# Library mathcomp.ssreflect.generic_quotient

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

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

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 provide 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).
See "Pragmatic Quotient Types in Coq", proceedings of ITP2013, by Cyril Cohen.

### Generic Quotienting ***

QuotClass (reprK : cancel repr pi) == builds the quotient which canonical surjection function is pi and which representative selection function is repr. QuotType Q class == packs the quotClass class to build a quotType You may declare such elements as Canonical \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 [quotType of Q] == clone of the canonical quotType structure of Q on T 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

### 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 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} [eqQuotType of Q] == clones the canonical eqQuotType structure of Q

### 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.

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.

Section QuotientDef.

Variable T : Type.

Record quot_mixin_of qT := QuotClass {
quot_repr : qT T;
quot_pi : T qT;
_ : cancel quot_repr quot_pi
}.

Notation quot_class_of := quot_mixin_of.

Record quotType := QuotTypePack {
quot_sort :> Type;
quot_class : quot_class_of quot_sort
}.

Variable qT : quotType.
Definition pi_phant of phant qT := quot_pi (quot_class qT).
Definition repr_of := quot_repr (quot_class qT).

Lemma repr_ofK : cancel repr_of \pi.

Definition QuotType_clone (Q : Type) qT cT
of phant_id (quot_class qT) cT := @QuotTypePack Q cT.

End QuotientDef.

Protecting some symbols.

Module Type PiSig.
Parameter f : (T : Type) (qT : quotType T), phant qT T qT.
Axiom E : f = pi_phant.
End PiSig.

Module Pi : PiSig.
Definition f := pi_phant.
Definition E := erefl f.
End Pi.

Module MPi : PiSig.
Definition f := pi_phant.
Definition E := erefl f.
End MPi.

Module Type ReprSig.
Parameter f : (T : Type) (qT : quotType T), qT T.
Axiom E : f = repr_of.
End ReprSig.

Module Repr : ReprSig.
Definition f := repr_of.
Definition E := erefl f.
End Repr.

Fancy Notations

Notation repr := Repr.f.
Notation "\pi_ Q" := (@Pi.f _ _ (Phant Q)) : quotient_scope.
Notation "\pi" := (@Pi.f _ _ (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.

Canonical mpi_unlock := Unlockable MPi.E.
Canonical pi_unlock := Unlockable Pi.E.
Canonical repr_unlock := Unlockable Repr.E.

Notation quot_class_of := quot_mixin_of.
Notation QuotType Q m := (@QuotTypePack _ Q m).
Notation "[ 'quotType' 'of' Q ]" := (@QuotType_clone _ Q _ _ id)
(at level 0, format "[ 'quotType' 'of' Q ]") : form_scope.

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.

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 _).

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.

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 _) _)).

Section EqQuotTypeStructure.

Variable T : Type.
Variable eq_quot_op : rel T.

Definition eq_quot_mixin_of (Q : Type) (qc : quot_class_of T Q)
(ec : Equality.class_of Q) :=
{mono \pi_(QuotTypePack qc) : x y /
eq_quot_op x y >-> @eq_op (Equality.Pack ec) x y}.

Record eq_quot_class_of (Q : Type) : Type := EqQuotClass {
eq_quot_quot_class :> quot_class_of T Q;
eq_quot_eq_mixin :> Equality.class_of Q;
pi_eq_quot_mixin :> eq_quot_mixin_of eq_quot_quot_class eq_quot_eq_mixin
}.

Record eqQuotType : Type := EqQuotTypePack {
eq_quot_sort :> Type;
_ : eq_quot_class_of eq_quot_sort;

}.

Implicit Type eqT : eqQuotType.

Definition eq_quot_class eqT : eq_quot_class_of eqT :=
let: EqQuotTypePack _ cT as qT' := eqT return eq_quot_class_of qT' in cT.

Canonical eqQuotType_eqType eqT := EqType eqT (eq_quot_class eqT).
Canonical eqQuotType_quotType eqT := QuotType eqT (eq_quot_class eqT).

Coercion eqQuotType_eqType : eqQuotType >-> eqType.
Coercion eqQuotType_quotType : eqQuotType >-> quotType.

