lean2/hott/hit/coeq.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.coeq
Authors: Floris van Doorn
Declaration of the coequalizer
-/
import .type_quotient
open type_quotient eq equiv
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 (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},
{ intros [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
definition 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) = sorry ⬝ Pcp x ⬝ sorry :=
sorry
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
definition 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) = sorry ⬝ Pcp x ⬝ sorry :=
sorry
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
definition 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) = sorry /-Pcp x-/ :=
sorry
end
end coeq