# Library mathcomp.ssreflect.generic_quotient

(* (c) Copyright 2006-2016 Microsoft Corporation and Inria.
Distributed under the terms of CeCILL-B.                                  *)

From HB Require Import structures.
From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice.
From mathcomp Require Import seq fintype.

Quotient Types NB: See CONTRIBUTING.md for an introduction to HB concepts and commands. Provided a base type T, this files defines an interface for quotients Q of the type T with explicit functions for canonical surjection (\pi : T -> Q) and for choosing a representative (repr : Q -> T). It then provides a helper to quotient T by a decidable equivalence relation (e : rel T) if T is a choiceType (or encodable as a choiceType modulo e). Reference: Cyril Cohen, Pragmatic Quotient Types in Coq, ITP 2013

### Generic Quotienting ***

quotType T == the type of quotient types based on T The HB class is called Quotient. The quotType interface supports these operations (in quotient_scope): \pi_Q x == the class in Q of the element x of T \pi x == the class of x where Q is inferred from the context repr c == canonical representative in T of the class c x = y % [mod Q] := \pi_Q x = \pi_Q y <-> x and y are equal modulo Q x <> y % [mod Q] := \pi_Q x <> \pi_Q y x == y % [mod Q] := \pi_Q x == \pi_Q y x != y % [mod Q] := \pi_Q x != \pi_Q y The quotient_scope is delimited by %qT, The most useful lemmas are piE and reprK. List of factories: isQuotient.Build T Q (reprK : cancel repr pi) == builds the quotient whose canonical surjection function is (pi : T -> Q) and whose representative selection function is repr

### Morphisms ***

One may declare existing functions and predicates as liftings of some morphisms for a quotient. PiMorph1 pi_f == where pi_f : {morph \pi : x / f x >-> fq x} declares fq : Q -> Q as the lifting of f : T -> T PiMorph2 pi_g == idem with pi_g : {morph \pi : x y / g x y >-> gq x y} PiMono1 pi_p == idem with pi_p : {mono \pi : x / p x >-> pq x} PiMono2 pi_r == idem with pi_r : {morph \pi : x y / r x y >-> rq x y} PiMorph11 pi_f == idem with pi_f : {morph \pi : x / f x >-> fq x} where fq : Q -> Q' and f : T -> T'. PiMorph eq == Most general declaration of compatibility, /!\ use with caution /!\ One can use the following helpers to build the liftings which may or may not satisfy the above properties (but if they do not, it is probably not a good idea to define them): lift_op1 Q f := lifts f : T -> T lift_op2 Q g := lifts g : T -> T -> T lift_fun1 Q p := lifts p : T -> R lift_fun2 Q r := lifts r : T -> T -> R lift_op11 Q Q' f := lifts f : T -> T' There is also the special case of constants and embedding functions that one may define and declare as compatible with Q using: lift_cst Q x := lifts x : T to Q PiConst c := declare the result c of the previous construction as compatible with Q lift_embed Q e := lifts e : R -> T to R -> Q PiEmbed f := declare the result f of the previous construction as compatible with Q

### Quotients that have an eqType structure ***

Having a canonical (eqQuotType e) structure enables piE to replace terms of the form (x == y) by terms of the form (e x' y') if x and y are canonical surjections of some x' and y'. eqQuotType e == the type of quotients types on T which mirror the equivalence relation (e : rel T) the HB class is called EqQuotient. The most useful property is that an eqQuotType is an eqType. List of factories: isEqQuotient.Build T e Q m == builds an (eqQuotType e) structure on Q from the morphism property m where m : {mono \pi : x y / e x y >-> x == y}

### Equivalence and quotient by an equivalence ***

EquivRel r er es et == builds an equiv_rel structure based on the reflexivity, symmetry and transitivity property of a boolean relation. {eq_quot e} == builds the quotType of T by equiv where e : rel T is an equiv_rel and T is a choiceType or a (choiceTypeMod e) it is canonically an eqType, a choiceType, a quotType and an eqQuotType x = y % [mod_eq e] := x = y % [mod {eq_quot e} ] <-> x and y are equal modulo e ...

Set Implicit Arguments.

Declare Scope quotient_scope.

