lean2/hott/types/fiber.hlean
2015-03-12 18:07:55 -07:00

51 lines
1.3 KiB
Text

/-
Copyright (c) 2014 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Module: types.fiber
Author: Floris van Doorn
Ported from Coq HoTT
Theorems about fibers
-/
import types.sigma types.eq
structure fiber {A B : Type} (f : A → B) (b : B) :=
(point : A)
(point_eq : f point = b)
open equiv sigma sigma.ops eq
namespace fiber
variables {A B : Type} {f : A → B} {b : B}
definition sigma_char (f : A → B) (b : B) : fiber f b ≃ (Σ(a : A), f a = b) :=
begin
fapply equiv.MK,
{intro x, exact ⟨point x, point_eq x⟩},
{intro x, exact (fiber.mk x.1 x.2)},
{intro x, cases x, apply idp},
{intro x, cases x, apply idp},
end
definition equiv_fiber_eq (x y : fiber f b)
: (x = y) ≃ (Σ(p : point x = point y), point_eq x = ap f p ⬝ point_eq y) :=
begin
apply equiv.trans,
{apply eq_equiv_fn_eq_of_equiv, apply sigma_char},
apply equiv.trans,
{apply equiv.symm, apply equiv_sigma_eq},
apply sigma_equiv_sigma_id,
intro p,
apply equiv_of_equiv_of_eq,
rotate 1,
apply inv_con_eq_equiv_eq_con,
{apply (ap (λx, x = _)), rewrite transport_eq_Fl}
end
definition eq_mk {x y : fiber f b} (p : point x = point y) (q : point_eq x = ap f p ⬝ point_eq y)
: x = y :=
to_inv !equiv_fiber_eq ⟨p, q⟩
end fiber