2017-02-20 21:23:57 +00:00
|
|
|
|
import types.trunc types.sum types.lift types.unit
|
2017-02-18 21:56:38 +00:00
|
|
|
|
|
2017-02-20 21:23:57 +00:00
|
|
|
|
open pi prod sum unit bool trunc is_trunc is_equiv eq equiv lift pointed
|
2017-02-18 21:56:38 +00:00
|
|
|
|
|
|
|
|
|
namespace choice
|
2018-10-10 01:27:27 +00:00
|
|
|
|
universe variables u v
|
2017-02-18 21:56:38 +00:00
|
|
|
|
|
|
|
|
|
-- the following brilliant name is from Agda
|
|
|
|
|
definition unchoose [unfold 4] (n : ℕ₋₂) {X : Type} (A : X → Type) : trunc n (Πx, A x) → Πx, trunc n (A x) :=
|
|
|
|
|
trunc.elim (λf x, tr (f x))
|
|
|
|
|
|
2018-10-10 01:27:27 +00:00
|
|
|
|
definition has_choice [class] (n : ℕ₋₂) (X : Type.{u}) : Type.{max u (v+1)} :=
|
|
|
|
|
Π(A : X → Type.{v}), is_equiv (unchoose n A)
|
2017-02-18 21:56:38 +00:00
|
|
|
|
|
2018-10-10 01:27:27 +00:00
|
|
|
|
definition choice_equiv [constructor] {n : ℕ₋₂} {X : Type} [H : has_choice.{u v} n X]
|
|
|
|
|
(A : X → Type) : trunc n (Πx, A x) ≃ (Πx, trunc n (A x)) :=
|
2017-02-18 21:56:38 +00:00
|
|
|
|
equiv.mk _ (H A)
|
|
|
|
|
|
2018-10-10 01:27:27 +00:00
|
|
|
|
definition has_choice_of_succ (X : Type) (H : Πk, has_choice.{_ v} (k.+1) X) (n : ℕ₋₂) :
|
|
|
|
|
has_choice.{_ v} n X :=
|
2017-02-18 21:56:38 +00:00
|
|
|
|
begin
|
|
|
|
|
cases n with n,
|
2018-09-11 15:06:46 +00:00
|
|
|
|
{ intro A, exact is_equiv_of_is_contr _ _ _ },
|
2017-02-18 21:56:38 +00:00
|
|
|
|
{ exact H n }
|
|
|
|
|
end
|
|
|
|
|
|
2018-10-10 01:27:27 +00:00
|
|
|
|
/- currently we prove it using univalence, which means we cannot apply it to lift. -/
|
|
|
|
|
definition has_choice_equiv_closed (n : ℕ₋₂) {A B : Type} (f : A ≃ B) (hA : has_choice.{u v} n B)
|
|
|
|
|
: has_choice.{u v} n A :=
|
|
|
|
|
begin
|
|
|
|
|
induction f using rec_on_ua_idp, exact hA
|
|
|
|
|
end
|
|
|
|
|
|
|
|
|
|
definition has_choice_empty [instance] (n : ℕ₋₂) : has_choice.{_ v} n empty :=
|
2017-02-18 21:56:38 +00:00
|
|
|
|
begin
|
|
|
|
|
intro A, fapply adjointify,
|
|
|
|
|
{ intro f, apply tr, intro x, induction x },
|
|
|
|
|
{ intro f, apply eq_of_homotopy, intro x, induction x },
|
|
|
|
|
{ intro g, induction g with g, apply ap tr, apply eq_of_homotopy, intro x, induction x }
|
|
|
|
|
end
|
|
|
|
|
|
2018-10-10 01:27:27 +00:00
|
|
|
|
definition has_choice_unit [instance] : Πn, has_choice.{_ v} n unit :=
|
2017-02-18 21:56:38 +00:00
|
|
|
|
begin
|
|
|
|
|
intro n A, fapply adjointify,
