lean2/hott/types/fiber.hlean

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/-
2015-04-19 19:58:13 +00:00
Copyright (c) 2015 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Author: Floris van Doorn
Ported from Coq HoTT
Theorems about fibers
-/
2015-04-19 19:58:13 +00:00
import .sigma .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 fiber_eq_equiv (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 pathover_eq_equiv_Fl,
end
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definition fiber_eq {x y : fiber f b} (p : point x = point y)
(q : point_eq x = ap f p ⬝ point_eq y) : x = y :=
to_inv !fiber_eq_equiv ⟨p, q⟩
end fiber