Reserved Notation "\pi_ Q" (at level 0, format "\pi_ Q").
Reserved Notation "\pi" (at level 0, format "\pi").
Reserved Notation "{pi_ Q a }"
(at level 0, Q at next level, format "{pi_ Q a }").
Reserved Notation "{pi a }" (at level 0, format "{pi a }").
Reserved Notation "x == y %[mod_eq e ]" (at level 70, y at next level,
no associativity, format "'[hv ' x '/' == y '/' %[mod_eq e ] ']'").
Reserved Notation "x = y %[mod_eq e ]" (at level 70, y at next level,
no associativity, format "'[hv ' x '/' = y '/' %[mod_eq e ] ']'").
Reserved Notation "x != y %[mod_eq e ]" (at level 70, y at next level,
no associativity, format "'[hv ' x '/' != y '/' %[mod_eq e ] ']'").
Reserved Notation "x <> y %[mod_eq e ]" (at level 70, y at next level,
no associativity, format "'[hv ' x '/' <> y '/' %[mod_eq e ] ']'").
Reserved Notation "{eq_quot e }"
(at level 0, e at level 0, format "{eq_quot e }").

Delimit Scope quotient_scope with qT.
Local Open Scope quotient_scope.

Definition of the quotient interface.

#[short(type="quotType")]
HB.structure Definition Quotient T := { qT of isQuotient T qT }.
Arguments repr_of [T qT] : rename.

Section QuotientDef.

Variable T : Type.
Variable qT : quotType T.
Definition pi_phant of phant qT := @quot_pi_subdef _ qT.
Local Notation "\pi" := (pi_phant (Phant qT)).

Lemma repr_ofK : cancel (@repr_of _ _) \pi.

End QuotientDef.
Arguments repr_ofK {T qT}.

Protecting some symbols.

Fancy Notations

Notation "\pi_ Q" := (@pi _ _ (Phant Q)) : quotient_scope.
Notation "\pi" := (@pi _ _ (Phant _)) (only parsing) : quotient_scope.
Notation "x == y %[mod Q ]" := (\pi_Q x == \pi_Q y) : quotient_scope.
Notation "x = y %[mod Q ]" := (\pi_Q x = \pi_Q y) : quotient_scope.
Notation "x != y %[mod Q ]" := (\pi_Q x != \pi_Q y) : quotient_scope.
Notation "x <> y %[mod Q ]" := (\pi_Q x \pi_Q y) : quotient_scope.

Local Notation "\mpi" := (@mpi _ _ (Phant _)).
Canonical mpi_unlock := Unlockable mpi.unlock.
Canonical pi_unlock := Unlockable pi.unlock.
Canonical repr_unlock := Unlockable repr.unlock.

Notation "[ 'quotType' 'of' Q ]" := (Quotient.clone _ Q%type _)
(at level 0, format "[ 'quotType' 'of' Q ]") : form_scope.

Arguments repr {T qT} x.

Exporting the theory

Section QuotTypeTheory.

Variable T : Type.
Variable qT : quotType T.

Lemma reprK : cancel repr \pi_qT.

Variant pi_spec (x : T) : T Type :=
PiSpec y of x = y %[mod qT] : pi_spec x y.

Lemma piP (x : T) : pi_spec x (repr (\pi_qT x)).

Lemma mpiE : \mpi =1 \pi_qT.

Lemma quotW P : ( y : T, P (\pi_qT y)) x : qT, P x.

Lemma quotP P : ( y : T, repr (\pi_qT y) = y P (\pi_qT y))
x : qT, P x.

End QuotTypeTheory.

Arguments reprK {T qT} x.

About morphisms This was pi_morph T (x : T) := PiMorph { pi_op : T; _ : x = pi_op }.
Structure equal_to T (x : T) := EqualTo {
equal_val : T;
_ : x = equal_val
}.
Lemma equal_toE (T : Type) (x : T) (m : equal_to x) : equal_val m = x.

Notation piE := (@equal_toE _ _).

Canonical equal_to_pi T (qT : quotType T) (x : T) :=
@EqualTo _ (\pi_qT x) (\pi x) (erefl _).

Arguments EqualTo {T x equal_val}.

Section Morphism.

Variables T U : Type.
Variable (qT : quotType T).
Variable (qU : quotType U).

Variable (f : T T) (g : T T T) (p : T U) (r : T T U).
Variable (fq : qT qT) (gq : qT qT qT) (pq : qT U) (rq : qT qT U).
Variable (h : T U) (hq : qT qU).
Hypothesis pi_f : {morph \pi : x / f x >-> fq x}.
Hypothesis pi_g : {morph \pi : x y / g x y >-> gq x y}.
Hypothesis pi_p : {mono \pi : x / p x >-> pq x}.
Hypothesis pi_r : {mono \pi : x y / r x y >-> rq x y}.
Hypothesis pi_h : (x : T), \pi_qU (h x) = hq (\pi_qT x).
Variables (a b : T) (x : equal_to (\pi_qT a)) (y : equal_to (\pi_qT b)).

