87 lines
2.8 KiB
Text
87 lines
2.8 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.coeq
|
|
Authors: Floris van Doorn
|
|
|
|
Declaration of the coequalizer
|
|
-/
|
|
|
|
import .type_quotient
|
|
|
|
open type_quotient eq equiv equiv.ops
|
|
|
|
namespace coeq
|
|
section
|
|
|
|
universe u
|
|
parameters {A B : Type.{u}} (f g : A → B)
|
|
|
|
inductive coeq_rel : B → B → Type :=
|
|
| Rmk : Π(x : A), coeq_rel (f x) (g x)
|
|
open coeq_rel
|
|
local abbreviation R := coeq_rel
|
|
|
|
definition coeq : Type := type_quotient coeq_rel -- TODO: define this in root namespace
|
|
|
|
definition coeq_i (x : B) : coeq :=
|
|
class_of R x
|
|
|
|
/- cp is the name Coq uses. I don't know what it abbreviates, but at least it's short :-) -/
|
|
definition cp (x : A) : coeq_i (f x) = coeq_i (g x) :=
|
|
eq_of_rel coeq_rel (Rmk f g x)
|
|
|
|
protected definition rec {P : coeq → Type} (P_i : Π(x : B), P (coeq_i x))
|
|
(Pcp : Π(x : A), cp x ▸ P_i (f x) = P_i (g x)) (y : coeq) : P y :=
|
|
begin
|
|
fapply (type_quotient.rec_on y),
|
|
{ intro a, apply P_i},
|
|
{ intro a a' H, cases H, apply Pcp}
|
|
end
|
|
|
|
protected definition rec_on [reducible] {P : coeq → Type} (y : coeq)
|
|
(P_i : Π(x : B), P (coeq_i x)) (Pcp : Π(x : A), cp x ▸ P_i (f x) = P_i (g x)) : P y :=
|
|
rec P_i Pcp y
|
|
|
|
theorem rec_cp {P : coeq → Type} (P_i : Π(x : B), P (coeq_i x))
|
|
(Pcp : Π(x : A), cp x ▸ P_i (f x) = P_i (g x))
|
|
(x : A) : apd (rec P_i Pcp) (cp x) = Pcp x :=
|
|
!rec_eq_of_rel
|
|
|
|
protected definition elim {P : Type} (P_i : B → P)
|
|
(Pcp : Π(x : A), P_i (f x) = P_i (g x)) (y : coeq) : P :=
|
|
rec P_i (λx, !tr_constant ⬝ Pcp x) y
|
|
|
|
protected definition elim_on [reducible] {P : Type} (y : coeq) (P_i : B → P)
|
|
(Pcp : Π(x : A), P_i (f x) = P_i (g x)) : P :=
|
|
elim P_i Pcp y
|
|
|
|
theorem elim_cp {P : Type} (P_i : B → P) (Pcp : Π(x : A), P_i (f x) = P_i (g x))
|
|
(x : A) : ap (elim P_i Pcp) (cp x) = Pcp x :=
|
|
begin
|
|
apply (@cancel_left _ _ _ _ (tr_constant (cp x) (elim P_i Pcp (coeq_i (f x))))),
|
|
rewrite [-apd_eq_tr_constant_con_ap,↑elim,rec_cp],
|
|
end
|
|
|
|
protected definition elim_type (P_i : B → Type)
|
|
(Pcp : Π(x : A), P_i (f x) ≃ P_i (g x)) (y : coeq) : Type :=
|
|
elim P_i (λx, ua (Pcp x)) y
|
|
|
|
protected definition elim_type_on [reducible] (y : coeq) (P_i : B → Type)
|
|
(Pcp : Π(x : A), P_i (f x) ≃ P_i (g x)) : Type :=
|
|
elim_type P_i Pcp y
|
|
|
|
theorem elim_type_cp (P_i : B → Type) (Pcp : Π(x : A), P_i (f x) ≃ P_i (g x))
|
|
(x : A) : transport (elim_type P_i Pcp) (cp x) = Pcp x :=
|
|
by rewrite [tr_eq_cast_ap_fn,↑elim_type,elim_cp];apply cast_ua_fn
|
|
|
|
end
|
|
|
|
end coeq
|
|
|
|
attribute coeq.coeq_i [constructor]
|
|
attribute coeq.rec coeq.elim [unfold-c 8]
|
|
attribute coeq.elim_type [unfold-c 7]
|
|
attribute coeq.rec_on coeq.elim_on [unfold-c 6]
|
|
attribute coeq.elim_type_on [unfold-c 5]
|