r ?v ?w
|- related ?r ?u ?w
=> apply: (transitivity_e H (symmetricity_e H H1) H2)
| H:equivalence ?r, H1:related ?r ?u ?v, H2: related ?r ?w ?v
|- related ?r ?u ?w
=> apply: (transitivity_e H H1 (symmetricity_e H H2))
| H: equivalence ?r, H1: inc (J ?u ?v) ?r, H2: inc (J ?v ?w) ?r |-
inc (J ?u ?w) ?r
=> apply: (transitivity_e H H1 H2)
end.

Comparison of the two sets of definitions

Lemma equivalence_equivalence r:
equivalence r -> equivalence_re (related r)(substrate r).
Proof.
move => [gr ra rb rc]; split => //.
by move=> y; split => p; [ apply: ra | substr_tac ].
Qed.

We say that g is the graph of r if related g is the same function as r. We define the graph of r on x as the set of pairs of x related by r

Definition graph_on (r:relation) x:=
Zo(coarse x)(fun w => r (P w)(Q w)).

Lemma graph_on_graph r x: sgraph (graph_on r x).
Proof. by move => y /Zo_S /setX_P [ok _]. Qed.

Lemma graph_on_P0 r x a b:
inc (J a b) (graph_on r x) <-> [/\ inc a x, inc b x & r a b].
Proof.
split; first by move/Zo_P => [] /setX_P; aw ; move => [pa pb pc] pd;split.
move => [pa pb pc]; apply: Zo_i; [by apply: setXp_i | aw].
Qed.

Lemma graph_on_P1 r x a b:
related (graph_on r x) a b <-> [/\ inc a x, inc b x & r a b].
Proof. apply/graph_on_P0. Qed.

Lemma graph_on_P2 r x : equivalence_re r x -> forall u v,
(related (graph_on r x) u v <-> r u v).
Proof.
move=> [[rs rt] rr] u v; split; first by move/Zo_hi; aw.
move => ruv; apply/ graph_on_P1.
move: (rs _ _ ruv) => rvu; move: (rt _ _ _ ruv rvu) => ruu.
by move: (rt _ _ _ rvu ruv) => rvv; split => //; apply rr.
Qed.

Lemma graph_on_P3 r x: order_re r x -> forall u v,
(related (graph_on r x) u v <-> r u v).
Proof.
move=> [[_ _ cc] rr]; split; first by move /Zo_hi; aw.
move => ruv; move: (cc _ _ ruv) => [ruu rvv].
apply: Zo_i ; [by apply:setXp_i; apply /rr | aw].
Qed.

Lemma graph_on_sr1 r x: sub (substrate (graph_on r x)) x.
Proof.
by move=> y /(substrate_P (@graph_on_graph r x))[] [z] /graph_on_P0 [pa pb _].
Qed.

Lemma order_preorder r: order r -> preorder r.
Proof. by move => [gr tr ar rr]. Qed.

Lemma preorder_from_rel r x:
preorder_r r -> preorder (graph_on r x).
Proof.
move=> pr.
have gg: sgraph (graph_on r x) by apply: graph_on_graph.
move: pr=> [tr rr].
split=>//.
by move=> y /(substrate_P gg) [] [z] /graph_on_P0 [yx zx ryz];
apply /graph_on_P0; split => // ;move: (rr _ _ ryz) => [].
move=> a b c /graph_on_P1 [ax bx rab] /graph_on_P1 [_ cx rbc].
apply /graph_on_P1;split => //; apply: tr rab rbc.
Qed.

Lemma order_from_rel r x:
order_r r -> order (graph_on r x).
Proof.
move=> [ta tb tc].
have [tr pa pb]: preorder (graph_on r x) by apply: preorder_from_rel; split.
split => //.
by move=> a b /graph_on_P1 [ax bx rab] /graph_on_P1 [_ _ rba]; apply: tb.
Qed.

Lemma equivalence_from_rel r x:
equivalence_r r -> equivalence (graph_on r x).
Proof.
move=> [pa pb].
have gg: sgraph (graph_on r x) by apply: graph_on_graph.
apply: (equivalence_relation_pr1 gg).
split.
move => a b /graph_on_P1 [ax bx rab]; apply /graph_on_P1.
by split => //; apply: pa.
move=> a b c /graph_on_P1 [ax bx rab] /graph_on_P1 [_ cx rbc].
apply /graph_on_P1;split => //; apply: pb rab rbc.
Qed.

Lemma graph_on_sr (r: relation) x:
(forall a, inc a x -> r a a) ->
substrate (graph_on r x) = x.
Proof.
move=> rr.
apply: extensionality; first by apply: graph_on_sr1.
move => t xt; move: (rr _ xt) => rtt.
apply: substr_i; apply: Zo_i; [by apply:setXp_i | aw].
Qed.

Hint Rewrite graph_on_sr: bw.

Finest relation on a set: an element is only related to itself The correspondence is the identity, the graph is the diagonal

Definition restricted_eq x := fun u v => inc u x /\ u = v.

Lemma diagonal_graph_on x: graph_on (restricted_eq x) x = diagonal x.
Proof.
set_extens t.
move /Zo_P => [pa [pb pc]]; apply /diagonal_i_P;split=> //.
by move /setX_P: pa =>[].
move /diagonal_i_P=> [pa pb pc];apply/Zo_i => //; apply /setX_P; split => //;ue.
Qed.

Lemma diagonal_equivalence x: equivalence (diagonal x).
Proof.
rewrite - diagonal_graph_on; apply: equivalence_from_rel;split.
by move=> a b /= [ax] <-.
by move=> a b c [au ab] [bu bc]; split =>//; ue.
Qed.

Lemma diagonal_osr x: order_on (diagonal x) x.
Proof.
rewrite -diagonal_graph_on; split; last by bw => t tx.
apply: order_from_rel;split.
- by move=> a b c [au ab] [bu bc]; split =>//; ue.
- by move=> a b /= [ax <-].
- by move=> a b [aa <-].
Qed.

Example of an equivalence without graph: equipotency

Lemma equipotent_equivalence: equivalence_r equipotent.
Proof.
split; first by move=> x y;apply: equipotentS.
move => x y z; apply: equipotentT.
Qed.

The coarsest relation on a set: everything is related

Lemma coarse_sr u: substrate (coarse u) = u.
Proof.
rewrite /coarse/substrate.
transitivity (u \cup u); last by apply: setU2_id.
case: (emptyset_dichot u); [ move ->; rewrite setX_0l; bw | move=> h; bw ].
Qed.

Lemma coarse_related u x y:
related (coarse u) x y <-> (inc x u /\ inc y u).
Proof.
by split; [ move/setX_P => [_]; aw | move => [pa pb];apply : setXp_i ].
Qed.

Lemma coarse_graph x: sgraph (coarse x).
Proof. apply: setX_graph. Qed.

Lemma coarse_equivalence u: equivalence (coarse u).
Proof.
split.
- by apply: coarse_graph.
- by move => y; rewrite coarse_sr => yu;apply/ coarse_related.
move=> x y z /coarse_related [pa pb]/coarse_related [pc pd].
- by apply/coarse_related.
- by move=> x y /coarse_related [pc pd]; apply/coarse_related.
Qed.

Lemma sub_graph_coarse_substrate r:
sgraph r -> sub r (coarse (substrate r)).
Proof.
rewrite /coarse=> gr t tr; apply: setX_i; first (by apply: gr); substr_tac.
Qed.

Example 5 page E II.114

Lemma equivalence_relation_bourbaki_ex5 A E
(r := (fun x y => (inc x (E -s A) /\ (x = y) \/ (inc x A /\ inc y A)))):
sub A E ->
(equivalence (graph_on r E) /\ substrate (graph_on r E) = E).
Proof.
move => sa; split; last first.
apply graph_on_sr; move => a ae.
by case: (inc_or_not a A) => h; [right | left]; split => //; apply /setC_P.
apply: equivalence_from_rel; split.
move=> x y; case=> [] [pa pb]; [left | right ]; split => //; ue.
move => y x z; case => [] [pa pb]; first by rewrite pb.
move => h;right; case:h => [] [qa qb]; split => //; ue.
Qed.

Intersection of equivalence relations is an equivalence

Lemma setIrel_graph z:
(alls z sgraph) -> sgraph (intersection z).
Proof.
case: (emptyset_dichot z).
move => -> _; rewrite setI_0; apply: sgraph_set0.
move => [t tz] alg y yi; apply: (alg _ tz); exact: (setI_hi yi tz).
Qed.

Lemma setIrel_P z: nonempty z -> forall x y,
(related (intersection z) x y <->
(forall r, inc r z -> related r x y)).
Proof. move=> nez x y; apply/(setI_P nez). Qed.

Lemma setIrelR z:
(alls z reflexivep) -> reflexivep (intersection z).
Proof.
move => alr.
case: (emptyset_dichot z).
move => -> x; rewrite setI_0 /substrate domain_set0 range_set0.
by case /setU2_P => /in_set0.
move => nez y ys; apply/(setIrel_P nez) => r rz; apply (alr _ rz).
apply: (substrate_sub (setI_s1 rz) ys).
Qed.

Lemma setIrel_sr z e:
nonempty z -> (alls z reflexivep) ->
(forall r, inc r z -> substrate r = e) ->
substrate (intersection z) = e.
Proof.
move=> nez allr alls.
move: (setIrelR allr)=> ir.
set_extens t => ts.
move: (ir _ ts); rewrite /related=> aux.
move: nez => [u uz]; move: (setI_hi aux uz) => Ju.
by rewrite - (alls _ uz); apply: (arg1_sr Ju).
have rtt: related (intersection z) t t.
apply/(setIrel_P nez) => r rz; rewrite - (alls _ rz) in ts.
by apply: (allr _ rz).
substr_tac.
Qed.

Lemma setIrelT z:
(alls z transitivep) -> transitivep (intersection z).
Proof.
move=> allt; case: (emptyset_dichot z).
move => ->; rewrite setI_0 => a b c;case; case.
move=> neX y x u /(setIrel_P neX) rxy /(setIrel_P neX) ryz.
apply/(setIrel_P neX) => r rz.
exact: ((allt _ rz) _ _ _ (rxy _ rz) (ryz _ rz)).
Qed.

Lemma setIrelS z:
(alls z symmetricp) -> symmetricp (intersection z).
Proof.
move=> alls; case: (emptyset_dichot z).
move => ->; rewrite setI_0 => a b; case; case.
move=>nez x y /(setIrel_P nez) rxy; apply/(setIrel_P nez) => r rz.
apply: ((alls _ rz) _ _ (rxy _ rz)).
Qed.

Lemma setIrel_equivalence z:
(alls z equivalence) -> equivalence (intersection z).
Proof.
move=> alleq; apply: symmetric_transitive_equivalence.
- by apply: setIrel_graph; move => r /alleq [].
- by apply: setIrelS=>//; move => r /alleq [_ _].
- by apply: setIrelT=>//; move => r /alleq [_ _ ].
Qed.

Lemma setIrel_or z: (alls z order) -> order (intersection z).
Proof.
move=> alleq;split.
- by apply: setIrel_graph; move => r /alleq [].
- by apply: setIrelR; move => r /alleq [_ ].
- by apply: setIrelT=>//; move => r /alleq [_ _ ].
- case: (emptyset_dichot z).
move => ->; rewrite setI_0 => a b; case; case.
move => [t te] x y pa pb; move: (alleq _ te)=> [_ _ _]; apply.
apply:(setI_hi pa te).
apply:(setI_hi pb te).
Qed.

The set of all relations or all equivalences of a set

Definition equivalences x :=
Zo (powerset (coarse x)) (fun r => (equivalence r)
/\ (substrate r = x)).

Lemma equivalencesP r x:
inc r (equivalences x) <-> (equivalence r /\ (substrate r = x)).
Proof.
split; first by move /Zo_hi.
move => pa; apply:Zo_i => //; apply/setP_P.
move: pa => [[xx _ _ _] <-].
by move: (sub_graph_coarse_substrate xx).
Qed.

Lemma inc_coarse_all_equivalence_relations u:
inc (coarse u) (equivalences u).
Proof.
by apply/equivalencesP; split; [apply: coarse_equivalence | rewrite coarse_sr].
Qed.

We show that an equivalence is a self-inverse projector

Lemma selfinverse_graph_symmetric r: sgraph r ->
(symmetricp r <-> (r = inverse_graph r)).
Proof.
move => gr; split => sp.
set_extens t => tr.
by move: (gr _ tr) => aux;
rewrite - aux; apply/igraph_pP; apply: sp; red; rewrite aux.
by move /igraphP: tr => [pt pa]; rewrite -pt; apply: sp.
by move => x y Jr; rewrite sp; apply/igraph_pP.
Qed.

Lemma idempotent_graph_transitive r:
sgraph r -> (transitivep r <-> sub (r \cg r) r).
Proof.
move=> gr; split.
move=> tr t /compg_P [pt [y J1r J2r]].
rewrite - pt; move: tr J1r J2r; apply.
by move => h x y z Jxy Jyz; apply: h; apply/compg_pP; exists x.
Qed.

Theorem equivalence_pr r:
equivalence r <-> ((r \cg r) = r /\ r = inverse_graph r).
Proof.
split => hyp.
move: (hyp) => [pa pb pc pd].
split; last by apply /selfinverse_graph_symmetric.
apply: extensionality.
by apply /idempotent_graph_transitive.
move=> x xr; move: (pa _ xr) => px.
rewrite - px; apply /compg_pP; exists (P x); rewrite ? px //.
equiv_tac =>//; substr_tac.
move: hyp=> [crr ri].
have gr: (sgraph r) by rewrite ri; fprops.
apply: symmetric_transitive_equivalence => //.
by apply /selfinverse_graph_symmetric.
apply/idempotent_graph_transitive => //; rewrite crr; fprops.
Qed.

## EII-6-2 Equivalence classes; quotient set

Equivalence associated to a function f by f x = f y

Definition eq_rel_associated f x y :=
[/\ inc x (source f), (inc y (source f)) & (Vf f x = Vf f y)].

Definition equivalence_associated f :=
(inverse_graph (graph f)) \cg (graph f).

Section EquivalenceAssociated.
Variable (f: Set).
Hypothesis (ff : function f).