Internal Lemmas : do not use directly
Lemma pi_morph1 : \pi (f a) = fq (equal_val x).
Lemma pi_morph2 : \pi (g a b) = gq (equal_val x) (equal_val y).
Lemma pi_mono1 : p a = pq (equal_val x).
Lemma pi_mono2 : r a b = rq (equal_val x) (equal_val y).
Lemma pi_morph11 : \pi (h a) = hq (equal_val x).

End Morphism.

Arguments pi_morph1 {T qT f fq}.
Arguments pi_morph2 {T qT g gq}.
Arguments pi_mono1 {T U qT p pq}.
Arguments pi_mono2 {T U qT r rq}.
Arguments pi_morph11 {T U qT qU h hq}.

Notation "{pi_ Q a }" := (equal_to (\pi_Q a)) : quotient_scope.
Notation "{pi a }" := (equal_to (\pi a)) : quotient_scope.

Declaration of morphisms
Notation PiMorph pi_x := (EqualTo pi_x).
Notation PiMorph1 pi_f :=
(fun a (x : {pi a}) ⇒ EqualTo (pi_morph1 pi_f a x)).
Notation PiMorph2 pi_g :=
(fun a b (x : {pi a}) (y : {pi b}) ⇒ EqualTo (pi_morph2 pi_g a b x y)).
Notation PiMono1 pi_p :=
(fun a (x : {pi a}) ⇒ EqualTo (pi_mono1 pi_p a x)).
Notation PiMono2 pi_r :=
(fun a b (x : {pi a}) (y : {pi b}) ⇒ EqualTo (pi_mono2 pi_r a b x y)).
Notation PiMorph11 pi_f :=
(fun a (x : {pi a}) ⇒ EqualTo (pi_morph11 pi_f a x)).

lifting helpers
Notation lift_op1 Q f := (locked (fun x : Q\pi_Q (f (repr x)) : Q)).
Notation lift_op2 Q g :=
(locked (fun x y : Q\pi_Q (g (repr x) (repr y)) : Q)).
Notation lift_fun1 Q f := (locked (fun x : Qf (repr x))).
Notation lift_fun2 Q g := (locked (fun x y : Qg (repr x) (repr y))).
Notation lift_op11 Q Q' f := (locked (fun x : Q\pi_Q' (f (repr x)) : Q')).

constant declaration
Notation lift_cst Q x := (locked (\pi_Q x : Q)).
Notation PiConst a := (@EqualTo _ _ a (lock _)).

embedding declaration, please don't redefine \pi
Notation lift_embed qT e := (locked (fun x\pi_qT (e x) : qT)).

Lemma eq_lock T T' e : e =1 (@locked (T T') (fun x : Te x)).

Notation PiEmbed e :=
(fun x ⇒ @EqualTo _ _ (e x) (eq_lock (fun _\pi _) _)).

#[short(type="eqQuotType")]
HB.structure Definition EqQuotient T eq_quot_op :=
{Q of isEqQuotient T eq_quot_op Q & Quotient T Q & hasDecEq Q}.

Canonical pi_eq_quot_mono T eq_quot_op eqT :=
PiMono2 (@pi_eq_quot T eq_quot_op eqT).

Notation "[ 'eqQuotType' e 'of' Q ]" := (EqQuotient.clone _ e Q%type _)
(at level 0, format "[ 'eqQuotType' e 'of' Q ]") : form_scope.

Even if a quotType is a natural subType, we do not make this subType canonical, to allow the user to define the subtyping he wants. However one can:
• get the hasDecEq and the hasChoice by subtyping
• get the subType structure and maybe declare it Canonical.

Definition quot_type_subdef T (qT : quotType T) of phant qT : Type := qT.
Notation quot_type_of T Q := (@quot_type_subdef T _ (Phant Q)).
Notation quot_type Q := (quot_type_subdef (Phant Q)).

Module QuotSubType.
Section QuotSubType.
Variable (T : eqType) (qT : quotType T).

Definition Sub x (px : repr (\pi_qT x) == x) := \pi_qT x.

Lemma qreprK x Px : repr (@Sub x Px) = x.

Lemma sortPx (x : qT) : repr (\pi_qT (repr x)) == repr x.