|
|
|
|
|
{ intro f, induction f ⋆ with a, apply tr, intro u, induction u, exact a },
|
|
|
|
|
{ intro f, apply eq_of_homotopy, intro u, induction u, esimp, generalize f ⋆, intro a,
|
|
|
|
|
induction a, reflexivity },
|
|
|
|
|
{ intro g, induction g with g, apply ap tr, apply eq_of_homotopy,
|
|
|
|
|
intro u, induction u, reflexivity }
|
|
|
|
|
end
|
|
|
|
|
|
2018-10-10 01:27:27 +00:00
|
|
|
|
definition has_choice_sum [instance] (n : ℕ₋₂) (A B : Type.{u})
|
|
|
|
|
[has_choice.{_ v} n A] [has_choice.{_ v} n B] : has_choice.{_ v} n (A ⊎ B) :=
|
2017-02-18 21:56:38 +00:00
|
|
|
|
begin
|
|
|
|
|
intro P, fapply is_equiv_of_equiv_of_homotopy,
|
|
|
|
|
{ exact calc
|
|
|
|
|
trunc n (Πx, P x) ≃ trunc n ((Πa, P (inl a)) × Πb, P (inr b))
|
|
|
|
|
: trunc_equiv_trunc n !equiv_sum_rec⁻¹ᵉ
|
|
|
|
|
... ≃ trunc n (Πa, P (inl a)) × trunc n (Πb, P (inr b)) : trunc_prod_equiv
|
|
|
|
|
... ≃ (Πa, trunc n (P (inl a))) × Πb, trunc n (P (inr b))
|
2017-02-20 21:23:57 +00:00
|
|
|
|
: by exact prod_equiv_prod (choice_equiv _) (choice_equiv _)
|
2017-02-18 21:56:38 +00:00
|
|
|
|
... ≃ Πx, trunc n (P x) : equiv_sum_rec },
|
|
|
|
|
{ intro f, induction f, apply eq_of_homotopy, intro x, esimp, induction x with a b: reflexivity }
|
|
|
|
|
end
|
|
|
|
|
|
2018-10-10 01:27:27 +00:00
|
|
|
|
definition has_choice_bool [instance] (n : ℕ₋₂) : has_choice.{_ v} n bool :=
|
2017-02-20 21:23:57 +00:00
|
|
|
|
has_choice_equiv_closed n bool_equiv_unit_sum_unit _
|
|
|
|
|
|
2018-10-10 01:27:27 +00:00
|
|
|
|
definition has_choice_lift.{u'} [instance] (n : ℕ₋₂) (A : Type) [has_choice.{_ v} n A] :
|
|
|
|
|
has_choice.{_ v} n (lift.{u u'} A) :=
|
2017-03-02 03:58:50 +00:00
|
|
|
|
sorry --has_choice_equiv_closed n !equiv_lift⁻¹ᵉ _
|
2017-02-20 21:23:57 +00:00
|
|
|
|
|
2018-10-10 01:27:27 +00:00
|
|
|
|
definition has_choice_punit [instance] (n : ℕ₋₂) : has_choice.{_ v} n punit := has_choice_unit n
|
|
|
|
|
definition has_choice_pbool [instance] (n : ℕ₋₂) : has_choice.{_ v} n pbool := has_choice_bool n
|
|
|
|
|
definition has_choice_plift [instance] (n : ℕ₋₂) (A : Type*) [has_choice.{_ v} n A]
|
|
|
|
|
: has_choice.{_ v} n (plift A) := has_choice_lift n A
|
|
|
|
|
definition has_choice_psum [instance] (n : ℕ₋₂) (A B : Type*) [has_choice.{_ v} n A] [has_choice.{_ v} n B]
|
|
|
|
|
: has_choice.{_ v} n (psum A B) := has_choice_sum n A B
|
2017-02-18 21:56:38 +00:00
|
|
|
|
|
|
|
|
|
end choice
|