Lemma ea_graph_on:
graph_on (eq_rel_associated f) (source f) = equivalence_associated f.
Proof.
set_extens t.
move /Zo_P => [pa [pb pc pd]]; move /setX_P:pa => [pe _ _].
rewrite - pe; apply /compg_pP; exists (Vf f (P t)) => //; first Wtac.
apply/igraph_pP; rewrite pd; Wtac.
move/compg_P => [pt [z pa /igraph_pP pb]].
have pc: inc (P t) (source f) by Wtac.
have pd: inc (Q t) (source f) by Wtac.
apply: Zo_i; first by apply /setX_P.
by split => // ; rewrite - (Vf_pr ff pa) - (Vf_pr ff pb).
Qed.

Lemma ea_relatedP x y:
related (equivalence_associated f) x y <->
[/\ inc x (source f), inc y (source f) & Vf f x = Vf f y].
Proof.
rewrite - ea_graph_on; split; first by move/graph_on_P1 => [_ _].
by move =>[pa pb pc]; red; apply /graph_on_P1.
Qed.

Lemma graph_ea_equivalence:
equivalence (equivalence_associated f).
Proof.
rewrite - ea_graph_on; apply: equivalence_from_rel; split.
by move => a b [pa pb pc].
move => a b c [pa pb pc] [pd pe pf]; split => //; ue.
Qed.

Lemma graph_ea_substrate:
substrate (equivalence_associated f) = source f.
Proof.
rewrite - ea_graph_on graph_on_sr //.
Qed.

End EquivalenceAssociated.

The class of x is the set of elements related to x

Definition class r x := fun_image (Zo r (fun z => P z = x)) Q.
Definition quotient r := fun_image (substrate r) (class r).
Definition classp r x := inc (rep x) (substrate r) /\ x = class r (rep x).

Section Class.
Variable (r:Set).
Hypothesis (er: equivalence r).

Lemma class_P x y: inc y (class r x) <-> related r x y.
Proof.
have gr: (sgraph r) by fprops.
rewrite /related; split.
by move/funI_P => [z /Zo_P [pa <-] ->]; rewrite (gr _ pa).
move=> h; apply/funI_P; exists (J x y); [ apply: Zo_i |]; aw.
Qed.

Lemma class_is_im_of_singleton x:
class r x = im_of_singleton r x.
Proof.
set_extens t; first by move/class_P/dirim_set1_P.
by move/dirim_set1_P/class_P.
Qed.

Lemma sub_class_substrate x: sub (class r x) (substrate r).
Proof. move=> t /class_P rt; substr_tac. Qed.

Lemma class_eq1 u v: related r u v -> class r u = class r v.
Proof.
move=> ruv; set_extens x; move/class_P => h; apply/class_P; equiv_tac.
Qed.

Lemma class_eq2 u v: inc u (class r v) -> class r u = class r v.
Proof.
by move => pb; symmetry;apply: class_eq1; apply/(class_P).
Qed.

Lemma setQ_ne x: inc x (quotient r) -> nonempty x.
Proof.
by move/funI_P => [z zr ->]; exists z; apply/class_P; equiv_tac.
Qed.

Lemma setQ_repi x: inc x (quotient r) -> inc (rep x) x.
Proof. by move => /setQ_ne; apply: rep_i. Qed.

Lemma inc_class_setQ x:
inc x (substrate r) -> inc (class r x) (quotient r).
Proof. move => ta; apply/funI_P; ex_tac. Qed.

Lemma class_class x: inc x (substrate r) -> classp r (class r x).
Proof.
move=> xs; move: (setQ_repi (inc_class_setQ xs)) =>/class_P rxr.
by split =>//; [substr_tac | apply: class_eq1].
Qed.

Lemma setQ_P x: inc x (quotient r) <-> classp r x.
Proof.
apply: (iff_trans (funI_P _ _ _)); split.
by move => [z zs ->]; apply: class_class.
move => [pa pb]; ex_tac.
Qed.

Lemma class_rep x: inc x (quotient r) -> class r (rep x) = x.
Proof. by move /setQ_P => [_]. Qed.

Lemma in_class_relatedP y z:
related r y z <-> (exists x, [/\ classp r x, inc y x & inc z x]).
Proof.
split.
move=> ryx;exists (class r y).
have ys: inc y (substrate r) by substr_tac.
split; first (by apply: class_class); apply/class_P => //; equiv_tac.
move =>[w [ [_ ->]]] /class_P pb /class_P pc; equiv_tac.
Qed.

Lemma related_rep_in_class x y:
inc x (quotient r) -> inc y x -> related r (rep x) y.
Proof.
move=> xq; move: (xq) =>/setQ_P cx yx; apply/in_class_relatedP; exists x.
split => //;by apply: setQ_repi.
Qed.

Lemma rep_in_class x: classp r x -> inc (rep x) x.
Proof. move /setQ_P; apply: setQ_repi. Qed.

Lemma rel_in_class x y: classp r x -> inc y x -> related r (rep x) y.
Proof. by move /setQ_P => xq yx; apply: related_rep_in_class. Qed.

Lemma sub_class_sr x: classp r x -> sub x (substrate r).
Proof. move => [pa ->]; apply: sub_class_substrate. Qed.

Lemma rel_in_class2 x y: classp r x -> related r (rep x) y -> inc y x.
Proof. by move => pa pb; rewrite (proj2 pa); apply /class_P. Qed.

Lemma class_dichot x y: classp r x -> classp r y -> disjointVeq x y.
Proof.
move=> cx cy; mdi_tac xy.
move => u ux uy; case: xy.
move: (rel_in_class cx ux) => pa.
move: (rel_in_class cy uy) => pb.
rewrite (proj2 cx) (proj2 cy); apply: class_eq1; equiv_tac.
Qed.

Lemma inc_in_setQ_sr x y:
inc x y -> inc y (quotient r) -> inc x (substrate r).
Proof. move=> xy /setQ_P [pa pb].
apply: (@sub_class_substrate (rep y)); ue.
Qed.

Hint Resolve inc_class_setQ: fprops.

Lemma setU_setQ: union (quotient r) = substrate r.
Proof.
set_extens t => ts.
move: (setU_hi ts) => [z tz zu]; apply: (inc_in_setQ_sr tz zu).
apply: (@setU_i _ (class r t)); [ apply/class_P; equiv_tac | fprops].
Qed.

Lemma rep_i_sr x: inc x (quotient r) -> inc (rep x) (substrate r).
Proof. by move /setQ_P => [g _]. Qed.

Lemma inc_itself_class x: inc x (substrate r) -> inc x (class r x).
Proof. move => xsr; apply/class_P; equiv_tac. Qed.

Lemma related_rep_class x: inc x (substrate r) -> related r x (rep (class r x)).
Proof.
move=> xs.
suff: related r (rep (class r x)) x by move=> h; equiv_tac.
by apply: related_rep_in_class; fprops; apply: inc_itself_class.
Qed.

Lemma related_rr_P u v:
inc u (quotient r) -> inc v (quotient r) ->
(related r (rep u) (rep v) <-> (u = v)).
Proof.
move => uq vq; split.
move=> h; rewrite - (class_rep uq) -(class_rep vq).
by apply: class_eq1.
by move=> ->; apply: (reflexivity_e er); apply: rep_i_sr.
Qed.

Lemma related_equiv_P u v:
related r u v <->
[/\ inc u (substrate r), inc v (substrate r) & class r u = class r v].
Proof.
split.
by move=> ruv; split; [substr_tac | substr_tac| apply: class_eq1].
move=> [us vs cuv].
apply /class_P; rewrite cuv; by apply: inc_itself_class.
Qed.

Lemma is_class_pr x y:
inc x y -> inc y (quotient r) -> y = class r x.
Proof.
move=> xy yq.
have <-: class r (rep y) = y by apply: class_rep.
apply: class_eq1=>// ; apply: (related_rep_in_class yq xy).
Qed.

End Class.

Hint Resolve rep_i_sr inc_itself_class inc_class_setQ : fprops.

Canonical projection on the quotient

Definition canon_proj r := Lf (class r) (substrate r) (quotient r).

Section CanonProj.
Variable (r:Set).
Hypothesis (er: equivalence r).

Lemma canon_proj_f : function (canon_proj r).
Proof. apply: lf_function; move=> t ts /=; fprops. Qed.

Lemma canon_proj_s : source (canon_proj r) = substrate r.
Proof. rewrite /canon_proj; aw. Qed.

Lemma canon_proj_t : target (canon_proj r) = quotient r.
Proof. rewrite /canon_proj; aw. Qed.

Hint Rewrite canon_proj_s canon_proj_t: aw.
Hint Resolve canon_proj_f: fprops.

Lemma canon_proj_V x:
inc x (substrate r) -> Vf (canon_proj r) x = class r x.
Proof. rewrite /canon_proj=> xs; aw; move=> t ts /=; fprops. Qed.

Hint Rewrite canon_proj_V : aw.

Lemma canon_proj_setQ_i x:
inc x (substrate r) -> inc (Vf (canon_proj r) x) (quotient r).
Proof. move=> xs; aw; fprops. Qed.

Lemma rel_gcp_P x y:
inc x (substrate r) -> inc y (quotient r) ->
(inc (J x y) (graph (canon_proj r)) <-> inc x y).
Proof.
move=> xs yq.
have fc: (function (canon_proj r)) by fprops.
split => h.
move: (Vf_pr fc h); aw => ->; apply/class_P => //; equiv_tac.
have: (y = Vf (canon_proj r) x).
aw; move: (class_rep er yq) h => <- /(class_P er) rryx.
apply: class_eq1 =>//.
move=> ->; Wtac; aw.
Qed.

Lemma canon_proj_fs: surjection (canon_proj r).
Proof.
split;[ fprops | move=> y; aw => yr; exists (rep y); fprops].
aw; [ by apply: class_rep | fprops ].
Qed.

Lemma sub_im_canon_proj_quotient a x:
sub a (substrate r) ->
inc x (image_by_fun (canon_proj r) a) ->
inc x (quotient r).
Proof.
move=> sas xi.
have fc:function (canon_proj r) by fprops.
have sa: sub a (source (canon_proj r)) by aw.
move /(Vf_image_P fc sa): xi => [u ua ->]; aw; fprops.
Qed.

Criterion 55 of Bourbaki

Lemma related_e_P u v:
related r u v <->
[/\ inc u (source (canon_proj r)),
inc v (source (canon_proj r)) &
Vf (canon_proj r) u = Vf (canon_proj r) v].
Proof.
rewrite canon_proj_s.
split.
by move/(related_equiv_P er) => [us vs cuv]; aw.
move => [us vs h]; apply/(related_equiv_P er); split => //; move: h; aw.
Qed.

End CanonProj.

Hint Rewrite canon_proj_s canon_proj_t: aw.
Hint Resolve canon_proj_f: fprops.
Hint Rewrite canon_proj_V : aw.

The diagonal is the graph of equality

Lemma diagonal_class x u:
inc u x -> class (diagonal x) u = singleton u.
Proof.
move: (diagonal_equivalence x) => h.
move=> ux; apply: set1_pr; first by apply/(class_P h) /diagonal_pi_P.
by move => z /(class_P h) /diagonal_pi_P [_ ->].
Qed.

Lemma canon_proj_diagonal_fb x:
bijection (canon_proj (diagonal x)).
Proof.
have h:=(diagonal_equivalence x).
have cps:= (canon_proj_fs h).
split => //; split;first by fct_tac.
aw; move=> a b ax bx /=; aw.
move: ax bx;rewrite (proj2 (diagonal_osr x)) => ax bx.
rewrite ! diagonal_class //; apply:set1_inj.
Qed.

Lemma canon_proj_diagonal_fb_contra r:
equivalence r -> bijection (canon_proj r) ->
r = diagonal (substrate r).
Proof.
move => pa pb.
set_extens t; last first.
by move /diagonal_i_P => [Px px qx]; rewrite -Px -qx; apply: reflexivity_e.
move => tr; move: (pa) => [sa _ _ _]; move: (sa _ tr) => pt.
have ptsr: inc (P t) (substrate r) by substr_tac.
have qtsr: inc (Q t) (substrate r) by substr_tac.
apply /diagonal_i_P; split => //.
move: (bij_inj pb); rewrite canon_proj_s => h; move: (h _ _ ptsr qtsr).
by aw; apply; apply: class_eq1 => //;rewrite /related pt.
Qed.

Relation associated to P in a product

Definition first_proj_eqr x y :=
eq_rel_associated (first_proj (x \times y)).

Definition first_proj_eq x y :=
equivalence_associated (first_proj (x \times y)).

Lemma first_proj_equivalence x y:
equivalence (first_proj_eq x y).
Proof. apply: graph_ea_equivalence; apply: first_proj_f. Qed.

Lemma first_proj_sr x y:
substrate (first_proj_eq x y) = x \times y.
Proof.
rewrite graph_ea_substrate; [apply:lf_source | apply:first_proj_f].
Qed.

Lemma first_proj_eq_related_P x y a b:
related (first_proj_eq x y) a b <->
[/\ inc a (x \times y), inc b (x \times y) & P a = P b].
Proof.
have ff: function (first_proj (x \times y)) by apply:first_proj_f.
split.
move/(ea_relatedP ff); rewrite lf_source; move => [pa pb].
rewrite ! first_proj_V => //.
move => [pa pb pc]; apply/(ea_relatedP ff).
by rewrite lf_source ! first_proj_V.
Qed.

Lemma first_proj_classP x y: nonempty y -> forall z,
(classp (first_proj_eq x y) z <->
exists2 u, inc u x & z = (singleton u) \times y).
Proof.
move=> [y0 y0y].
set (r:=first_proj_eq x y).
have er: equivalence r by apply: first_proj_equivalence.
move: (first_proj_sr x y); rewrite -/r=> sr.
move => z; split.
move => cz; move: (proj1 cz); rewrite sr; move /setX_P => [sa sb sc].
exists (P (rep z)) => //.
set_extens t => ta.
move: (rel_in_class er cz ta) => /first_proj_eq_related_P [qa qb qc].
by move /setX_P: qb => [rq rb rc]; apply /indexedrP.
rewrite (proj2 cz); apply /(class_P er); apply /first_proj_eq_related_P.
move/indexedrP: ta => [pa pb pc]; split => //; apply /setX_P; split => //; ue.
move=> [u uz zp]; rewrite zp.
have nep: (nonempty (singleton u \times y)).
by exists (J u y0); apply: setXp_i; fprops.
move: (rep_i nep); set w:= rep (singleton u \times y) => pa.
move /indexedrP: (pa) => [ra rb rc].
have wp: inc w (x \times y) by apply /setX_P;split => //; ue.
red; rewrite - /w; split; first by ue.
set_extens t.
move /indexedrP => [sa sb sc]; apply /(class_P er).
by apply /first_proj_eq_related_P;split; [exact | apply /setX_P |];rewrite sb.
move /(class_P er) => /first_proj_eq_related_P [_ /setX_P [qa qb qc] qd].
apply /indexedrP;split => //; ue.
Qed.