Lemma sort_Sub (x : qT) : x = Sub (sortPx x).

Lemma reprP K (PK : x Px, K (@Sub x Px)) u : K u.

#[export]
HB.instance Definition _ := isSub.Build _ _ (quot_type qT) reprP qreprK.
#[export]
HB.instance Definition _ := [Equality of quot_type qT by <:].
End QuotSubType.
Module Exports. End Exports.
End QuotSubType.
Export QuotSubType.Exports.

Notation "[ 'Sub' Q 'of' T 'by' %/ ]" :=
(SubType.copy Q%type (quot_type_of T Q%type))
(at level 0, format "[ 'Sub' Q 'of' T 'by' %/ ]") : form_scope.

Notation "[ 'Sub' Q 'by' %/ ]" :=
(SubType.copy Q%type (quot_type Q))
(at level 0, format "[ 'Sub' Q 'by' %/ ]") : form_scope.

Notation "[ 'Equality' 'of' Q 'by' <:%/ ]" :=
(Equality.copy Q%type (quot_type Q))
(at level 0, format "[ 'Equality' 'of' Q 'by' <:%/ ]") : form_scope.

Notation "[ 'Choice' 'of' Q 'by' <:%/ ]" := (Choice.copy Q%type (quot_type Q))
(at level 0, format "[ 'Choice' 'of' Q 'by' <:%/ ]") : form_scope.

Notation "[ 'Countable' 'of' Q 'by' <:%/ ]" := (Countable.copy Q%type (quot_type Q))
(at level 0, format "[ 'Countable' 'of' Q 'by' <:%/ ]") : form_scope.

Notation "[ 'Finite' 'of' Q 'by' <:%/ ]" := (Finite.copy Q%type (quot_type Q))
(at level 0, format "[ 'Finite' 'of' Q 'by' <:%/ ]") : form_scope.

Definition of a (decidable) equivalence relation

Section EquivRel.

Variable T : Type.

Lemma left_trans (e : rel T) :
symmetric e transitive e left_transitive e.

Lemma right_trans (e : rel T) :
symmetric e transitive e right_transitive e.

Variant equiv_class_of (equiv : rel T) :=
EquivClass of reflexive equiv & symmetric equiv & transitive equiv.

Record equiv_rel := EquivRelPack {
equiv :> rel T;
_ : equiv_class_of equiv
}.

Variable e : equiv_rel.

Definition equiv_class :=
let: EquivRelPack _ ce as e' := e return equiv_class_of e' in ce.

Definition equiv_pack (r : rel T) ce of phant_id ce equiv_class :=
@EquivRelPack r ce.

Lemma equiv_refl x : e x x.
Lemma equiv_sym : symmetric e.
Lemma equiv_trans : transitive e.

Lemma eq_op_trans (T' : eqType) : transitive (@eq_op T').

Lemma equiv_ltrans: left_transitive e.

Lemma equiv_rtrans: right_transitive e.

End EquivRel.

#[global] Hint Resolve equiv_refl : core.

Notation EquivRel r er es et := (@EquivRelPack _ r (EquivClass er es et)).
Notation "[ 'equiv_rel' 'of' e ]" := (@equiv_pack _ _ e _ id)
(at level 0, format "[ 'equiv_rel' 'of' e ]") : form_scope.

Encoding to another type modulo an equivalence

Section EncodingModuloRel.

Variables (D E : Type) (ED : E D) (DE : D E) (e : rel D).

Variant encModRel_class_of (r : rel D) :=
EncModRelClassPack of ( x, r x x r (ED (DE x)) x) & (r =2 e).

Record encModRel := EncModRelPack {
enc_mod_rel :> rel D;
_ : encModRel_class_of enc_mod_rel
}.

Variable r : encModRel.

Definition encModRelClass :=
let: EncModRelPack _ c as r' := r return encModRel_class_of r' in c.

Definition encModRelP (x : D) : r x x r (ED (DE x)) x.

Definition encModRelE : r =2 e.

Definition encoded_equiv : rel E := [rel x y | r (ED x) (ED y)].

End EncodingModuloRel.

Notation EncModRelClass m :=
(EncModRelClassPack (fun x _m x) (fun _ _erefl _)).
Notation EncModRel r m := (@EncModRelPack _ _ _ _ _ r (EncModRelClass m)).

Section EncodingModuloEquiv.

Variables (D E : Type) (ED : E D) (DE : D E) (e : equiv_rel D).
Variable (r : encModRel ED DE e).

