- -

Library mathcomp.ssreflect.generic_quotient

- -

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

- -

- 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 "Pragamatic 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 }", only parsing).
- -
-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.
- -
- -
-

- -

- 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 _).
- -
- -
-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 Lemmmas : 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)).
- -
-

- -

- lifiting 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 : Q ⇒ f (repr x))).
-Notation lift_fun2 Q g := (locked (fun x y : Q ⇒ g (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 : T ⇒ e x)).
- -
-Notation PiEmbed e :=
- (fun x ⇒ @EqualTo _ _ (e x) (eq_lock (fun _ ⇒ \pi _) _)).
- -
-

- -

- About eqQuotType -

- -
-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 m ⇒ EqQuotTypePack (@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 x ⇒ EquivQuotient (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 _ rxx ⇒ rxx) (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.
- -
-