Lemma first_proj_equiv_proj x y:
nonempty y->
bijection (Lf (fun u => (singleton u) \times y)
x (quotient (first_proj_eq x y))).
Proof.
move=> ney.
set (f:=fun u => (singleton u) \times y).
set (g:= (Lf f x (quotient (first_proj_eq x y)))).
have efp:equivalence (first_proj_eq x y) by apply: first_proj_equivalence.
have ta: lf_axiom f x (quotient (first_proj_eq x y)).
move => u ux ; apply /(setQ_P efp) /(first_proj_classP _ ney); ex_tac.
rewrite /g; apply: lf_bijective => //.
rewrite /f => u v ux vx sp.
apply: set1_inj; rewrite - (setX_domain (singleton u) ney).
by rewrite sp setX_domain.
move=>z /(setQ_P efp) /(first_proj_classP _ ney) [u ux ep]; ex_tac.
Qed.

Lemma sub_quotient_powerset r:
equivalence r ->
sub (quotient r) (powerset (substrate r)).
Proof.
by move=> er x /(setQ_P er) => cr; apply /setP_P; apply: sub_class_sr.
Qed.

Lemma partition_from_equivalence r:
equivalence r ->
partition_s (quotient r)(substrate r).
Proof.
move=> er.
split; last by move=> t tq; apply: (setQ_ne er tq).
split; first by apply: setU_setQ.
by move=> a b /(setQ_P er) ca /(setQ_P er) cd;apply: (class_dichot er).
Qed.

Partition of a set and induced equivalence

Definition in_same_coset f x y:=
exists i, [/\ inc i (source f) , inc x (Vf f i) & inc y (Vf f i)].

Definition partition_relation f x :=
graph_on (in_same_coset f) x.

Section InSameCoset.
Variables (f x: Set).
Hypothesis (ff: function f).
Hypothesis fpa: partition_w_fam (graph f) x.

Lemma partition_inc_unique1 i j y:
inc i (source f) -> inc y (Vf f i) ->
inc j (source f) -> inc y (Vf f j) -> i = j.
Proof.
rewrite (proj33 ff) => p1 p2 p3 p4.
by rewrite -(cover_at_pr fpa p1 p2) - (cover_at_pr fpa p3 p4).
Qed.

Lemma isc_hi a b: (in_same_coset f a b) -> (inc a x /\ inc b x).
Proof.
move => [t [tsf p1 p2]]; rewrite (proj33 ff) in tsf.
rewrite -(proj33 fpa); split; union_tac.
Qed.

Lemma isc_rel_P a b:
(related (partition_relation f x) a b <-> in_same_coset f a b).
Proof.
move: fpa => [qa qb qc].
split; first by move /graph_on_P1 => [_ _].
by move => h; move: (isc_hi h) => [ax bx]; apply /graph_on_P1.
Qed.

Lemma isc_rel1P a b: inc a x -> inc b x ->
((in_same_coset f a b) <-> (cover_at (graph f) a = cover_at (graph f) b)).
Proof.
move => ax bx; split.
move => [t [tsf pa pb]].
rewrite (proj33 ff) in tsf.
by rewrite(cover_at_pr fpa tsf pa) (cover_at_pr fpa tsf pb).
move => h.
move: (cover_at_in fpa ax) (cover_at_in fpa bx); rewrite -h - (proj33 ff).
set s := (cover_at (graph f) a); move => [pa pb] [pc pd].
by exists s.
Qed.

Lemma isc_rel_sr: substrate (partition_relation f x) = x.
Proof. by apply: graph_on_sr => a ax; apply /isc_rel1P. Qed.

Lemma isc_rel_equivalence:
equivalence (partition_relation f x).
Proof.
apply: equivalence_from_rel; split.
by move => u v [i [isf Wx Wy]]; exists i.
move => b a c pa pb.
move: (isc_hi pa) (isc_hi pb) => [ax bx] [_ cx].
apply/(isc_rel1P ax cx).
by move /(isc_rel1P ax bx): pa => ->; move /(isc_rel1P bx cx): pb => ->.
Qed.

Lemma isc_rel_class a:
classp (partition_relation f x) a
-> exists2 u, inc u (source f) & a = Vf f u.
Proof.
move=> [rs ac].
move: isc_rel_equivalence => ec.
rewrite isc_rel_sr in rs.
move: (cover_at_in fpa rs); set u := cover_at _ _;move => [pa pb].
rewrite ac /Vf (proj33 ff); ex_tac.
set_extens t.
move /(class_P ec) /isc_rel_P => h; move: (proj2 (isc_hi h)) => tx.
by move: (cover_at_in fpa tx)=> []; move /(isc_rel1P rs tx): h => <-.
move => ts.
have tx: inc t x by rewrite - (proj33 fpa); union_tac.
apply /(class_P ec) /isc_rel_P /(isc_rel1P rs tx).
by symmetry; apply:(same_cover_at fpa rs ts).
Qed.

Lemma isc_rel_class2 u:
inc u (source f) -> nonempty (Vf f u)
-> classp (partition_relation f x) (Vf f u) .
Proof.
move=> us neW.
move: isc_rel_equivalence => ec.
rewrite (proj33 ff) in us.
have sWx: (sub (Vf f u) x) by rewrite -(proj33 fpa) => t tW; union_tac.
move: (rep_i neW) => rw;move: (sWx _ rw) => rs.
move:(cover_at_pr fpa us rw) => pa.
red; rewrite isc_rel_sr //; split=> //.
set_extens t.
move => ts.
have tx: inc t x by rewrite - (proj33 fpa); union_tac.
apply /(class_P ec) /isc_rel_P /(isc_rel1P rs tx).
by symmetry; apply:(same_cover_at fpa rs);rewrite pa.
move /(class_P ec) /isc_rel_P => h; move: (proj2 (isc_hi h)) => tx.
rewrite -pa; move: (proj1 (cover_at_in fpa tx)).
by move /(isc_rel1P rs tx): h => ->.
Qed.

Lemma partition_fun_fb:
(allf (graph f) nonempty) ->
bijection (Lf (Vf f)
(source f) (quotient (partition_relation f x))).
Proof.
move: isc_rel_equivalence => er.
rewrite /allf - (proj33 ff); move=> alne; apply: lf_bijective.
- move=> t tf; apply/(setQ_P er).
apply: isc_rel_class2 =>//; apply: (alne _ tf).
- move=> u v us vs sW.
move: (alne _ us)=> [y yu]; have yv: (inc y (Vf f v)) by rewrite - sW.
by apply: (partition_inc_unique1 us yu vs yv).
- move=> y; move/(setQ_P er) => ic.
move: (isc_rel_class ic) => [u us] ->; ex_tac.
Qed.

End InSameCoset.

A system of representatives for the partition f of x as a set f or a function g

Definition representative_system s f x :=
[/\ function f, partition_w_fam (graph f) x, sub s x
& forall i, inc i (source f) -> singletonp ((Vf f i) \cap s)].

Definition representative_system_function g f x :=
injection g /\ (representative_system (range (graph g)) f x).

Lemma rep_sys_function_pr g f x i:
representative_system_function g f x -> inc i (source f)
-> exists! a, (inc a (source g) /\ inc (Vf g a) (Vf f i)).
Proof.
move=> [[fg ig] [ff pf gg alsi]] ins.
move: (alsi _ ins) => [u isu].
apply /unique_existence; split.
move: (set1_1 u).
rewrite - isu; move /setI2_P => [pa] /(range_fP fg) [a asg pb].
by exists a; rewrite - pb.
move=> a b [asg Wa][bsg Wb]; apply: ig => //.
have aux: (forall a, inc a (source g) -> inc (Vf g a) (Vf f i) -> Vf g a = u).
move=> c csg Wc.
apply: set1_eq; rewrite -isu; apply: setI2_i => //; Wtac.
rewrite aux // aux //.
Qed.

Lemma rep_sys_function_pr2 g f x:
injection g -> function f -> partition_w_fam (graph f) x
-> sub (target g) x
-> (forall i, inc i (source f)
-> exists! a, (inc a (source g) /\ inc (Vf g a) (Vf f i)))
-> representative_system_function g f x.
Proof.
move=> ig ff pf stx exu; move:(ig)=> [fg _].
split =>//; split=>//.
by apply: (@sub_trans (target g)); [apply: f_range_graph |].
move=> i isf; move: (exu _ isf) => [a [[asg Wa] un]].
exists (Vf g a); set_extens t.
move/setI2_P => [tv /(range_fP fg) [z zsg eq]]; apply/set1_P.
by rewrite eq;congr (Vf g _); symmetry;apply: un;rewrite - eq.
move /set1_P => ->; apply/setI2_P; split => //; Wtac.
Qed.

Lemma section_canon_proj_pr g f x y r:
r = partition_relation f x -> function f -> partition_w_fam (graph f) x
-> is_right_inverse (canon_proj r) g->
inc y x ->
related r y (Vf g (class r y)).
Proof.
move=> pr ff pf ri yx.
have eq: equivalence r by rewrite pr; apply: isc_rel_equivalence.
have sr: substrate r = x by rewrite pr isc_rel_sr //.
rewrite - sr in yx.
have cyq: inc (class r y) (quotient r) by fprops.
have sg: source g = target (canon_proj r) by apply: right_i_source.
move: sg; aw => sg; rewrite - sg in cyq.
have yri: inc y (class r y) by apply: inc_itself_class => //.
apply: (symmetricity_e eq).
apply /(class_P eq).
have <- //: (class r y) = class r (Vf g (class r y)).
move: ri => [pa pb].
set t := (class r y).
have Wcs: inc (Vf g t) (substrate r).
by move: pa=> [_ fg]; aw; move=>->; Wtac.
have: Vf (canon_proj r \co g) t = Vf (canon_proj r) (Vf g t)by aw.
rewrite pb; bw; aw; rewrite /t;fprops.
Qed.

Lemma section_is_representative_system_function g f x:
function f -> partition_w_fam (graph f) x
-> is_right_inverse (canon_proj (partition_relation f x)) g ->
(forall u, inc u (source f) -> nonempty (Vf f u)) ->
representative_system_function g f x.
Proof.
move=> ff pf ti alne.
have ig: injection g by apply: (right_inverse_fi ti).
apply: rep_sys_function_pr2=>//.
move: ti =>[ [_ _ <-] _]; aw.
by rewrite isc_rel_sr; fprops.
move=> i isf.
set (r:=partition_relation f x) in *.
have sf: domain (graph f) =source f by apply: f_domain_graph.
have er: equivalence r by rewrite /r; apply: isc_rel_equivalence.
have sr: substrate r = x by rewrite /r isc_rel_sr.
rewrite (right_i_source ti); aw.
apply /unique_existence;split.
set z:= (rep (Vf f i)).
have zW: inc z (Vf f i) by rewrite /z; apply: rep_i; apply: alne.
have zx: inc z x by move: pf=>[_ md <-]; rewrite - sf in isf; union_tac.
exists (class r z); split.
aw; rewrite - sr in zx; fprops.
move: (section_canon_proj_pr (refl_equal r) ff pf ti zx).
move /(isc_rel_P ff pf) => [ u [usf zW' qW]].
by rewrite (partition_inc_unique1 ff pf isf zW usf zW').
aw; move=> a b [atx Wa] [btx Wb].
have Wiq: inc (Vf f i) (quotient r).
apply/(setQ_P er); apply (isc_rel_class2 ff pf isf); ex_tac.
have aux: (forall u, inc u (quotient r) -> inc (Vf g u) (Vf f i) -> Vf f i = u).
move=> u /(setQ_P er) [rs eq] iWW; rewrite sr in rs.
rewrite eq; apply: is_class_pr=> //.
move: (section_canon_proj_pr (refl_equal r) ff pf ti rs).
rewrite - eq; move /(isc_rel_P ff pf) => [v [usf zW' qW]].
by rewrite (partition_inc_unique1 ff pf isf iWW usf qW).
by rewrite -(aux _ atx Wa) (aux _ btx Wb).
Qed.

## EII-6-3 Relations compatible with an equivalence relation

A relation is compoatible if the set of objects satisfying the relation is a union of classes

Definition compatible_with_equiv_p (p: property) (r:Set) :=
forall x x', p x -> related r x x' -> p x'.

Lemma trivial_equiv p x:
compatible_with_equiv_p p (diagonal x).
Proof. by move=> u v pa /diagonal_pi_P [_] <-. Qed.

A compatible relation induces a relation on the quotient

Definition relation_on_quotient p r :=
fun t => inc t (quotient r) /\ exists2 x, inc x t & p x.

Lemma rel_on_quoP p r:
equivalence r -> compatible_with_equiv_p p r -> forall t,
(relation_on_quotient p r t <->
(inc t (quotient r) /\ forall x, inc x t -> p x)).
Proof.
move=> er co t; split.
move=> [tq [x xt px]]; split=>// u ut; apply: (co _ _ px).
by move: tq => /(setQ_P er) h; apply /(in_class_relatedP er); exists t.
move => [tq xtpx];move: (setQ_ne er tq) => net.
have rtt: (inc (rep t) t) by apply: rep_i.
split =>//; ex_tac.
Qed.

Lemma rel_on_quoP2 p r:
equivalence r -> compatible_with_equiv_p p r -> forall y,
( (inc y (substrate r) /\ relation_on_quotient p r (Vf (canon_proj r) y)) <->
(inc y (substrate r) /\ p y)).
Proof.
move=> er co; split; move => [ysr ]; rewrite (canon_proj_V ysr).
move => [tq [x xt px]].
have yx: related r x y by move : xt; move /(class_P er) => h; equiv_tac.
by split =>//; apply: (co _ _ px yx).
move => py; split =>//; split; fprops.
exists y =>//; apply /(class_P er);equiv_tac.
Qed.