Lemma enc_mod_rel_is_equiv : equiv_class_of (enc_mod_rel r).

Definition enc_mod_rel_equiv_rel := EquivRelPack enc_mod_rel_is_equiv.

Definition encModEquivP (x : D) : r (ED (DE x)) x.

Local Notation e' := (encoded_equiv r).

Lemma encoded_equivE : e' =2 [rel x y | e (ED x) (ED y)].
Local Notation e'E := encoded_equivE.

Lemma encoded_equiv_is_equiv : equiv_class_of e'.

Canonical encoded_equiv_equiv_rel := EquivRelPack encoded_equiv_is_equiv.

Lemma encoded_equivP x : e' (DE (ED x)) x.

End EncodingModuloEquiv.

Quotient by a equivalence relation

Module EquivQuot.
Section EquivQuot.

Variables (D : Type) (C : choiceType) (CD : C D) (DC : D C).
Variables (eD : equiv_rel D) (encD : encModRel CD DC eD).
Notation eC := (encoded_equiv encD).

Definition canon x := choose (eC x) (x).

Record equivQuotient := EquivQuotient {
erepr : C;
_ : (frel canon) erepr erepr
}.

Definition type_of of (phantom (rel _) encD) := equivQuotient.

Lemma canon_id : x, (invariant canon canon) x.

Definition pi := locked (fun xEquivQuotient (canon_id x)).

Lemma ereprK : cancel erepr pi.

Local Notation encDE := (encModRelE encD).
Local Notation encDP := (encModEquivP encD).
Canonical encD_equiv_rel := EquivRelPack (enc_mod_rel_is_equiv encD).

Lemma pi_CD (x y : C) : reflect (pi x = pi y) (eC x y).

Lemma pi_DC (x y : D) :
reflect (pi (DC x) = pi (DC y)) (eD x y).

Lemma equivQTP : cancel (CD \o erepr) (pi \o DC).

Local Notation qT := (type_of (Phantom (rel D) encD)).
#[export]
HB.instance Definition _ := isQuotient.Build D qT equivQTP.

Lemma eqmodP x y : reflect (x = y %[mod qT]) (eD x y).

#[export]
HB.instance Definition _ := Choice.copy qT (can_type ereprK).

Lemma eqmodE x y : x == y %[mod qT] = eD x y.

#[export]
HB.instance Definition _ := isEqQuotient.Build _ eD qT eqmodE.

End EquivQuot.
Module Exports. End Exports.
End EquivQuot.
Export EquivQuot.Exports.

Arguments EquivQuot.ereprK {D C CD DC eD encD}.

Notation "{eq_quot e }" :=
(@EquivQuot.type_of _ _ _ _ _ _ (Phantom (rel _) e)) : quotient_scope.
Notation "x == y %[mod_eq r ]" := (x == y %[mod {eq_quot r}]) : quotient_scope.
Notation "x = y %[mod_eq r ]" := (x = y %[mod {eq_quot r}]) : quotient_scope.
Notation "x != y %[mod_eq r ]" := (x != y %[mod {eq_quot r}]) : quotient_scope.
Notation "x <> y %[mod_eq r ]" := (x y %[mod {eq_quot r}]) : quotient_scope.

If the type is directly a choiceType, no need to encode

Section DefaultEncodingModuloRel.

Variables (D : choiceType) (r : rel D).

Definition defaultEncModRelClass :=
@EncModRelClassPack D D id id r r (fun _ rxxrxx) (fun _ _erefl _).

Canonical defaultEncModRel := EncModRelPack defaultEncModRelClass.

End DefaultEncodingModuloRel.

Recovering a potential countable type structure

Section CountEncodingModuloRel.

Variables (D : Type) (C : countType) (CD : C D) (DC : D C).
Variables (eD : equiv_rel D) (encD : encModRel CD DC eD).
Notation eC := (encoded_equiv encD).

End CountEncodingModuloRel.

Section EquivQuotTheory.

Variables (T : choiceType) (e : equiv_rel T) (Q : eqQuotType e).

Lemma eqmodE x y : x == y %[mod_eq e] = e x y.

Lemma eqmodP x y : reflect (x = y %[mod_eq e]) (e x y).

End EquivQuotTheory.

Section EqQuotTheory.

Variables (T : Type) (e : rel T) (Q : eqQuotType e).

Lemma eqquotE x y : x == y %[mod Q] = e x y.

Lemma eqquotP x y : reflect (x = y %[mod Q]) (e x y).

End EqQuotTheory.