40086d0084
examples:
foo_eq : Pi {A B : foo}, _ -> A = B
foo_mk_eq : Pi _, foo.mk _ = foo.mk _ (if constructor is called "bar", then this becomes "bar_eq")
foo_eq_equiv : Pi {A B : foo}, (A = B) ≃ _
also changed priority of some instances of is_trunc
139 lines
5.2 KiB
Text
139 lines
5.2 KiB
Text
/-
|
|
Copyright (c) 2015 Jakob von Raumer. All rights reserved.
|
|
Released under Apache 2.0 license as described in the file LICENSE.
|
|
|
|
Module: types.trunc
|
|
Authors: Jakob von Raumer, Floris van Doorn
|
|
|
|
Properties of is_trunc
|
|
-/
|
|
|
|
import types.pi types.eq
|
|
|
|
open sigma sigma.ops pi function eq equiv eq funext
|
|
namespace is_trunc
|
|
|
|
definition is_contr.sigma_char (A : Type) :
|
|
(Σ (center : A), Π (a : A), center = a) ≃ (is_contr A) :=
|
|
begin
|
|
fapply equiv.mk,
|
|
{intro S, apply is_contr.mk, exact S.2},
|
|
{fapply is_equiv.adjointify,
|
|
{intro H, apply sigma.mk, exact (@contr A H)},
|
|
{intro H, apply (is_trunc.rec_on H), intro Hint,
|
|
apply (contr_internal.rec_on Hint), intros [H1, H2],
|
|
apply idp},
|
|
{intro S, cases S, apply idp}}
|
|
end
|
|
|
|
definition is_trunc.pi_char (n : trunc_index) (A : Type) :
|
|
(Π (x y : A), is_trunc n (x = y)) ≃ (is_trunc (n .+1) A) :=
|
|
begin
|
|
fapply equiv.MK,
|
|
{intro H, apply is_trunc_succ_intro},
|
|
{intros [H, x, y], apply is_trunc_eq},
|
|
{intro H, apply (is_trunc.rec_on H), intro Hint, apply idp},
|
|
{intro P, apply eq_of_homotopy, intro a, apply eq_of_homotopy, intro b,
|
|
esimp [function.id,compose,is_trunc_succ_intro,is_trunc_eq],
|
|
generalize (P a b), intro H, apply (is_trunc.rec_on H), intro H', apply idp},
|
|
end
|
|
|
|
definition is_hprop_is_trunc [instance] (n : trunc_index) :
|
|
Π (A : Type), is_hprop (is_trunc n A) :=
|
|
begin
|
|
apply (trunc_index.rec_on n),
|
|
{intro A,
|
|
apply is_trunc_is_equiv_closed, apply equiv.to_is_equiv,
|
|
apply is_contr.sigma_char,
|
|
apply (@is_hprop.mk), intros,
|
|
fapply sigma_eq, apply x.2,
|
|
apply (@is_hprop.elim),
|
|
apply is_trunc_pi, intro a,
|
|
apply is_hprop.mk, intros [w, z],
|
|
have H : is_hset A,
|
|
begin
|
|
apply is_trunc_succ, apply is_trunc_succ,
|
|
apply is_contr.mk, exact y.2
|
|
end,
|
|
fapply (@is_hset.elim A _ _ _ w z)},
|
|
{intros [n', IH, A],
|
|
apply is_trunc_is_equiv_closed,
|
|
apply equiv.to_is_equiv,
|
|
apply is_trunc.pi_char},
|
|
end
|
|
|
|
definition is_trunc_succ_of_imp_is_trunc_succ {A : Type} {n : trunc_index} (H : A → is_trunc (n.+1) A)
|
|
: is_trunc (n.+1) A :=
|
|
@is_trunc_succ_intro _ _ (λx y, @is_trunc_eq _ _ (H x) x y)
|
|
|
|
definition is_trunc_of_imp_is_trunc_of_leq {A : Type} {n : trunc_index} (Hn : -1 ≤ n)
|
|
(H : A → is_trunc n A) : is_trunc n A :=
|
|
trunc_index.rec_on n (λHn H, empty.rec _ Hn)
|
|