## EII-6-4 saturated subsets

A set is satured if it is a union of classes

Definition saturated r x := compatible_with_equiv_p (fun y=> inc y x) r.

If f is the canonical projection this defines that smallest saturated set that contains x

Definition inverse_direct_value f x :=
image_by_fun (inverse_fun f) (image_by_fun f x).

Lemma idvalue_P f x: function f -> sub x (source f) ->forall y,
inc y (inverse_direct_value f x) <->
(inc y (source f) /\ (exists2 z, inc z x & Vf f y = Vf f z)).
Proof.
move => ff sxf.
split.
move /dirim_P => [u ]; aw; move /dirim_P => [z zx pa] /igraph_pP pb.
split; [Wtac | ex_tac].
have ->: Vf f y = u by Wtac.
Wtac.
move => [ysf [z zx pa]].
apply/dirim_P; exists (Vf f z).
apply/dirim_P;ex_tac; Wtac.
aw; apply/igraph_pP; rewrite -pa; Wtac.
Qed.

Lemma idvalue_cprojP r x:
equivalence r -> sub x (substrate r) ->forall y,
inc y (inverse_direct_value (canon_proj r) x) <->
(inc y (substrate r) /\ (exists2 z, inc z x & class r y = class r z)).
Proof.
move: (canon_proj_f r) => ff.
move => pa pb; move: (pb); rewrite - canon_proj_s => pc y.
split.
move/(idvalue_P ff pc) => [qa [v qb vx]]; split; first by exact.
have ysr: inc y (substrate r) by rewrite - canon_proj_s.
by ex_tac; move: vx; aw ;apply: pb.
move => [qa [v qb qc]]; apply/(idvalue_P ff pc).
have vsr: inc v (substrate r) by apply: pb.
have ysr: inc y (substrate r) by move: qa; aw.
split => //;exists v; aw.
Qed.

Lemma saturatedP r x:
equivalence r -> sub x (substrate r) ->
((saturated r x) <-> (forall y, inc y x -> sub (class r y) x)).
Proof.
move=> er sxs; split.
move=> sr y yx t; move/(class_P er); apply: (sr _ _ yx).
by move=> yw u v ux ruv; apply: (yw _ ux); apply/(class_P er).
Qed.

Lemma saturated2P r x:
equivalence r -> sub x (substrate r) ->
((saturated r x) <->
exists2 y, (forall z, inc z y -> classp r z) & x = union y).
Proof.
move => er sxs; move: (saturatedP er sxs) => aux;split.
move => srx.
exists (Zo (powerset x)(fun z => exists2 a, inc a x & z = class r a)).
move=> z /Zo_P [] /setP_P zx [a ax ->].
by apply: class_class =>//; apply: sxs.
set_extens t => ts.
apply: (@setU_i _ (class r t)).
by move: (sxs _ ts)=>h; apply/(class_P er);equiv_tac.
apply: Zo_i; [ by apply/setP_P;move /aux: srx; apply | ex_tac].
by move /setU_P: ts => [y ty /Zo_P [] /setP_P yx [a ax yc]]; apply: yx.
move => [y yp xu]; apply/aux; rewrite xu => z zu.
move: (setU_hi zu) =>[t zt ty]; move: (yp _ ty) => ct.
move => u /(class_P er) => rz; apply:(setU_i _ ty).
move: (rel_in_class er ct zt) => pa.
rewrite (proj2 (yp _ ty)); apply /(class_P er); equiv_tac.
Qed.

Lemma class_is_inv_direct_value r x:
equivalence r -> inc x (substrate r) ->
class r x = inverse_direct_value (canon_proj r) (singleton x).
Proof.
move=> er xs.
move: (set1_sub xs) => xs1.
set_extens t => ts.
apply/(idvalue_cprojP er xs1); split.
apply: (sub_class_substrate er ts).
exists x; fprops; apply :(class_eq2 er ts).
move/(idvalue_cprojP er xs1): ts => [v [pa /set1_P <- <-]]; fprops.
Qed.

Lemma saturated_P3 r x:
equivalence r -> sub x (substrate r) ->
(saturated r x <-> (x= inverse_direct_value (canon_proj r) x)).
Proof.
move=> er xs.
set (p:=canon_proj r).
have fp: function p by rewrite /p; fprops.
split => h.
apply: extensionality.
by apply: inverse_direct_image =>//; rewrite /p; aw.
move => y /(idvalue_cprojP er xs) [z [pa pb pc]].
move /(saturatedP er xs): h => q; move: (q _ pb); rewrite -pc.
apply; fprops.
apply/(saturatedP er xs)=> y yx; rewrite h => t tc.
apply /(idvalue_cprojP er xs); split.
apply: (sub_class_substrate er tc).
ex_tac;apply :(class_eq2 er tc).
Qed.

Lemma saturated_P4 r x:
equivalence r -> sub x (substrate r) ->
(saturated r x <-> (exists2 b, sub b (quotient r)
& x = image_by_fun (inverse_fun (canon_proj r)) b)).
Proof.
move=> er xs.
have qr: quotient r = target (canon_proj r) by aw.
have sj := (canon_proj_fs er).
split.
move/(saturated_P3 er xs) => hyp.
exists (image_by_fun (canon_proj r) x) =>//.
rewrite qr;apply: sub_trans (f_range_graph (proj1 sj)); apply: dirim_Sr.
move=> [b bq xeq].
apply/(saturated_P3 er xs).
rewrite qr in bq; move: (direct_inv_im_surjective sj bq).
by rewrite /inverse_direct_value -xeq => ->.
Qed.

Links between saturation and union, intersection, complement

Lemma saturated_setU r x:
equivalence r ->
(alls x (sub^~ (substrate r))) -> (alls x (saturated r)) ->
(sub (union x) (substrate r) /\ saturated r (union x)).
Proof.
move=> er als alsa.
have r1: (sub (union x) (substrate r)).
by move => t /setU_P [z tz zx]; apply: (als _ zx).
split=>//;apply /saturatedP => // y /setU_P [z yz zx].
apply: sub_trans (setU_s1 zx).
by move: (alsa _ zx) => /(saturatedP er (als _ zx)); apply.
Qed.

Lemma saturated_setI r x:
equivalence r -> nonempty x ->
(alls x (sub^~ (substrate r))) -> (alls x (saturated r)) ->
(sub (intersection x) (substrate r) /\ saturated r (intersection x)).
Proof.
move=> er nex als alsa.
have aux: (sub (intersection x) (substrate r)).
by move: nex => [y yx]; apply: sub_trans (als _ yx); apply: setI_s1.
split =>//; apply /(saturatedP er aux) => y yi t tc; apply: (setI_i nex).
move=> z zx; move:(alsa _ zx).
move /(saturatedP er (als _ zx))=> s; apply: (s _ (setI_hi yi zx) _ tc).
Qed.

Lemma saturated_setC r a:
equivalence r -> sub a (substrate r) -> saturated r a ->
saturated r ((substrate r) -s a).
Proof.
move=> er ar.
move /(saturated_P4 er ar) => [b sq aib].
apply /(saturated_P4 er);first by apply: sub_setC.
exists ((quotient r) -s b); first by apply: sub_setC.
move: (iim_fun_C b (canon_proj_f r)).
by rewrite aib iim_fun_pr canon_proj_s canon_proj_t; symmetry.
Qed.

Definition saturation_of r x :=
inverse_direct_value (canon_proj r) x.

Lemma saturation_of_pr r x:
equivalence r -> sub x (substrate r) ->
saturation_of r x =
union (Zo (quotient r)(fun z=> exists2 i, inc i x & z = class r i)).
Proof.
move=> er sr.
have fc: function (canon_proj r) by fprops.
rewrite /saturation_of/inverse_direct_value.
set_extens y.
move /(idvalue_cprojP er sr) => [v [pa pb pc]].
apply: (@setU_i _ (class r y)); fprops; apply: Zo_i; fprops.
by ex_tac; rewrite pd.
move /setU_P => [z zy/Zo_P [zq [i ix sc]]].
move: zy; rewrite sc; move/(class_P er) => pc.
apply/(idvalue_cprojP er sr); split;first by substr_tac.
ex_tac;symmetry;by apply:class_eq1.
Qed.

Lemma saturation_of_smallest r x:
equivalence r -> sub x (substrate r) ->
[/\ saturated r (saturation_of r x),
sub x (saturation_of r x)
& (forall y, sub y (substrate r) -> saturated r y -> sub x y
-> sub (saturation_of r x) y)].
Proof.
move=> er xsr.
have fc: function (canon_proj r) by fprops.
split.
- apply /(saturated_P4 er).
by move => t /(idvalue_cprojP er xsr) [pa _].
exists (image_by_fun (canon_proj r) x) => //.
rewrite - canon_proj_t.
apply: sub_trans (f_range_graph fc); apply: dirim_Sr.
- move => t tx; apply/(idvalue_cprojP er xsr); split;first by apply: xsr.
ex_tac.
- move=> y ys sy sxy t.
move /(idvalue_cprojP er xsr) => [tsr [z tx sc]].
move /(saturated_P3 er ys): sy => ->.
by apply/(idvalue_cprojP er ys); split => //; exists z => //; apply: sxy.
Qed.

Definition union_image x f:=
union (Zo x (fun z=> exists2 i, inc i (source f) & z = Vf f i)).

Lemma saturation_of_union r f g:
equivalence r -> function f -> function g ->
(forall i, inc i (source f) -> sub (Vf f i) (substrate r)) ->
source f = source g ->
(forall i, inc i (source f) -> saturation_of r (Vf f i) = Vf g i)
-> saturation_of r (union_image (powerset(substrate r)) f) =
union_image (powerset(substrate r)) g.
Proof.
move=> er ff fg als sfsg alsat.
have fc: function (canon_proj r) by fprops.
have U1sr: sub (union_image (powerset (substrate r)) f) (substrate r).
by move => t /setU_P [z zt /Zo_P [/setP_P h _]]; apply: h.
set_extens x.
move/(idvalue_cprojP er U1sr) => [zu [z /setU_P [v zv]] /Zo_P []
/setP_P vsr [i isf fi] sc].
move:(als _ isf) => pa.
apply/setU_P; exists (Vf g i); rewrite -(alsat _ isf).
apply/(idvalue_cprojP er pa); split => //; exists z => //; ue.
apply: Zo_i; first by apply/setP_P=> t /(idvalue_cprojP er pa) [ ok _].
rewrite - sfsg; ex_tac.
move => /setU_P [z xz /Zo_P [/setP_P zr [i isr zv]]].
move: xz; rewrite - sfsg in isr; rewrite zv - (alsat _ isr).
move/(idvalue_cprojP er (als _ isr)) => [pa [t pb pc]].
apply/(idvalue_cprojP er U1sr); split => //; exists t => //.
apply /setU_P; exists (Vf f i) => //; apply:Zo_i; last by ex_tac.
by apply /setP_P; apply: als.
Qed.

## EII-6-5 Mappings compatible with equivalence relations

The canonical projection is surjective, thus has a section

Definition section_canon_proj r :=
Lf rep (quotient r) (substrate r).

Lemma section_canon_proj_axioms r:
equivalence r ->
lf_axiom rep (quotient r) (substrate r).
Proof. move=> er t ta; fprops. Qed.

Lemma section_canon_proj_V r x:
equivalence r ->
inc x (quotient r) -> Vf (section_canon_proj r) x = (rep x).
Proof.
rewrite/section_canon_proj=> er iq; aw.
by apply: section_canon_proj_axioms.
Qed.

Lemma section_canon_proj_f r:
equivalence r -> function (section_canon_proj r).
Proof.
rewrite/section_canon_proj=> er.
by apply: lf_function; apply: section_canon_proj_axioms.
Qed.

Lemma right_inv_canon_proj r:
equivalence r ->
is_right_inverse (canon_proj r) (section_canon_proj r).
Proof.
move=> er.
have ss: source (section_canon_proj r) = quotient r.
by rewrite/section_canon_proj; aw.
have fs: function (section_canon_proj r) by apply: section_canon_proj_f.
have fc: function (canon_proj r) by fprops.
have cpo: (canon_proj r) \coP (section_canon_proj r).
split => //; rewrite/ section_canon_proj; aw.
split=> //.
apply: function_exten; try fct_tac; fprops; aw.
rewrite ss=> x xs.
have aux: (Vf (section_canon_proj r) x) = rep x.
by rewrite section_canon_proj_V //.
aw; try ue.
rewrite identity_V // aux; apply: class_rep =>//.
Qed.

A mapping is compatible with an equivalence if it is constant on classes; it is compatible with two equivalences is the image of a class is a subset of a class

Definition compatible_with_equiv f r :=
[/\ function f, source f = substrate r &
forall x x', related r x x' -> Vf f x = Vf f x'].

Definition compatible_with_equivs f r r' :=
[/\ function f, target f = substrate r' &
compatible_with_equiv ((canon_proj r') \co f) r].

Lemma compatible_with_equiv_pr f r:
function f -> source f = substrate r ->
(compatible_with_equiv f r <->
(forall y, compatible_with_equiv_p (fun x => y = Vf f x) r)).
Proof.
move=> ff sf; rewrite/compatible_with_equiv/compatible_with_equiv_p.
split.
by move=> [_ _ hyp] y x x' =>->; apply: hyp.
by move=> hyp; split => // x x'; apply: (hyp (Vf f x) x x').
Qed.

Lemma compatible_constant_on_classes f r x y:
equivalence r ->
compatible_with_equiv f r -> inc y (class r x) -> Vf f x = Vf f y.
Proof.
by move=> eq [ff sf hyp] yc; apply: hyp; move/(class_P eq): yc.
Qed.

Lemma compatible_constant_on_classes2 f r x:
equivalence r -> compatible_with_equiv f r ->
constantfp (restriction f (class r x)).
Proof.
move=> re [ff sf hyp].
have scs: sub (class r x) (substrate r) by apply: sub_class_substrate.
rewrite - sf in scs.
split; first by apply: (proj31 (restriction_prop ff scs)).
have sr: source (restriction f (class r x)) = class r x.
by rewrite /restriction; aw.
rewrite sr.
move=> a b ac bc /=; rewrite ! restriction_V //; apply: hyp.
move: ac bc => /(class_P re) ac /(class_P re) bc; equiv_tac.
Qed.

