lean2/hott/types/function.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.
Author: Floris van Doorn
Ported from Coq HoTT
Theorems about embeddings and surjections
-/
import hit.trunc .pi .fiber .equiv
open equiv sigma sigma.ops eq trunc is_trunc pi is_equiv fiber prod
variables {A B : Type} {f : A → B} {b : B}
structure is_embedding [class] (f : A → B) :=
(elim : Π(a a' : A), is_equiv (ap f : a = a' → f a = f a'))
structure is_surjective [class] (f : A → B) :=
(elim : Π(b : B), ∥ fiber f b ∥)
structure is_split_surjective [class] (f : A → B) :=
(elim : Π(b : B), fiber f b)
structure is_retraction [class] (f : A → B) :=
(sect : B → A)
(right_inverse : Π(b : B), f (sect b) = b)
namespace function
attribute is_embedding.elim [instance]
definition is_surjective_rec_on {P : Type} (H : is_surjective f) (b : B) [Pt : is_hprop P]
(IH : fiber f b → P) : P :=
trunc.rec_on (is_surjective.elim f b) IH
definition is_surjective_of_is_split_surjective [instance] [H : is_split_surjective f]
: is_surjective f :=
is_surjective.mk (λb, tr (is_split_surjective.elim f b))
definition is_injective_of_is_embedding [reducible] [H : is_embedding f] {a a' : A}
: f a = f a' → a = a' :=
(ap f)⁻¹
definition is_embedding_of_is_injective [HA : is_hset A] [HB : is_hset B]
(H : Π(a a' : A), f a = f a' → a = a') : is_embedding f :=
begin
fapply is_embedding.mk,
intro a a',
fapply adjointify,
{exact (H a a')},
{intro p, apply is_hset.elim},
{intro p, apply is_hset.elim}
end
definition is_hprop_is_embedding [instance] (f : A → B) : is_hprop (is_embedding f) :=
begin
have H : (Π(a a' : A), is_equiv (@ap A B f a a')) ≃ is_embedding f,
begin
fapply equiv.MK,
{exact is_embedding.mk},
{intro h, cases h, exact elim},
{intro h, cases h, apply idp},
{intro p, apply idp},
end,
apply is_trunc_equiv_closed,
exact H,
end
definition is_hprop_is_surjective [instance] (f : A → B) : is_hprop (is_surjective f) :=
begin
have H : (Π(b : B), merely (fiber f b)) ≃ is_surjective f,
begin
fapply equiv.MK,
{exact is_surjective.mk},
{intro h, cases h, exact elim},
{intro h, cases h, apply idp},
{intro p, apply idp},
end,
apply is_trunc_equiv_closed,
exact H,
end
definition is_embedding_of_is_equiv [instance] (f : A → B) [H : is_equiv f] : is_embedding f :=
is_embedding.mk _
definition is_equiv_of_is_surjective_of_is_embedding (f : A → B)
[H : is_embedding f] [H' : is_surjective f] : is_equiv f :=
@is_equiv_of_is_contr_fun _ _ _
(λb, is_surjective_rec_on H' b
(λa, is_contr.mk a
(λa',
fiber_eq ((ap f)⁻¹ ((point_eq a) ⬝ (point_eq a')⁻¹))
(by rewrite (right_inv (ap f)); rewrite inv_con_cancel_right))))
end function