(λn IH Hn, is_trunc_succ_of_imp_is_trunc_succ)
|
|
Hn H
|
|
|
|
definition is_hset_of_axiom_K {A : Type} (K : Π{a : A} (p : a = a), p = idp) : is_hset A :=
|
|
is_hset.mk _ (λa b p q, eq.rec_on q K p)
|
|
|
|
theorem is_hset_of_relation.{u} {A : Type.{u}} (R : A → A → Type.{u})
|
|
(mere : Π(a b : A), is_hprop (R a b)) (refl : Π(a : A), R a a)
|
|
(imp : Π{a b : A}, R a b → a = b) : is_hset A :=
|
|
is_hset_of_axiom_K
|
|
(λa p,
|
|
have H2 : transport (λx, R a x → a = x) p (@imp a a) = @imp a a, from !apd,
|
|
have H3 : Π(r : R a a), transport (λx, a = x) p (imp r)
|
|
= imp (transport (λx, R a x) p r), from
|
|
to_fun (equiv.symm !heq_pi) H2,
|
|
have H4 : imp (refl a) ⬝ p = imp (refl a), from
|
|
calc
|
|
imp (refl a) ⬝ p = transport (λx, a = x) p (imp (refl a)) : transport_eq_r
|
|
... = imp (transport (λx, R a x) p (refl a)) : H3
|
|
... = imp (refl a) : is_hprop.elim,
|
|
cancel_left H4)
|
|
|
|
definition relation_equiv_eq {A : Type} (R : A → A → Type)
|
|
(mere : Π(a b : A), is_hprop (R a b)) (refl : Π(a : A), R a a)
|
|
(imp : Π{a b : A}, R a b → a = b) (a b : A) : R a b ≃ a = b :=
|
|
@equiv_of_is_hprop _ _ _
|
|
(@is_trunc_eq _ _ (is_hset_of_relation R mere refl @imp) a b)
|
|
imp
|
|
(λp, p ▹ refl a)
|
|
|
|
local attribute not [reducible]
|
|
definition is_hset_of_double_neg_elim {A : Type} (H : Π(a b : A), ¬¬a = b → a = b)
|
|
: is_hset A :=
|
|
is_hset_of_relation (λa b, ¬¬a = b) _ (λa n, n idp) H
|
|
|
|
section
|
|
open decidable
|
|
--this is proven differently in init.hedberg
|
|
definition is_hset_of_decidable_eq (A : Type)
|
|
[H : decidable_eq A] : is_hset A :=
|
|
is_hset_of_double_neg_elim (λa b, by_contradiction)
|
|
end
|
|
|
|
definition is_trunc_of_axiom_K_of_leq {A : Type} (n : trunc_index) (H : -1 ≤ n)
|
|
(K : Π(a : A), is_trunc n (a = a)) : is_trunc (n.+1) A :=
|
|
@is_trunc_succ_intro _ _ (λa b, is_trunc_of_imp_is_trunc_of_leq H (λp, eq.rec_on p !K))
|
|
|
|
open trunctype equiv equiv.ops
|
|
|
|
protected definition trunctype.sigma_char.{l} (n : trunc_index) :
|
|
(trunctype.{l} n) ≃ (Σ (A : Type.{l}), is_trunc n A) :=
|
|
begin
|
|
fapply equiv.MK,
|
|
/--/ intro A, exact (⟨carrier A, struct A⟩),
|
|
/--/ intro S, exact (trunctype.mk S.1 S.2),
|
|
/--/ intro S, apply (sigma.rec_on S), intros [S1, S2], apply idp,
|
|
intro A, apply (trunctype.rec_on A), intros [A1, A2], apply idp,
|
|
end
|
|
|
|
protected definition trunctype_eq_equiv (n : trunc_index) (A B : n-Type) :
|
|
(A = B) ≃ (carrier A = carrier B) :=
|
|
calc
|
|
(A = B) ≃ (trunctype.sigma_char n A = trunctype.sigma_char n B) : eq_equiv_fn_eq_of_equiv
|
|
... ≃ ((trunctype.sigma_char n A).1 = (trunctype.sigma_char n B).1) : equiv.symm (!equiv_subtype)
|
|
... ≃ (carrier A = carrier B) : equiv.refl
|
|
|
|
|
|
end is_trunc
|