Lemma compatible_with_proj f r x y:
equivalence r -> compatible_with_equiv f r ->
inc x (substrate r) -> inc y (substrate r) ->
Vf (canon_proj r) x = Vf (canon_proj r) y -> Vf f x = Vf f y.
Proof.
move=> er cgt xs ys; aw => sx.
apply: (@compatible_constant_on_classes _ r) =>//; rewrite sx; fprops.
Qed.

Lemma compatible_with_pr r r' f x y:
equivalence r -> equivalence r' ->
compatible_with_equivs f r r' ->
related r x y -> related r' (Vf f x) (Vf f y).
Proof.
move=> er er' [ff sf [fc sc hyp]] rxy.
have cc: ((canon_proj r') \coP f) by split; first (by fprops); aw.
move: sc; aw => sc.
have xs: inc x (source f) by rewrite sc; substr_tac.
have ys: inc y (source f) by rewrite sc; substr_tac.
have px: (inc (Vf f x) (substrate r')) by Wtac.
have py: (inc (Vf f y) (substrate r')) by Wtac.
move: (hyp _ _ rxy); aw =>// sc1.
by apply/related_equiv_P.
Qed.

Lemma compatible_with_pr2 r r' f:
equivalence r -> equivalence r' ->
function f ->
target f = substrate r'-> source f = substrate r->
(forall x y, related r x y -> related r' (Vf f x) (Vf f y)) ->
compatible_with_equivs f r r'.
Proof.
move=> er er' ff tf sf rr'.
have cc: (canon_proj r') \coP f by split; fprops; symmetry; aw.
split =>//; split=>//; aw;first by fct_tac.
move=> x x' rxx'; move: (rr' _ _ rxx'); move /(related_equiv_P er').
move => [Ws Ws' cxx']; aw; rewrite sf; substr_tac.
Qed.

Lemma compatible_with_proj3 r r' f x y:
equivalence r -> equivalence r' ->
compatible_with_equivs f r r'->
inc x (substrate r) -> inc y (substrate r) ->
Vf (canon_proj r) x = Vf (canon_proj r) y ->
class r' (Vf f x) = class r' (Vf f y).
Proof.
move=> er er' co xs ys; aw => sc.
apply: class_eq1 =>//; apply: (compatible_with_pr er er' co).
by apply /(related_equiv_P er).
Qed.

A compatible mapping induces a mapping on the quotient

Definition fun_on_quotient r f :=
f \co (section_canon_proj r).

Lemma exists_fun_on_quotient f r:
equivalence r -> function f -> source f = substrate r ->
(compatible_with_equiv f r <->
(exists h, h \coP (canon_proj r) /\ h \co (canon_proj r) = f)).
Proof.
move=> er ff sf.
rewrite exists_left_composable; aw; try apply: canon_proj_fs=>//.
split.
by move => co x y xs ys sW; apply: (compatible_with_proj er co xs ys sW).
move=> hyp; split=>//.
move=> x x' /(related_equiv_P er) [xs ys cc];apply: hyp; aw.
Qed.

Lemma exists_unique_fun_on_quotient f r h:
equivalence r -> compatible_with_equiv f r ->
h \coP (canon_proj r) -> h \co (canon_proj r) = f ->
h = fun_on_quotient r f.
Proof.
move=> er cfr chc chcf.
have sc: (surjection (canon_proj r)) by apply: canon_proj_fs.
move: (right_inv_canon_proj er) => ri.
have ff: function f by move: cfr=> [].
have sf: source f = source (canon_proj r) by aw; move: cfr=> [ _].
apply: (exists_left_composable_aux ff sf ri chc chcf).
Qed.

Lemma compose_foq_proj f r:
equivalence r -> compatible_with_equiv f r ->
(fun_on_quotient r f) \co (canon_proj r) = f.
Proof.
move=> er co; move: (co)=> [ff sf hyp].
set (g:= canon_proj r).
have sg: surjection g by rewrite/g; apply: canon_proj_fs.
have sfsg: source f = source g by rewrite /g; aw.
have sW: forall x y, inc x (source g) -> inc y (source g) ->
Vf g x = Vf g y -> Vf f x = Vf f y.
rewrite - sfsg sf; move=> x y xr yr h.
by apply: (compatible_with_proj er co xr yr).
apply: (left_composable_value ff sg sfsg sW (right_inv_canon_proj er)
(refl_equal (fun_on_quotient r f))).
Qed.

A mapping compatible with two relations induces a mapping on the two quotients

Definition fun_on_rep f: Set -> Set := fun x=> f(rep x).
Definition fun_on_reps r' f := fun x=> Vf (canon_proj r') (f(rep x)) .

Definition function_on_quotient r f b :=
Lf (fun_on_rep f)(quotient r)(b).

Definition function_on_quotients r r' f :=
Lf (fun_on_reps r' f)(quotient r)(quotient r').

Definition fun_on_quotients r r' f :=
((canon_proj r') \co f) \co (section_canon_proj r).

Lemma foq_axioms r f b:
equivalence r ->
lf_axiom f (substrate r) b ->
lf_axiom (fun_on_rep f) (quotient r) b.
Proof. move=> er h x xs;apply: h; fprops. Qed.

Lemma foqs_axioms r r' f:
equivalence r ->
equivalence r' ->
lf_axiom f (substrate r)(substrate r') ->
lf_axiom (fun_on_reps r' f) (quotient r) (quotient r').
Proof.
move => er er' h x xs.
have w: (inc (rep x) (substrate r)) by fprops.
rewrite /fun_on_reps; aw; fprops.
Qed.

Lemma foqc_axioms r f:
equivalence r ->
function f -> source f = substrate r ->
f \coP (section_canon_proj r).
Proof.
move=> er ff sf.
split =>//; first by apply: section_canon_proj_f.
rewrite /section_canon_proj; aw.
Qed.

Lemma foqcs_axioms r r' f:
equivalence r ->
equivalence r' ->
function f -> source f = substrate r -> target f = substrate r' ->
(canon_proj r' \co f) \coP (section_canon_proj r).
Proof.
move=> er er' ff sf tf.
have fc: (function (canon_proj r')) by fprops.
split; first by fct_tac => //; symmetry; aw.
by apply: section_canon_proj_f.
rewrite/section_canon_proj; aw.
Qed.

Lemma foq_f r f b:
equivalence r ->
lf_axiom f (substrate r) b ->
function (function_on_quotient r f b).
Proof.
by rewrite /function_on_quotient=> *; apply: lf_function; apply: foq_axioms.
Qed.

Lemma foqs_f r r' f:
equivalence r ->
equivalence r' ->
lf_axiom f (substrate r)(substrate r') ->
function (function_on_quotients r r' f).
Proof.
by rewrite /function_on_quotients=> *;apply: lf_function; apply: foqs_axioms.
Qed.

Lemma foqc_f r f:
equivalence r -> function f ->
source f = substrate r ->
function (fun_on_quotient r f).
Proof.
rewrite/fun_on_quotient=> er ff sf; fct_tac.
by apply: section_canon_proj_f.
by rewrite /section_canon_proj; aw.
Qed.

Lemma foqcs_f r r' f:
equivalence r ->
equivalence r' ->
function f -> source f = substrate r -> target f = substrate r' ->
function (fun_on_quotients r r' f).
Proof.
rewrite/fun_on_quotients=> er er' ff sf tf.
by apply: compf_f; apply: foqcs_axioms.
Qed.

Lemma foq_V r f b x:
equivalence r ->
lf_axiom f (substrate r) b ->
inc x (quotient r) ->
Vf (function_on_quotient r f b) x = f (rep x).
Proof.
by rewrite /function_on_quotient=> er ta xq; aw; apply: foq_axioms.
Qed.

Lemma foqc_V r f x:
equivalence r -> function f ->
source f = substrate r -> inc x (quotient r) ->
Vf (fun_on_quotient r f) x = Vf f (rep x).
Proof.
rewrite/fun_on_quotient=> er ff sf xq; aw.
by rewrite section_canon_proj_V.
by apply: foqc_axioms.
by rewrite /section_canon_proj; aw.
Qed.

Lemma foqs_V r r' f x:
equivalence r ->
equivalence r' ->
lf_axiom f (substrate r)(substrate r') ->
inc x (quotient r) ->
Vf (function_on_quotients r r' f) x = class r' (f (rep x)).
Proof.
rewrite /function_on_quotients=> er er' ta xq; aw.
by rewrite /fun_on_reps; aw; apply: ta; fprops.
by apply: foqs_axioms.
Qed.

Lemma foqcs_V r r' f x:
equivalence r ->
equivalence r' ->
function f -> source f = substrate r -> target f = substrate r' ->
inc x (quotient r) ->
Vf (fun_on_quotients r r' f) x = class r' (Vf f (rep x)).
Proof.
move=> er er' ff sf tf xq.
rewrite /fun_on_quotients.
have fs: function(section_canon_proj r) by apply: section_canon_proj_f.
have ccf: (canon_proj r') \coP f by split; fprops; symmetry;aw.
have cc2: ( (canon_proj r') \co f) \coP (section_canon_proj r).
by split; first (by fct_tac); rewrite /section_canon_proj; aw.
have Wt: inc (Vf (section_canon_proj r) x) (target (section_canon_proj r)).
by Wtac; rewrite /section_canon_proj; aw.
have irs: inc (rep x) (source f) by ue.
rewrite compf_V //; last by rewrite/section_canon_proj; aw.
rewrite section_canon_proj_V // /section_canon_proj.
aw; Wtac.
Qed.

Lemma fun_on_quotient_pr r f x:
Vf f x = fun_on_rep (fun _ => Vf f x) (Vf (canon_proj r)x).
Proof. done. Qed.

Lemma fun_on_quotient_pr2 r r' f x:
Vf (canon_proj r') (Vf f x) =
fun_on_reps r' (fun _ => Vf f x) (Vf (canon_proj r) x).
Proof. trivial. Qed.

Lemma composable_fun_proj r f b:
equivalence r ->
lf_axiom f (substrate r) b ->
(function_on_quotient r f b) \coP (canon_proj r).
Proof.
move=> er ta.
split; [by apply: foq_f | fprops | ].
rewrite/function_on_quotient; aw.
Qed.

Lemma composable_fun_projs r r' f:
equivalence r ->
equivalence r' ->
lf_axiom f (substrate r) (substrate r') ->
(function_on_quotients r r' f) \coP (canon_proj r).
Proof.
move=> er er' ta.
split; first (by apply: foqs_f); fprops.
rewrite/function_on_quotients; aw.
Qed.

Lemma composable_fun_projc r f:
equivalence r ->
compatible_with_equiv f r ->
(fun_on_quotient r f) \coP (canon_proj r).
Proof.
move=> er [ff sf h].
split; [ by apply: foqc_f | fprops | ].
rewrite/fun_on_quotient/section_canon_proj; aw.
Qed.

Lemma composable_fun_projcs r r' f:
equivalence r -> equivalence r' ->
compatible_with_equivs f r r'->
(fun_on_quotients r r' f) \coP (canon_proj r).
Proof.
move=> er er' [ff sf [fg sg h]].
move: sg; aw => sg.
split; fprops;first by apply: foqcs_f.
rewrite/fun_on_quotients/section_canon_proj; aw.
Qed.

Lemma fun_on_quotient_pr3 r f x:
equivalence r -> inc x (substrate r) ->
compatible_with_equiv f r ->
Vf f x = Vf (fun_on_quotient r f) (Vf (canon_proj r) x).
Proof.
move=> er xs co; rewrite - compf_V; last by aw.
by rewrite compose_foq_proj.
by apply: composable_fun_projc.
Qed.

Figure 3 (French version only)

Lemma fun_on_quotient_pr4 r r' f:
equivalence r -> equivalence r' ->
compatible_with_equivs f r r'->
(canon_proj r') \co f = (fun_on_quotients r r' f) \co (canon_proj r).
Proof.
rewrite/fun_on_quotients; move=> er er' [ff sf h].
symmetry; apply: (compose_foq_proj er h).
Qed.

Lemma fun_on_quotient_pr5 r r' f x:
equivalence r -> equivalence r' ->
compatible_with_equivs f r r'->
inc x (substrate r) ->
Vf (canon_proj r') (Vf f x) =
Vf (fun_on_quotients r r' f) (Vf (canon_proj r) x).
Proof.
move => er er' co xs.
move: (co)=> [ff sf [fg sg h]]; move: sg; rewrite compf_s => sg.
have cc: composable (canon_proj r') f by split; fprops;symmetry; aw.
rewrite - compf_V //; try ue.
rewrite - compf_V //; last by aw.
by rewrite -fun_on_quotient_pr4 //.
by apply: composable_fun_projcs=>//.
Qed.

Lemma compose_fun_proj_ev r f b x:
equivalence r ->
compatible_with_equiv (Lf f (substrate r) b) r ->
inc x (substrate r) ->
lf_axiom f (substrate r) b ->
Vf (function_on_quotient r f b \co canon_proj r) x = f x.
Proof.
move=> er co xr ta.
have cq: (inc (class r x) (quotient r)) by fprops.
aw; try apply: composable_fun_proj =>//; last by aw.
have rrx: related r (rep (class r x)) x.
by apply: symmetricity_e=>//; apply: related_rep_class=>//.
rewrite foq_V //.
by move: co=> [_ _ h]; move: (h _ _ rrx); aw; fprops.
Qed.

Lemma compose_fun_proj_ev2 r r' f x:
equivalence r ->
equivalence r' ->
compatible_with_equivs (Lf f (substrate r) (substrate r')) r r' ->
lf_axiom f (substrate r) (substrate r') ->
inc x (substrate r) ->
inc (f x) (substrate r') ->
Vf (canon_proj r') (f x) =
Vf (function_on_quotients r r' f \co canon_proj r) x.
Proof.
move=> er er' co ta xs fxs.
aw; try apply: composable_fun_projs =>//; last by aw.
set (y:= (Vf (canon_proj r) x)).
have : (Vf (function_on_quotients r r' f) y
= (Vf (fun_on_quotients r r' (Lf f (substrate r) (substrate r'))))y).
have ys: inc y (quotient r) by rewrite /y; aw; fprops.
rewrite foqs_V // foqcs_V //; aw.
by apply: rep_i_sr.
by apply: lf_function.
rewrite - fun_on_quotient_pr5 // /y; aw.
by move => ->.
Qed.

