lean2/hott/hit/quotient.hlean

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/-
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
-/
import .type_quotient .trunc
open eq is_trunc trunc type_quotient
namespace quotient
section
parameters {A : Type} (R : A → A → hprop)
-- set-quotients are just truncations of type-quotients
definition quotient : Type := trunc 0 (type_quotient (λa a', trunctype.carrier (R a a')))
definition class_of (a : A) : quotient :=
tr (class_of _ a)
definition eq_of_rel {a a' : A} (H : R a a') : class_of a = class_of a' :=
ap tr (eq_of_rel _ H)
theorem is_hset_quotient : is_hset quotient :=
begin unfold quotient, exact _ end
protected definition rec {P : quotient → Type} [Pt : Πaa, is_hset (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 (type_quotient.rec_on y),
{ exact Pc},
{ intro a a' H,
apply concat, apply transport_compose;apply Pp}}
end
protected definition rec_on [reducible] {P : quotient → Type} (x : quotient)
[Pt : Πaa, is_hset (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
theorem rec_eq_of_rel {P : quotient → Type} [Pt : Πaa, is_hset (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) = Pp H :=
!is_hset.elim
protected definition elim {P : Type} [Pt : is_hset 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_hset P]
(Pc : A → P) (Pp : Π⦃a a' : A⦄ (H : R a a'), Pc a = Pc a') : P :=
elim Pc Pp x
theorem elim_eq_of_rel {P : Type} [Pt : is_hset 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) = Pp H :=
begin
apply (@cancel_left _ _ _ _ (tr_constant (eq_of_rel H) (elim Pc Pp (class_of a)))),
rewrite [-apd_eq_tr_constant_con_ap,↑elim,rec_eq_of_rel],
end
/-
there are no theorems to eliminate to the universe here,
because the universe is generally not a set
-/
end
end quotient