81 lines
2.6 KiB
Text
81 lines
2.6 KiB
Text
|
/-
|
||
|
|
Copyright (c) 2015 Floris van Doorn. All rights reserved.
|
||
|
|
Released under Apache 2.0 license as described in the file LICENSE.
|
||
|
|
|
||
|
|
Module: hit.quotient
|
||
|
|
Authors: Floris van Doorn
|
||
|
|
|
||
|
|
Declaration of set-quotients
|
||
|
|
-/
|
||
|
|
|
||
|
|
/-
|
||
|
|
We aim to model
|
||
|
|
HIT quotient : Type :=
|
||
|
|
| class_of : A -> quotient
|
||
|
|
| eq_of_rel : Πa a', R a a' → class_of a = class_of a'
|
||
|
|
| is_hset_quotient : is_hset quotient
|
||
|
|
-/
|
||
|
|
|
||
|
|
import .coeq .trunc
|
||
|
|
|
||
|
|
open coeq eq is_trunc trunc
|
||
|
|
|
||
|
|
namespace quotient
|
||
|
|
context
|
||
|
|
universe u
|
||
|
|
parameters {A : Type.{u}} (R : A → A → hprop.{u})
|
||
|
|
|
||
|
|
structure total : Type := {pt1 : A} {pt2 : A} (rel : R pt1 pt2)
|
||
|
|
open total
|
||
|
|
|
||
|
|
definition quotient : Type :=
|
||
|
|
trunc 0 (coeq (pt1 : total → A) (pt2 : total → A))
|
||
|
|
|
||
|
|
definition class_of (a : A) : quotient :=
|
||
|
|
tr (coeq_i _ _ a)
|
||
|
|
|
||
|
|
definition eq_of_rel {a a' : A} (H : R a a') : class_of a = class_of a' :=
|
||
|
|
ap tr (cp _ _ (total.mk H))
|
||
|
|
|
||
|
|
definition is_hset_quotient : is_hset quotient :=
|
||
|
|
by unfold quotient; exact _
|
||
|
|
|
||
|
|
protected definition rec {P : quotient → Type} [Pt : Πaa, is_trunc 0 (P aa)]
|
||
|
|
(Pc : Π(a : A), P (class_of a)) (Pp : Π⦃a a' : A⦄ (H : R a a'), eq_of_rel H ▹ Pc a = Pc a')
|
||
|
|
(x : quotient) : P x :=
|
||
|
|
begin
|
||
|
|
apply (@trunc.rec_on _ _ P x),
|
||
|
|
{ intro x', apply Pt},
|
||
|
|
{ intro y, fapply (coeq.rec_on _ _ y),
|
||
|
|
{ exact Pc},
|
||
|
|
{ intro H, cases H with [a, a', H], esimp [is_typeof],
|
||
|
|
--rewrite (transport_compose P tr (cp pt1 pt2 (total.mk H)) (Pc a))
|
||
|
|
apply concat, apply transport_compose;apply Pp}}
|
||
|
|
end
|
||
|
|
|
||
|
|
protected definition rec_on [reducible] {P : quotient → Type} (x : quotient)
|
||
|
|
[Pt : Πaa, is_trunc 0 (P aa)] (Pc : Π(a : A), P (class_of a))
|
||
|
|
(Pp : Π⦃a a' : A⦄ (H : R a a'), eq_of_rel H ▹ Pc a = Pc a') : P x :=
|
||
|
|
rec Pc Pp x
|
||
|
|
|
||
|
|
definition rec_eq_of_rel {P : quotient → Type} [Pt : Πaa, is_trunc 0 (P aa)]
|
||
|
|
(Pc : Π(a : A), P (class_of a)) (Pp : Π⦃a a' : A⦄ (H : R a a'), eq_of_rel H ▹ Pc a = Pc a')
|
||
|
|
{a a' : A} (H : R a a') : apD (rec Pc Pp) (eq_of_rel H) = sorry ⬝ Pp H ⬝ sorry :=
|
||
|
|
sorry
|
||
|
|
|
||
|
|
protected definition elim {P : Type} [Pt : is_trunc 0 P] (Pc : A → P)
|
||
|
|
(Pp : Π⦃a a' : A⦄ (H : R a a'), Pc a = Pc a') (x : quotient) : P :=
|
||
|
|
rec Pc (λa a' H, !tr_constant ⬝ Pp H) x
|
||
|
|
|
||
|
|
protected definition elim_on [reducible] {P : Type} (x : quotient) [Pt : is_trunc 0 P]
|
||
|
|
(Pc : A → P) (Pp : Π⦃a a' : A⦄ (H : R a a'), Pc a = Pc a') : P :=
|
||
|
|
elim Pc Pp x
|
||
|
|
|
||
|
|
protected definition elim_eq_of_rel {P : Type} [Pt : is_trunc 0 P] (Pc : A → P)
|
||
|
|
(Pp : Π⦃a a' : A⦄ (H : R a a'), Pc a = Pc a') {a a' : A} (H : R a a')
|
||
|
|
: ap (elim Pc Pp) (eq_of_rel H) = sorry ⬝ Pp H ⬝ sorry :=
|
||
|
|
sorry
|
||
|
|
|
||
|
|
end
|
||
|
|
end quotient
|