Lemma compose_fun_proj_eq r f b:
equivalence r ->
compatible_with_equiv (Lf f (substrate r) b) r ->
lf_axiom f (substrate r) b ->
(function_on_quotient r f b) \co (canon_proj r) =
Lf f (substrate r) b.
Proof.
move=> er co ta.
apply: function_exten; try solve [ rewrite/function_on_quotient; aw].
- apply: compf_f; apply: composable_fun_proj=>//.
- apply: lf_function =>//.
- rewrite /function_on_quotient; aw.
by move =>x xs /=; rewrite compose_fun_proj_ev=>//; aw.
Qed.

Lemma compose_fun_proj_eq2 r r' f:
equivalence r ->
equivalence r' ->
lf_axiom f (substrate r) (substrate r') ->
compatible_with_equivs (Lf f (substrate r) (substrate r')) r r'->
(function_on_quotients r r' f) \co (canon_proj r) =
(canon_proj r') \co (Lf f (substrate r) (substrate r')).
Proof.
move=> er er' ta co.
have ffoq:function (function_on_quotients r r' f) by apply: foqs_f.
have fa: function (Lf f (substrate r) (substrate r')) by apply: lf_function.
have cn: (function_on_quotients r r' f) \coP (canon_proj r).
by split; fprops; rewrite/function_on_quotients; aw.
have cc: (canon_proj r') \coP (Lf f (substrate r) (substrate r')).
by split; fprops; aw.
apply: function_exten; try fct_tac; aw.
rewrite /function_on_quotients; aw.
move => x xs.
have aux: (inc (f x) (substrate r')) by apply: ta.
rewrite -compose_fun_proj_ev2 //; aw.
Qed.

Lemma compatible_ea f:
function f ->
compatible_with_equiv f (equivalence_associated f).
Proof.
move=> ff.
split=>//; first by rewrite graph_ea_substrate.
by move=> x x' => /(ea_relatedP ff) [_ _].
Qed.

Lemma ea_foq_fi f:
function f ->
injection (fun_on_quotient (equivalence_associated f) f).
Proof.
move=> ff.
set (g:= equivalence_associated f).
have eg: equivalence g by rewrite /g; apply: graph_ea_equivalence.
have sf:source f = substrate g by rewrite /g graph_ea_substrate.
split; first by apply: foqc_f.
have: source (fun_on_quotient g f) = (quotient g)
by rewrite/fun_on_quotient/section_canon_proj; aw.
move => -> x y xq yq.
rewrite foqc_V // foqc_V // => h.
move: (rep_i_sr eg xq) (rep_i_sr eg yq) => xs ys.
have: (related g (rep x) (rep y)) by apply/(ea_relatedP ff);rewrite sf.
by move/related_rr_P; apply.
Qed.

Lemma ea_foq_on_im_fb f:
function f ->
bijection (restriction2 (fun_on_quotient (equivalence_associated f) f)
(quotient (equivalence_associated f)) (range (graph f))).
Proof.
move=> ff.
set (g:= equivalence_associated f).
set (h:= fun_on_quotient g f).
have eg: equivalence g by rewrite /g; apply: graph_ea_equivalence.
have sfsg : source f = substrate g by rewrite /g graph_ea_substrate.
have ih: injection h by rewrite /h/g; apply: ea_foq_fi.
have fh: function h by fct_tac.
have qg: quotient g = source h.
by rewrite /h/fun_on_quotient/section_canon_proj; aw.
have aux: sub (quotient g) (source h) by rewrite qg.
suff rg: range (graph f) = image_by_fun h (quotient g).
rewrite rg qg restriction1_pr //.
apply: restriction1_fb; fprops.
set_extens t.
move /(range_fP ff) => [x xsf ->].
rewrite sfsg in xsf.
have gxq: inc (class g x) (quotient g) by fprops.
apply /(Vf_image_P fh aux); exists (class g x) => //.
have : (related g x (rep (class g x))) by apply: related_rep_class.
move / (ea_relatedP ff x (rep (class g x))) => [pa pb ->].
rewrite - (foqc_V eg) //;Wtac.
move /(Vf_image_P fh aux)=> [u us ->].
by rewrite foqc_V //; Wtac; rewrite sfsg; apply: rep_i_sr.
Qed.

Canonical decomposition of a function as a surjection, a bijection and an ionjection

Lemma canonical_decompositiona f (r:= equivalence_associated f):
function f ->
function ((restriction2 (fun_on_quotient r f)
(quotient r) (range (graph f)))
\co (canon_proj r)).
Proof.
move=> ff.
set (g:= canon_proj r).
set (a:=range (graph f)).
set (j:=canonical_injection a (target f)).
set (k:=restriction2 (fun_on_quotient r f) (quotient r) a).
have er: equivalence r by rewrite /r; apply: graph_ea_equivalence.
have sr: source f = substrate r by rewrite /r graph_ea_substrate.
have sat: sub a (target f) by apply: f_range_graph.
have fj: function j by apply: ci_f.
have ffo: function (fun_on_quotient r f) by apply: foqc_f.
have ra: restriction2_axioms (fun_on_quotient r f) (quotient r) a.
red;rewrite{2 3}/fun_on_quotient/section_canon_proj; aw;split => //.
move => x /(dirim_P) [z zq zj].
rewrite (Vf_pr ffo zj) foqc_V //; apply/range_fP => //.
exists (rep z) => //; rewrite sr; fprops.
have fk:function k by rewrite /k; apply: (proj31 (restriction2_prop ra)).
have ckg: k \coP g.
rewrite /k/g; split; fprops; rewrite /restriction2;aw.
fct_tac.
Qed.

Lemma canonical_decomposition f (r:= equivalence_associated f):
function f ->
f = (canonical_injection (range (graph f))(target f))
\co (restriction2 (fun_on_quotient r f) (quotient r) (range (graph f))
\co (canon_proj r)).
Proof.
move=> ff.
set (g:= canon_proj r).
set (a:=range (graph f)).
set (j:=canonical_injection a (target f)).
set (k:=restriction2 (fun_on_quotient r f) (quotient r) a).
have er: equivalence r by rewrite/r; apply: graph_ea_equivalence.
have sr: source f = substrate r by rewrite /r graph_ea_substrate.
have sat: sub a (target f) by rewrite /a ; apply: f_range_graph.
have fj:function j by rewrite /j;apply: ci_f.
have foq: function (fun_on_quotient r f) by apply: foqc_f.
have ra: restriction2_axioms (fun_on_quotient r f) (quotient r) a.
red;rewrite{2 3}/fun_on_quotient/section_canon_proj; aw;split => //.
move => x /(dirim_P) [z zq zj].
rewrite (Vf_pr foq zj) foqc_V //; apply/range_fP => //.
exists (rep z) => //; rewrite sr; fprops.
move: (proj31 (restriction2_prop ra)) => fr.
have fk:function k by rewrite /k.
have ckg:k \coP g.
rewrite /k/g; split; fprops; rewrite /restriction2;aw.
have sfg: source f = source g by rewrite /g; aw.
have fkg: function (k \co g) by fct_tac.
have cjkg: j \coP (k \co g).
by split => //; rewrite/j/k/canonical_injection/restriction2; aw.
apply: function_exten=>//; try fct_tac; aw.
by rewrite/j/canonical_injection; aw.
move=> x xsf.
move: (xsf); rewrite sr=> xsr; rewrite sfg in xsf.
have a1: (quotient r = target g) by rewrite /g;aw.
have W1q: inc (Vf g x) (quotient r) by Wtac; fct_tac.
have p1: inc (Vf k (Vf g x)) a.
have: (a = target k) by rewrite/k/restriction2; aw.
by move=> ->; Wtac; rewrite /k/restriction2; aw; ue.
aw.
rewrite /j ci_V // /k restriction2_V // foqc_V // /g; aw.
have : (related r x (rep (class r x))) by apply: related_rep_class.
by move/(ea_relatedP ff) => [_ _].
Qed.

Lemma surjective_pr7 f:
surjection f ->
canonical_injection (range (graph f))(target f) = identity (target f).
Proof.
move=> sf; rewrite surjective_pr3 //.
Qed.

Lemma canonical_decomposition_surj f (r:= equivalence_associated f):
surjection f ->
f = (restriction2 (fun_on_quotient r f) (quotient r) (target f))
\co (canon_proj r).
Proof.
move=> sf.
move: (surjective_pr7 sf) (surjective_pr3 sf) => p1 p2.
move: sf => [sf _].
move: (canonical_decomposition sf); simpl; rewrite p1 p2; fold r.
set (g:= (restriction2 (fun_on_quotient r f) (quotient r) (target f))).
set (h:= g \co (canon_proj r)).
move=> ->.
have: (target f = target h) by rewrite /h/g/restriction2; aw.
move =>->; apply: compf_id_l.
by rewrite /h/g/r -p2; apply: canonical_decompositiona.
Qed.

Lemma canonical_decompositionb f (r:= equivalence_associated f):
function f ->
restriction2 (fun_on_quotient r f) (quotient r) (target f) =
(fun_on_quotient r f).
Proof.
move=> ff.
set (k:=restriction2 (fun_on_quotient r f) (quotient r) (target f)).
have er: equivalence r by rewrite /r; apply: graph_ea_equivalence.
have sf: source f = substrate r by rewrite /r graph_ea_substrate.
have ffoq: function (fun_on_quotient r f) by apply: foqc_f.
have ra: restriction2_axioms (fun_on_quotient r f) (quotient r) (target f).
red;rewrite{2 3}/fun_on_quotient/section_canon_proj; aw;split => //.
move => x /(dirim_P) [z zq zj].
rewrite (Vf_pr ffoq zj) foqc_V //; Wtac; rewrite sf;fprops.
have fk: function k by rewrite /k; apply: (proj31(restriction2_prop ra)).
have sk: source k = quotient r by rewrite /k/restriction2; aw.
apply: function_exten =>//.
- rewrite /restriction2/fun_on_quotient/section_canon_proj; aw.
- by rewrite/k/restriction2/fun_on_quotient; aw.
- move=> x; rewrite sk => xq; rewrite restriction2_V //.
Qed.

Lemma canonical_decomposition_surj2 f (r:= equivalence_associated f):
surjection f ->
f = (fun_on_quotient r f) \co (canon_proj r).
Proof.
move=> sf; move: (canonical_decomposition_surj sf).
move: sf => [ff _]; move: (canonical_decompositionb ff).
by simpl; move => -> <-.
Qed.

## EII-6-6 Inverse image of an equivalence relation; induced equivalence relation

We have an equivalence on the source induced by the equivalence on the target

Definition inv_image_relation f r :=
equivalence_associated (canon_proj r \co f).

Definition iirel_axioms f r :=
[/\ function f, equivalence r & substrate r = target f].

Lemma iirel_f f r:
iirel_axioms f r -> function (canon_proj r \co f).
Proof.
move=> [ff er tf]; fct_tac; [ fprops | aw].
Qed.

Lemma iirel_relation f r:
iirel_axioms f r -> equivalence (inv_image_relation f r).
Proof.
move=> ia; rewrite /inv_image_relation.
by apply: graph_ea_equivalence; apply: iirel_f.
Qed.

Lemma iirel_substrate f r:
iirel_axioms f r -> substrate (inv_image_relation f r) = source f.
Proof.
move=> ia; rewrite /inv_image_relation graph_ea_substrate; aw.
by apply: iirel_f.
Qed.

Lemma iirel_relatedP f r: iirel_axioms f r -> forall x y,
(related (inv_image_relation f r) x y <->
[/\ inc x (source f), inc y (source f) & related r (Vf f x) (Vf f y)]).
Proof.
move=> ia.
have fc : function ((canon_proj r) \co f) by apply: iirel_f.
move: ia => [ff er tf].
rewrite /inv_image_relation.
have cct: ((canon_proj r) \coP f) by split; fprops; aw.
have iWs: forall u, inc u (source f) -> inc (Vf f u) (substrate r).
by move=> u; rewrite tf; fprops.
move => x y.
split.
move/(ea_relatedP fc); rewrite compf_s; move => [xs ys].
move: (iWs _ xs) (iWs _ ys)=> p1 p2; aw => p3;split => //.
by apply/(related_equiv_P er).
move => [pa pb pc].
apply /(ea_relatedP fc); split; aw; fprops;apply: class_eq1;fprops.
Qed.

Lemma iirel_classP f r:
iirel_axioms f r -> forall x,
(classp (inv_image_relation f r) x <->
exists y, [/\ classp r y,
nonempty (y \cap (range (graph f)))
& x = inv_image_by_fun f y]).
Proof.
move=> ia; move: (ia) => [ff er tf].
move :(iirel_relation ia) => pa.
move: (iirel_substrate ia) => sir.
move => x; split.
move => clix; move: (clix)=> []; rewrite sir => sa sb.
have iWs: inc (Vf f (rep x)) (substrate r) by rewrite tf; Wtac.
exists (class r (Vf f (rep x))); split.
- by apply: class_class.
- exists (Vf f (rep x)); apply: setI2_i; [ fprops | Wtac ].
- set_extens t.
rewrite {1} sb => /(class_P pa) /(iirel_relatedP ia) [ysf tsf ryt].
apply/(iim_fun_P); exists (Vf f t); [ by apply /class_P | Wtac].
move /(iim_fun_P) => [u uc Jg]; rewrite sb; apply /(class_P pa).
apply /(iirel_relatedP ia);split => //; first by Wtac.
by rewrite - (Vf_pr ff Jg); apply /(class_P er).
move=> [y [cy [u ui] xi]]; move: cy => [c1 c2].
move: ui => /setI2_P [uy] /(range_fP ff) [v vsf uw].
have vx: inc v x by rewrite xi; apply /(iim_fun_P); ex_tac; rewrite uw; Wtac.
have qc: related r (rep y) (Vf f v) by rewrite - uw;apply /class_P => //; ue.
suff : x = class (inv_image_relation f r) v.
move => ->; apply: class_class => //; ue.
set_extens t.
rewrite xi => /(iim_fun_P) [z' z'y Jg2].
apply/(class_P pa); apply /(iirel_relatedP ia);split => //; first by Wtac.
have pb: related r (rep y) z' by apply /(class_P er); ue.
rewrite - (Vf_pr ff Jg2); equiv_tac.
move /(class_P pa) => /(iirel_relatedP ia) [_ qa qb].
rewrite xi; apply /(iim_fun_P); exists (Vf f t) => //; last by Wtac.
rewrite c2; apply /class_P => //; equiv_tac.
Qed.

