lean2/hott/types/lift.hlean
2015-09-11 23:35:21 -07:00

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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
Theorems about lift
-/
import ..function
open eq equiv equiv.ops is_equiv is_trunc
namespace lift
universe variables u v
variables {A : Type.{u}} (z z' : lift.{u v} A)
protected definition eta : up (down z) = z :=
by induction z; reflexivity
protected definition code [unfold 2 3] : lift A → lift A → Type
| code (up a) (up a') := a = a'
protected definition decode [unfold 2 3] : Π(z z' : lift A), lift.code z z' → z = z'
| decode (up a) (up a') := λc, ap up c
variables {z z'}
protected definition encode [unfold 3 4 5] (p : z = z') : lift.code z z' :=
by induction p; induction z; esimp
variables (z z')
definition lift_eq_equiv : (z = z') ≃ lift.code z z' :=
equiv.MK lift.encode
!lift.decode
abstract begin
intro c, induction z with a, induction z' with a', esimp at *, induction c,
reflexivity
end end
abstract begin
intro p, induction p, induction z, reflexivity
end end
section
variables {a a' : A}
definition eq_of_up_eq_up [unfold 4] (p : up a = up a') : a = a' :=
lift.encode p
definition lift_transport {P : A → Type} (p : a = a') (z : lift (P a))
: p ▸ z = up (p ▸ down z) :=
by induction p; induction z; reflexivity
end
variables {A' : Type} (f : A → A') (g : lift A → lift A')
definition lift_functor [unfold 4] : lift A → lift A'
| lift_functor (up a) := up (f a)
definition is_equiv_lift_functor [constructor] [Hf : is_equiv f] : is_equiv (lift_functor f) :=
adjointify (lift_functor f)
(lift_functor f⁻¹)
abstract begin
intro z', induction z' with a',
esimp, exact ap up !right_inv
end end
abstract begin
intro z, induction z with a,
esimp, exact ap up !left_inv
end end
definition lift_equiv_lift_of_is_equiv [constructor] [Hf : is_equiv f] : lift A ≃ lift A' :=
equiv.mk _ (is_equiv_lift_functor f)
definition lift_equiv_lift [constructor] (f : A ≃ A') : lift A ≃ lift A' :=
equiv.mk _ (is_equiv_lift_functor f)
definition lift_equiv_lift_refl (A : Type) : lift_equiv_lift (erfl : A ≃ A) = erfl :=
by apply equiv_eq'; intro z; induction z with a; reflexivity
definition lift_inv_functor [unfold-full] (a : A) : A' :=
down (g (up a))
definition is_equiv_lift_inv_functor [constructor] [Hf : is_equiv g]
: is_equiv (lift_inv_functor g) :=
adjointify (lift_inv_functor g)
(lift_inv_functor g⁻¹)
abstract begin
intro z', rewrite [▸*,lift.eta,right_inv g],
end end
abstract begin
intro z', rewrite [▸*,lift.eta,left_inv g],
end end
definition equiv_of_lift_equiv_lift [constructor] (g : lift A ≃ lift A') : A ≃ A' :=
equiv.mk _ (is_equiv_lift_inv_functor g)
definition lift_functor_left_inv : lift_inv_functor (lift_functor f) = f :=
eq_of_homotopy (λa, idp)
definition lift_functor_right_inv : lift_functor (lift_inv_functor g) = g :=
begin
apply eq_of_homotopy, intro z, induction z with a, esimp, apply lift.eta
end
variables (A A')
definition is_equiv_lift_functor_fn [constructor]
: is_equiv (lift_functor : (A → A') → (lift A → lift A')) :=
adjointify lift_functor
lift_inv_functor
lift_functor_right_inv
lift_functor_left_inv
definition lift_imp_lift_equiv [constructor] : (lift A → lift A') ≃ (A → A') :=
(equiv.mk _ (is_equiv_lift_functor_fn A A'))⁻¹ᵉ
-- can we deduce this from lift_imp_lift_equiv?
definition lift_equiv_lift_equiv [constructor] : (lift A ≃ lift A') ≃ (A ≃ A') :=
equiv.MK equiv_of_lift_equiv_lift
lift_equiv_lift
abstract begin
intro f, apply equiv_eq, reflexivity
end end
abstract begin
intro g, apply equiv_eq, esimp, apply eq_of_homotopy, intro z,
induction z with a, esimp, apply lift.eta
end end
definition lift_eq_lift_equiv.{u1 u2} (A A' : Type.{u1})
: (lift.{u1 u2} A = lift.{u1 u2} A') ≃ (A = A') :=
!eq_equiv_equiv ⬝e !lift_equiv_lift_equiv ⬝e !eq_equiv_equiv⁻¹ᵉ
definition is_embedding_lift [instance] : is_embedding lift :=
begin
intro A A', fapply is_equiv.homotopy_closed,
exact to_inv !lift_eq_lift_equiv,
exact _,
{ intro p, induction p,
esimp [lift_eq_lift_equiv,equiv.trans,equiv.symm,eq_equiv_equiv],
rewrite [equiv_of_eq_refl,lift_equiv_lift_refl],
apply ua_refl}
end
-- is_trunc_lift is defined in init.trunc
end lift