lean2/hott/hit/set_quotient.hlean
Floris van Doorn c44ad80e4e feat(homotopy/torus): give recursion and induction principle for the torus
also change the surface of the torus to a square instead of an equality between paths
2015-11-22 18:29:37 -08:00

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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.
Authors: Floris van Doorn
Declaration of set-quotients, i.e. quotient of a mere relation which is then set-truncated.
-/
import function algebra.relation types.trunc types.eq
open eq is_trunc trunc quotient equiv
namespace set_quotient
section
parameters {A : Type} (R : A → A → hprop)
-- set-quotients are just set-truncations of (type) quotients
definition set_quotient : Type := trunc 0 (quotient R)
parameter {R}
definition class_of (a : A) : set_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_set_quotient [instance] : is_hset set_quotient :=
begin unfold set_quotient, exact _ end
protected definition rec {P : set_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'), Pc a =[eq_of_rel H] Pc a')
(x : set_quotient) : P x :=
begin
apply (@trunc.rec_on _ _ P x),
{ intro x', apply Pt},
{ intro y, fapply (quotient.rec_on y),
{ exact Pc},
{ intros, exact pathover_of_pathover_ap P tr (Pp H)}}
end
protected definition rec_on [reducible] {P : set_quotient → Type} (x : set_quotient)
[Pt : Πaa, is_hset (P aa)] (Pc : Π(a : A), P (class_of a))
(Pp : Π⦃a a' : A⦄ (H : R a a'), Pc a =[eq_of_rel H] Pc a') : P x :=
rec Pc Pp x
theorem rec_eq_of_rel {P : set_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'), Pc a =[eq_of_rel H] Pc a')
{a a' : A} (H : R a a') : apdo (rec Pc Pp) (eq_of_rel H) = Pp H :=
!is_hset.elimo
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 : set_quotient) : P :=
rec Pc (λa a' H, pathover_of_eq (Pp H)) x
protected definition elim_on [reducible] {P : Type} (x : set_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 eq_of_fn_eq_fn_inv !(pathover_constant (eq_of_rel H)),
rewrite [▸*,-apdo_eq_pathover_of_eq_ap,↑elim,rec_eq_of_rel],
end
protected definition rec_hprop {P : set_quotient → Type} [Pt : Πaa, is_hprop (P aa)]
(Pc : Π(a : A), P (class_of a)) (x : set_quotient) : P x :=
rec Pc (λa a' H, !is_hprop.elimo) x
protected definition elim_hprop {P : Type} [Pt : is_hprop P] (Pc : A → P) (x : set_quotient)
: P :=
elim Pc (λa a' H, !is_hprop.elim) x
end
end set_quotient
attribute set_quotient.class_of [constructor]
attribute set_quotient.rec set_quotient.elim [unfold 7] [recursor 7]
attribute set_quotient.rec_on set_quotient.elim_on [unfold 4]
open sigma relation function
namespace set_quotient
variables {A B C : Type} (R : A → A → hprop) (S : B → B → hprop) (T : C → C → hprop)
definition is_surjective_class_of : is_surjective (class_of : A → set_quotient R) :=
λx, set_quotient.rec_on x (λa, tr (fiber.mk a idp)) (λa a' r, !is_hprop.elimo)
/- non-dependent universal property -/
definition set_quotient_arrow_equiv (B : Type) [H : is_hset B] :
(set_quotient R → B) ≃ (Σ(f : A → B), Π(a a' : A), R a a' → f a = f a') :=
begin
fapply equiv.MK,
{ intro f, exact ⟨λa, f (class_of a), λa a' r, ap f (eq_of_rel r)⟩},
{ intro v x, induction v with f p, exact set_quotient.elim_on x f p},
{ intro v, induction v with f p, esimp, apply ap (sigma.mk f),
apply eq_of_homotopy3, intro a a' r, apply elim_eq_of_rel},
{ intro f, apply eq_of_homotopy, intro x, refine set_quotient.rec_on x _ _: esimp,
intro a, reflexivity,
intro a a' r, apply eq_pathover, apply hdeg_square, apply elim_eq_of_rel}
end
protected definition code [unfold 4] (a : A) (x : set_quotient R) [H : is_equivalence R]
: hprop :=
set_quotient.elim_on x (R a)
begin
intros a' a'' H1,
refine to_inv !trunctype_eq_equiv _, esimp,
apply ua,
apply equiv_of_is_hprop,
{ intro H2, exact is_transitive.trans R H2 H1},
{ intro H2, apply is_transitive.trans R H2, exact is_symmetric.symm R H1}
end
protected definition encode {a : A} {x : set_quotient R} (p : class_of a = x)
[H : is_equivalence R] : set_quotient.code R a x :=
begin
induction p, esimp, apply is_reflexive.refl R
end
definition rel_of_eq {a a' : A} (p : class_of a = class_of a' :> set_quotient R)
[H : is_equivalence R] : R a a' :=
set_quotient.encode R p
variables {R S T}
definition quotient_unary_map [unfold 7] (f : A → B) (H : Π{a a'} (r : R a a'), S (f a) (f a')) :
set_quotient R → set_quotient S :=
set_quotient.elim (class_of ∘ f) (λa a' r, eq_of_rel (H r))
definition quotient_binary_map [unfold 10 11] (f : A → B → C)
(H : Π{a a'} (r : R a a') {b b'} (s : S b b'), T (f a b) (f a' b'))
[HR : is_reflexive R] [HS : is_reflexive S] :
set_quotient R → set_quotient S → set_quotient T :=
begin
refine set_quotient.elim _ _,
{ intro a, refine set_quotient.elim _ _,
{ intro b, exact class_of (f a b)},
{ intro b b' s, apply eq_of_rel, apply H, apply is_reflexive.refl R, exact s}},
{ intro a a' r, apply eq_of_homotopy, refine set_quotient.rec _ _,
{ intro b, esimp, apply eq_of_rel, apply H, exact r, apply is_reflexive.refl S},
{ intro b b' s, apply eq_pathover, esimp, apply is_hset.elims}}
end
end set_quotient