Equivalence induced on a subset of the substrate

Definition induced_relation r a :=
inv_image_relation (canonical_injection a (substrate r)) r.

Definition induced_rel_axioms r a :=
equivalence r /\ sub a (substrate r).
Definition canonical_foq_induced_rel r a :=
restriction2 (fun_on_quotients (induced_relation r a) r
(canonical_injection a (substrate r)))
(quotient (induced_relation r a))
(image_by_fun (canon_proj r) a).

Section InducedRelation.
Variables (r a: Set).
Hypothesis ira: induced_rel_axioms r a.

Lemma induced_rel_iirel_axioms:
iirel_axioms (canonical_injection a (substrate r)) r.
Proof.
move: ira=> [er ar]; split; fprops.
rewrite/canonical_injection; aw.
Qed.

Lemma induced_rel_equivalence:
equivalence (induced_relation r a).
Proof. by apply: iirel_relation; apply: induced_rel_iirel_axioms. Qed.

Lemma induced_rel_substrate:
substrate (induced_relation r a) = a.
Proof.
rewrite /induced_relation iirel_substrate.
rewrite/canonical_injection; aw.
by apply: induced_rel_iirel_axioms.
Qed.

Lemma induced_rel_relatedP u v:
related (induced_relation r a) u v <->
[/\ inc u a, inc v a & related r u v].
Proof.
move: (induced_rel_iirel_axioms) => iia.
move: ira=> [er ar].
split.
move/(iirel_relatedP iia); rewrite lf_source; move => [pa pb].
rewrite ci_V //ci_V //.
move => [ua va ruv];apply/(iirel_relatedP iia); rewrite lf_source; split => //.
rewrite ci_V //ci_V //.
Qed.

Lemma induced_rel_classP x:
(classp (induced_relation r a) x <->
exists y, [/\ classp r y, nonempty (y \cap a) & x = (y \cap a)]).
Proof.
move: (induced_rel_iirel_axioms) => iia.
move: (ira)=> [er ar].
have p: forall y, inv_image_by_fun (canonical_injection a (substrate r)) y
= y \cap a.
rewrite/inv_image_by_fun /canonical_injection; aw.
rewrite - diagonal_is_identity.
move=> y;set_extens t.
move /iim_graph_P => [u uy] /diagonal_pi_P [pa pb]; apply/setI2_P.
split => //; ue.
move /setI2_P => [ty ta]; apply /iim_graph_P; ex_tac.
by apply /diagonal_pi_P.
split.
move/(iirel_classP iia); rewrite (ci_range ar); move => [y]; rewrite p.
by move => [pa pb pc]; exists y.
move => [y [cy ney xi]]; apply/(iirel_classP iia); exists y.
split; [ done | by rewrite (ci_range ar) | by rewrite p ].
Qed.

Lemma compatible_injection_induced_rel:
compatible_with_equivs (canonical_injection a (substrate r))
(induced_relation r a) r.
Proof.
move: (ira)=> [er sa].
apply: compatible_with_pr2 =>//.
- by apply: induced_rel_equivalence.
- by apply: ci_f.
- rewrite/canonical_injection; aw.
- by rewrite/canonical_injection;aw; rewrite induced_rel_substrate.
- move=> x y /induced_rel_relatedP [xa ya rxy]; rewrite ci_V // ci_V //.
Qed.

Lemma foq_induced_rel_fi:
injection (fun_on_quotients (induced_relation r a) r
(canonical_injection a (substrate r))).
Proof.
move: (ira)=> [er sa].
set (f := (canonical_injection a (substrate r))).
move: (induced_rel_equivalence) =>ei.
have ff: function f by rewrite /f;apply: ci_f.
have tf: target f = substrate r by rewrite /f /canonical_injection; aw.
have sf: source f = substrate (induced_relation r a).
rewrite /f induced_rel_substrate // /canonical_injection; aw.
split; first by apply: foqcs_f.
have : (source (fun_on_quotients (induced_relation r a) r f) =
quotient (induced_relation r a) ).
rewrite/fun_on_quotients /section_canon_proj; aw.
move=>->.
move=> x y xs ys /=; rewrite foqcs_V // foqcs_V //.
have eq: a = substrate (induced_relation r a) by rewrite induced_rel_substrate.
have rxa: inc (rep x) a by rewrite eq; fprops.
have rya: inc (rep y) a by rewrite eq; fprops.
rewrite ci_V // ci_V // => cc.
have: (related (induced_relation r a) (rep x) (rep y)).
apply /induced_rel_relatedP; split => //; apply/ related_equiv_P => //.
by split => //; apply: sa.
by move /(related_rr_P ei xs ys).
Qed.

Lemma foq_induced_rel_image :
image_by_fun (fun_on_quotients (induced_relation r a) r
(canonical_injection a (substrate r))) (quotient (induced_relation r a))
= image_by_fun (canon_proj r) a.
Proof.
move: (ira)=> [er sa].
set (ci := canonical_injection a (substrate r)).
have fci: function ci by apply: ci_f.
have sci: source ci = substrate (induced_relation r a).
by rewrite induced_rel_substrate /ci /canonical_injection; aw.
set (f:= fun_on_quotients (induced_relation r a) r ci).
move: (foq_induced_rel_fi); rewrite -/ci -/f; move => inf.
have ff: function f by fct_tac.
have ei: equivalence(induced_relation r a) by apply: induced_rel_equivalence.
have gf: sgraph (graph f) by fprops.
have tci: target ci = substrate r by rewrite /ci/canonical_injection; aw.
have fc: function (canon_proj r) by fprops.
set_extens x.
move/dirim_P => [y yq Jg].
move: (yq) => /(setQ_P ei) /induced_rel_classP [z [cz nei yi]].
move: (Vf_pr ff Jg); rewrite foqcs_V // => cx.
have: (inc (rep y) (z \cap a)) by rewrite -yi; apply: (setQ_repi ei).
move /setI2_P => [ryz rya]; rewrite ci_V // in cx.
move: nei=> [u] /setI2_P [uz ua]; apply /dirim_P; ex_tac.
have ryu: (related r (rep y) u) by apply/(in_class_relatedP er); exists z.
move: (sa _ ua) => us.
have ->: x = Vf (canon_proj r) u by aw; rewrite cx; apply: class_eq1.
by Wtac; aw.
move/dirim_P => [y ya Jg]; move: (sa _ ya)=> yu.
move: (Vf_pr fc Jg); aw; move => cx.
have yx: (inc y x) by rewrite cx; fprops.
set (z:= x \cap a).
have ney: nonempty z by rewrite /z; exists y; fprops.
have zq: inc z (quotient (induced_relation r a)).
apply/(setQ_P ei); apply/ induced_rel_classP; exists x; split => //.
rewrite cx; apply: class_class =>//.
apply /dirim_P; ex_tac.
have: (inc (rep z) (x \cap a)) by apply: (setQ_repi ei).
move /setI2_P => [rzx rza].
have -> : x = Vf f (x \cap a).
rewrite /f foqcs_V// ci_V // -/z cx.
have: (related r y (rep z)).
apply /(in_class_relatedP er); exists x; split => //.
by rewrite cx; apply: class_class.
by move /(related_equiv_P er) => [_ _].
rewrite -/z; Wtac; rewrite/f/fun_on_quotients/section_canon_proj; aw.
Qed.

Lemma canonical_foq_induced_rel_fb:
bijection (canonical_foq_induced_rel r a).
Proof.
set ci:= (canonical_injection a (substrate r)).
have sf: source (fun_on_quotients (induced_relation r a) r ci) =
quotient (induced_relation r a).
rewrite /fun_on_quotients/section_canon_proj; aw.
have ra:restriction2_axioms
(fun_on_quotients (induced_relation r a) r ci)
(quotient (induced_relation r a)) (image_by_fun (canon_proj r) a).
move: (foq_induced_rel_fi) => ifoq.
split; first by fct_tac.
by rewrite /fun_on_quotients/section_canon_proj; aw.
rewrite /fun_on_quotients; aw; move=> t ts; move: ira=> [er ar].
apply: (sub_im_canon_proj_quotient ar ts).
by rewrite foq_induced_rel_image; fprops.
split; first by apply: restriction2_fi=>//; apply: foq_induced_rel_fi.
apply: restriction2_fs=>//.
by rewrite - foq_induced_rel_image /image_of_fun sf.
Qed.

End InducedRelation.

## EII-6-7 Quotients of equivalence relations

Definition of a finer equivalence

Definition finer_equivalence s r:=
forall x y, related s x y -> related r x y.

Definition finer_axioms s r :=
[/\ equivalence s, equivalence r & substrate r = substrate s].

Lemma finest_equivalence r:
equivalence r -> finer_equivalence (diagonal (substrate r)) r.
Proof. move=> er x t /diagonal_pi_P [xs <-]; equiv_tac. Qed.

Lemma coarsest_equivalence r:
equivalence r -> finer_equivalence r (coarse (substrate r)).
Proof.
move=> er x y;rewrite coarse_related; move /(related_equiv_P er) => [pa pb _].
by split.
Qed.

Section FinerEquivalence.
Variable (r s: Set).
Hypothesis fa: finer_axioms s r.

Lemma finer_sub_equivP:
(finer_equivalence s r <-> sub s r).
Proof.
move: fa => [es er srss].
split.
move=> fe t ts.
have gs: (sgraph s) by fprops.
by rewrite -(gs _ ts); apply: fe; red; rewrite (gs _ ts).
move => t x y; rewrite /related; apply: t.
Qed.

Lemma finer_sub_equivP2:
(finer_equivalence s r <->
(forall x, exists y, sub (class s x) (class r y))).
Proof.
split.
move=> fsr x; exists x.
by move=> u /(class_P es) h; apply /(class_P er); apply:fsr.
move=> h x y rxy.
move:(h x)=> [z sxz].
have xs: inc x (substrate s) by substr_tac.
have ixc: inc x (class s x) by fprops.
have ys: inc y (class s x) by apply /(class_P es).
move: (sxz _ ixc)(sxz _ ys) => /(class_P er) pa /(class_P er) pb; equiv_tac.
Qed.

Lemma finer_sub_equivP3:
(finer_equivalence s r <-> forall y, saturated s (class r y)).
Proof.
split.
move=> fe y.
apply/(saturatedP es); first by rewrite - srss; apply: sub_class_substrate.
move => z zc; move: fe => /finer_sub_equivP2 fe.
move/(class_P er): (zc) => r1.
have zs: (inc z (substrate s)) by rewrite - srss; substr_tac.
move: (fe z)=> [x cxy].
suff: (class r x = class r y) by move=> <-.
have: (inc z (class r x)) by apply: cxy; fprops.
move => /(class_P er) => r2.
apply: (class_eq1 er); equiv_tac.
move=> p x y re; move: (p x).
move: (sub_class_substrate er (x:=x)); rewrite srss => aux.
move /(saturatedP es aux) => h.
have p1: inc x (substrate s) by substr_tac.
have p2: inc x (class r x) by rewrite - srss in p1;fprops.
have p3: inc y (class s x) by apply/class_P.
by move:((h _ p2) _ p3) => /(class_P er).
Qed.

Lemma compatible_with_finer:
finer_equivalence s r ->
compatible_with_equiv (canon_proj r) s.
Proof.
move :fa=> [es er fh] fe.
split; fprops; aw => x y rxy.
move: (fe _ _ rxy) => /(related_equiv_P er) [p1 p2 h]; aw.
Qed.

Lemma foq_finer_f:
finer_equivalence s r -> function(fun_on_quotient s (canon_proj r)).
Proof. move : fa=> [es er fh] fe; apply: foqc_f; fprops; aw. Qed.

Lemma foq_finer_V x:
finer_equivalence s r -> inc x (quotient s) ->
Vf (fun_on_quotient s (canon_proj r)) x = class r (rep x).
Proof.
move: fa => [es er fh] fe xq; rewrite foqc_V //; aw; fprops.
rewrite fh; fprops.
Qed.

Lemma foq_finer_fs:
finer_equivalence s r -> surjection(fun_on_quotient s (canon_proj r)).
Proof.
move=> fe.
split; first by apply: foq_finer_f.
move: fa=> [es er ss] y yt.
have yq:inc y (quotient r) by move:yt; rewrite/fun_on_quotient; aw.
have rys:(inc (rep y) (substrate r)) by fprops.
have sfo:source (fun_on_quotient s (canon_proj r)) = (quotient s).
by rewrite /fun_on_quotient/section_canon_proj; aw.
set(x := class s (rep y)).
have xq: (inc x (quotient s))by rewrite /x; rewrite ss in rys; fprops.
rewrite sfo; ex_tac.
rewrite foq_finer_V //.
have cry: (class r (rep y) = y) by apply: class_rep=>//.
rewrite -cry; apply: class_eq1 =>//;apply: fe.
apply: (symmetricity_e es).
apply /(class_P es); apply: rep_i; rewrite /x.
apply /(setQ_ne es); apply /(setQ_P es); apply/(class_class es); ue.
Qed.

End FinerEquivalence.

Quotient of a relation by a finer one

Definition quotient_of_relations r s:=
equivalence_associated (fun_on_quotient s (canon_proj r)).

Lemma cqr_aux s x y u:
equivalence s -> sub y (substrate s) ->
x = image_by_fun (canon_proj s) y ->
(inc u x <-> (exists2 v, inc v y & u = class s v)).
Proof.
move=> es ys xi.
split.
rewrite xi /image_by_fun => ui.
move:(sub_im_canon_proj_quotient ys ui) => uqs.
move: ui=> /dirim_P [z zy Jg]; move : (ys _ zy) => zs.
move: Jg => /(rel_gcp_P es zs uqs) zu.
ex_tac.
have aux: (class s (rep u) = u) by (apply: class_rep).
have : related s (rep u) z by apply /(class_P es); ue.
move /(related_equiv_P es) => [_ _]; ue.
have fc: function (canon_proj s) by fprops.
move=> [v vy ->]; move: (ys _ vy) => vs; rewrite xi.
apply/dirim_P; ex_tac; apply /(rel_gcp_P es vs); fprops.
Qed.