Definition EqQuotType_pack Q :=
fun (qT : quotType T) (eT : eqType) qc ec
of phant_id (quot_class qT) qc & phant_id (Equality.class eT) ec
fun mEqQuotTypePack (@EqQuotClass Q qc ec m).

Definition EqQuotType_clone (Q : Type) eqT cT
of phant_id (eq_quot_class eqT) cT := @EqQuotTypePack Q cT.

Lemma pi_eq_quot eqT : {mono \pi_eqT : x y / eq_quot_op x y >-> x == y}.

Canonical pi_eq_quot_mono eqT := PiMono2 (pi_eq_quot eqT).

End EqQuotTypeStructure.

Notation EqQuotType e Q m := (@EqQuotType_pack _ e Q _ _ _ _ id id m).
Notation "[ 'eqQuotType' e 'of' Q ]" := (@EqQuotType_clone _ e Q _ _ id)
(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 eqMixin and the choiceMixin by subtyping
• get the subType structure and maybe declare it Canonical.

Module QuotSubType.
Section SubTypeMixin.

Variable T : eqType.
Variable 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.

Canonical subType := SubType _ _ _ reprP qreprK.
Definition eqMixin := Eval hnf in [eqMixin of qT by <:].

Canonical eqType := EqType qT eqMixin.

End SubTypeMixin.

Definition choiceMixin (T : choiceType) (qT : quotType T) :=
Eval hnf in [choiceMixin of qT by <:].
Canonical choiceType (T : choiceType) (qT : quotType T) :=
ChoiceType qT (@choiceMixin T qT).

Definition countMixin (T : countType) (qT : quotType T) :=
Eval hnf in [countMixin of qT by <:].
Canonical countType (T : countType) (qT : quotType T) :=
CountType qT (@countMixin T qT).

Section finType.
Variables (T : finType) (qT : quotType T).
Canonical subCountType := [subCountType of qT].
Definition finMixin := Eval hnf in [finMixin of qT by <:].
End finType.

End QuotSubType.

Notation "[ 'subType' Q 'of' T 'by' %/ ]" :=
(@SubType T _ Q _ _ (@QuotSubType.reprP _ _) (@QuotSubType.qreprK _ _))
(at level 0, format "[ 'subType' Q 'of' T 'by' %/ ]") : form_scope.

Notation "[ 'eqMixin' 'of' Q 'by' <:%/ ]" :=
(@QuotSubType.eqMixin _ _: Equality.class_of Q)
(at level 0, format "[ 'eqMixin' 'of' Q 'by' <:%/ ]") : form_scope.

Notation "[ 'choiceMixin' 'of' Q 'by' <:%/ ]" :=
(@QuotSubType.choiceMixin _ _: Choice.mixin_of Q)
(at level 0, format "[ 'choiceMixin' 'of' Q 'by' <:%/ ]") : form_scope.

Notation "[ 'countMixin' 'of' Q 'by' <:%/ ]" :=
(@QuotSubType.countMixin _ _: Countable.mixin_of Q)
(at level 0, format "[ 'countMixin' 'of' Q 'by' <:%/ ]") : form_scope.

Notation "[ 'finMixin' 'of' Q 'by' <:%/ ]" :=
(@QuotSubType.finMixin _ _: Finite.mixin_of Q)
(at level 0, format "[ 'finMixin' '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.

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.

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

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.

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).

Definition quotClass := QuotClass equivQTP.
Canonical quotType := QuotType qT quotClass.

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

Fact eqMixin : Equality.mixin_of qT.
Canonical eqType := EqType qT eqMixin.
Definition choiceMixin := CanChoiceMixin ereprK.
Canonical choiceType := ChoiceType qT choiceMixin.

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

Canonical eqQuotType := EqQuotType eD qT eqmodE.

End EquivQuot.
End EquivQuot.

Canonical EquivQuot.quotType.
Canonical EquivQuot.eqType.
Canonical EquivQuot.choiceType.
Canonical EquivQuot.eqQuotType.

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).

Fact eq_quot_countMixin : Countable.mixin_of {eq_quot encD}.
Canonical eq_quot_countType := CountType {eq_quot encD} eq_quot_countMixin.

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.