Section QuotientRelations.
Variables (r s: Set).
Hypotheses (fa:finer_axioms s r) (fe: finer_equivalence s r).

Lemma quo_rel_equivalence:
equivalence (quotient_of_relations r s).
Proof.
apply: graph_ea_equivalence.
move: (foq_finer_fs fa fe)=> inj; fct_tac.
Qed.

Lemma quo_rel_substrate:
substrate (quotient_of_relations r s) = (quotient s).
Proof.
rewrite /quotient_of_relations graph_ea_substrate //.
rewrite/fun_on_quotient/section_canon_proj; aw.
apply:(surj_function (foq_finer_fs fa fe)).
Qed.

Lemma quo_rel_relatedP x y:
related (quotient_of_relations r s) x y <->
[/\ inc x (quotient s), inc y (quotient s) & related r (rep x) (rep y)].
Proof.
have foq: function (fun_on_quotient s (canon_proj r)).
by apply:(surj_function (foq_finer_fs fa fe)).
have fc: function (canon_proj r) by fprops.
have sf: source (fun_on_quotient s (canon_proj r)) = quotient s.
rewrite /fun_on_quotient/section_canon_proj; aw.
move: fa =>[es er ss].
have sc:source (canon_proj r) = substrate s by aw.
rewrite/quotient_of_relations; split.
move /(ea_relatedP foq); rewrite sf.
move => [xq yq h];split => //.
have p1: (inc (rep x) (substrate r)) by rewrite ss; fprops.
have p2: (inc (rep y) (substrate r)) by rewrite ss; fprops.
move: h;rewrite foqc_V // foqc_V //; aw => h.
by apply/(related_equiv_P er).
move => [xs ys rr]; apply /(ea_relatedP foq); rewrite sf; split => //.
move/(related_equiv_P er): rr => [p1 p2 p3].
rewrite foqc_V // foqc_V //; aw.
Qed.

Lemma quo_rel_related_bisP x y:
inc x (substrate s) -> inc y (substrate s) ->
(related (quotient_of_relations r s) (class s x) (class s y)
<-> related r x y).
Proof.
move=> xs ys.
move: fa =>[es er ss].
have p1:(related r x (rep (class s x))) by apply: fe; apply: related_rep_class.
have p2:(related r y (rep (class s y))) by apply: fe; apply: related_rep_class.
split.
move/quo_rel_relatedP=> [ixs iys rxy].
apply: (@transitivity_e _ (rep (class s x))) =>//.
apply: (@transitivity_e _ (rep (class s y))) =>//.
equiv_tac.
move=> h; apply/quo_rel_relatedP; split; fprops.
apply: (@transitivity_e _ x) =>//.
apply: symmetricity_e=>//.
equiv_tac.
Qed.

Lemma quo_rel_class_bisP x:
( inc x (quotient (quotient_of_relations r s)) <->
exists2 y, inc y (quotient r) & x = image_by_fun (canon_proj s) y).
Proof.
move: (quo_rel_equivalence)=> eq.
move: (fa) =>[es er ss].
have fc: function (canon_proj s) by fprops.
move: quo_rel_substrate => aux.
split.
move/(setQ_P eq) => cq; move: (cq)=> [ca cb].
set(y:= Zo (substrate r) (fun v => exists2 u, inc u x & inc v u)).
move: (rep_in_class eq cq) => rxx.
move: (sub_class_sr eq cq); rewrite aux => scx.
move: (scx _ rxx) => repxq; move: (setQ_repi es repxq) => rrx.
have rrsr: inc (rep (rep x)) (substrate r).
by rewrite ss; apply: (inc_in_setQ_sr es rrx repxq).
have ys: sub y (substrate r) by rewrite /y ss; apply: Zo_S.
have cy: y = class r (rep (rep x)).
set_extens u.
move /Zo_P => [usr [t tx ut]]; apply /(class_P er).
move: tx; rewrite {1} cb; move /(class_P eq).
move /quo_rel_relatedP => [uq vq ruv].
apply: (@transitivity_e _ (rep t))=>//.
by apply: fe; apply/(class_P es); rewrite class_rep.
move /(class_P er) => rab.
move: (arg2_sr rab) => bsr; move: rab.
rewrite ss in rrsr bsr.
move /(quo_rel_related_bisP rrsr bsr).
move /(class_P eq); rewrite (class_rep es repxq) - cb => h.
apply: Zo_i; [ ue | ex_tac].
exists y.
rewrite cy; apply/(setQ_P er); apply: (class_class er rrsr).
set (xx := image_by_fun (canon_proj s) y).
rewrite ss in ys.
set_extens u.
move => ux; move: (scx _ ux) => aux1;
apply/ (cqr_aux u es ys (refl_equal xx)).
move: (setQ_repi es aux1) => ur.
exists (rep u); first by apply: Zo_i; [ ue | ex_tac].
by symmetry; apply: class_rep.
move/ (cqr_aux u es ys (refl_equal xx)) => [v vy ->].
move: vy => /Zo_P [vr [w wx vw]].
have wq: (inc w (quotient s)) by apply: scx.
move: (related_rep_in_class es wq vw).
move => /(related_equiv_P es) [rws vd].
by move: (class_rep es wq) => -> <-.
move => [y] /(setQ_P er) cy xd; apply /(setQ_P eq).
move: (rep_in_class er cy) => ryy.
move: (sub_class_sr er cy); rewrite ss => ysr.
have sxq: sub x (quotient s).
by move => t tx; apply: (sub_im_canon_proj_quotient ysr); ue.
have nex: nonempty x by rewrite xd; apply: nonempty_image => //; aw; ex_tac.
move: (rep_i nex) => rrx.
move: (rrx) => /(cqr_aux (rep x) es ysr xd) [v vy asv].
split; [ue | set_extens t].
move => tx; apply/(class_P eq).
move/ (cqr_aux t es ysr xd): tx => [w wy asw].
rewrite asv asw ; apply /quo_rel_related_bisP; try apply: ysr => //.
by apply /(in_class_relatedP er); exists y.
move /(class_P eq) => /quo_rel_relatedP [aq bq rab].
apply/(cqr_aux t es ysr xd); exists (rep t); last by symmetry; apply: class_rep.
move: (rel_in_class er cy vy) => s1.
apply: (rel_in_class2 er cy).
apply: (transitivity_e er s1).
apply: (@transitivity_e _ (rep (rep x)))=>//.
apply: fe; apply /(in_class_relatedP es).
exists (rep x); split.
- by rewrite asv; apply: class_class =>//; apply: ysr.
- by rewrite asv; apply: inc_itself_class =>//; apply: ysr.
- by apply: (setQ_repi es).
Qed.

Lemma quotient_canonical_decomposition
(f := fun_on_quotient s (canon_proj r))
(qr := quotient_of_relations r s):
f = (fun_on_quotient qr f) \co (canon_proj qr).
Proof.
have sj: (surjection f) by rewrite /f;apply: foq_finer_fs.
apply: (canonical_decomposition_surj2 sj).
Qed.

End QuotientRelations.

Lemma quo_rel_pr s t
(r := inv_image_relation (canon_proj s) t):
equivalence s -> equivalence t -> substrate t = quotient s ->
t = quotient_of_relations r s.
Proof.
move=> es et st.
have iaa: iirel_axioms (canon_proj s) t.
by split;fprops; aw.
have er: equivalence r by rewrite /r; apply: iirel_relation.
have fa:finer_axioms s r.
by split=>//; rewrite /r iirel_substrate //; aw.
have fe: finer_equivalence s r.
move => x y sxy; apply / iirel_relatedP => //.
have xs: inc x (substrate s) by substr_tac.
have ys: inc y (substrate s) by substr_tac.
aw;split=>//.
have -> : (class s x =class s y) by apply: class_eq1.
by apply: reflexivity_e=>//; rewrite st; fprops.
have eq: (equivalence (quotient_of_relations r s)).
apply: quo_rel_equivalence=> //.
apply/sgraph_exten; fprops.
move=> u v; split.
move => tuv; apply/quo_rel_relatedP => //.
have us: (inc u (quotient s)) by rewrite - st; substr_tac.
have vs: (inc v (quotient s)) by rewrite - st; substr_tac.
have rus: (inc (rep u) (substrate s)) by fprops.
have rvs: (inc (rep v) (substrate s)) by fprops.
split=>//; apply /iirel_relatedP => //.
by split; aw; rewrite class_rep // class_rep.
move/(quo_rel_relatedP fa fe) => [uq vq].
move /(iirel_relatedP iaa) ;rewrite canon_proj_s.
move=> [ru rv]; aw; rewrite class_rep // class_rep //.
Qed.

## EII-6-8 Product of two equivalence relations

Definition prod_of_relation r r':=
graph_on
(fun x y=> inc (J(P x)(P y)) r /\ inc (J(Q x)(Q y)) r')
((substrate r) \times (substrate r')).

Lemma substrate_prod_of_rel r r':
equivalence r ->equivalence r' ->
substrate (prod_of_relation r r') = (substrate r)\times (substrate r').
Proof.
move => pa pb; rewrite graph_on_sr //.
by move => a /setX_P [_ ta tb]; split; equiv_tac.
Qed.

Lemma order_product2_sr1 f g:
preorder f -> preorder g ->
substrate (prod_of_relation f g) = (substrate f) \times (substrate g).
Proof.
move=> [_ rf _] [_ rg _ ]; apply: graph_on_sr.
move => a /setX_P [pa Ps Qs].
by split; [apply: rf | apply: rg].
Qed.

Lemma order_product2_sr f g:
order f -> order g ->
substrate (prod_of_relation f g) = (substrate f) \times (substrate g).
Proof.
move=> orf og; apply: order_product2_sr1; by apply: order_preorder.
Qed.

Lemma equivalence_prod_of_rel r r':
equivalence r -> equivalence r' ->
equivalence (prod_of_relation r r').
Proof.
move => pa pb.
apply: equivalence_from_rel; split.
move => a b [ra rb];split; equiv_tac.
move => a b c [ra rb] [rc rd];split; equiv_tac.
Qed.

Lemma order_product2_preorder f g:
preorder f -> preorder g -> preorder (prod_of_relation f g).
Proof.
move => [_ rf tf] [_ rg tg]; apply: preorder_from_rel; red; split.
move => x y z[pxy qxy][pyz qyz].
split; [ apply: tf pxy pyz | apply: tg qxy qyz].
move=> x y [pxy qxy]; split; split; try apply: rf; try apply: rg; substr_tac.
Qed.

Lemma order_product2_or f g:
order f -> order g -> order (prod_of_relation f g).
Proof.
move=> orf og.
have [gr rr tr]: preorder (prod_of_relation f g).
by apply: order_product2_preorder; apply: order_preorder.
move: orf og => [_ rf tf af] [_ rg tg ag].
split => //.
move => x y /Zo_P [pa [pb pc]] /Zo_P [qa [qb qc]].
move: pb pc qb qc; aw => pb pc qb qc.
move: pa => /setX_P [_ ]; aw ;move /setX_P => [px _ _] /setX_P [py _ _].
apply: pair_exten => //; [exact: (af _ _ pb qb) | exact: (ag _ _ pc qc)].
Qed.

Lemma prod_of_rel_P r r' a b:
related (prod_of_relation r r') a b <->
[/\ pairp a, pairp b, related r (P a) (P b) & related r' (Q a) (Q b)].
Proof.
rewrite /related /prod_of_relation; split.
by move /graph_on_P0 => [] /setX_P [pa _ _] /setX_P [pb _ _] [pc pd].
move => [pa pb pc pd]; apply /graph_on_P0;split => //;
apply/setX_P; split => //; substr_tac.
Qed.

Lemma related_prod_of_relP1 r r' x x' v:
related (prod_of_relation r r') (J x x') v <->
(exists y y', [/\ v = J y y', related r x y & related r' x' y']).
Proof.
split.
move /prod_of_rel_P; aw; move=> [_ pb pc pd].
by exists (P v); exists (Q v).
move=> [y [y' [eq r1 r2]]]; rewrite eq; apply /prod_of_rel_P; aw; split; fprops.
Qed.

Lemma related_prod_of_relP2 r r' x x' v:
(related (prod_of_relation r r') (J x x') v <->
inc v ((im_of_singleton r x) \times (im_of_singleton r' x'))).
Proof.
split.
move=> /related_prod_of_relP1 [y [y' [eq r1 r2]]]; rewrite eq.
by apply/setXp_i; apply /dirim_set1_P.
move /setX_P => [pv /dirim_set1_P j1r /dirim_set1_P j2r].
by apply/related_prod_of_relP1;exists (P v); exists (Q v);aw.
Qed.

Lemma class_prod_of_relP2 r r':
equivalence r -> equivalence r' -> forall x,
(classp (prod_of_relation r r') x <->
exists u v, [/\ classp r u, classp r' v & x = u \times v]).
Proof.
move=> er er'.
move:(equivalence_prod_of_rel er er')=> er''.
split.
move => cx; move: (cx) => [ca cb].
move: (ca) ;rewrite (substrate_prod_of_rel er er') => /setX_P [pa pb pc].
exists (class r (P (rep x))); exists (class r' (Q (rep x))); split => //.
by apply: (class_class er).
by apply: (class_class er').
set_extens v.
rewrite {1} cb; move /(class_P er'') => /prod_of_rel_P [qa qb qc qd].
apply /setX_P => //; split => //; apply /class_P => //.
move /setX_P => [qa qb qc]; rewrite cb; apply /(class_P er'').
by apply /prod_of_rel_P;split => //; apply /class_P.
move=> [u [v [pa pb pe]]].
move: (pa)(pb) => [pc _] [pd _].
suff : x = class (prod_of_relation r r') (J (rep u) (rep v)).
move => ->; apply: (class_class er'').
by rewrite (substrate_prod_of_rel er er'); apply: