lean2/hott/types/trunc.hlean

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/-
Copyright (c) 2015 Jakob von Raumer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
Properties of is_trunc and trunctype
-/
-- NOTE: the fact that (is_trunc n A) is a mere proposition is proved in .hprop_trunc
import types.pi types.eq types.equiv .function
open eq sigma sigma.ops pi function equiv is_trunc.trunctype
is_equiv prod is_trunc.trunc_index pointed nat
namespace is_trunc
variables {A B : Type} {n : trunc_index}
definition is_trunc_succ_of_imp_is_trunc_succ (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 (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
/- theorems about trunctype -/
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), intro S1 S2, apply idp},
{ intro A, apply (trunctype.rec_on A), intro A1 A2, apply idp},
end
definition trunctype_eq_equiv (n : trunc_index) (A B : n-Type) :
(A = B) ≃ (carrier A = carrier B) :=
calc
(A = B) ≃ (to_fun (trunctype.sigma_char n) A = to_fun (trunctype.sigma_char n) B)
: eq_equiv_fn_eq_of_equiv
... ≃ ((to_fun (trunctype.sigma_char n) A).1 = (to_fun (trunctype.sigma_char n) B).1)
: equiv.symm (!equiv_subtype)
... ≃ (carrier A = carrier B) : equiv.refl
definition is_trunc_is_embedding_closed (f : A → B) [Hf : is_embedding f] [HB : is_trunc n B]
(Hn : -1 ≤ n) : is_trunc n A :=
begin
cases n with n,
{exact !empty.elim Hn},
{apply is_trunc_succ_intro, intro a a',
fapply @is_trunc_is_equiv_closed_rev _ _ n (ap f)}
end
definition is_trunc_is_retraction_closed (f : A → B) [Hf : is_retraction f]
(n : trunc_index) [HA : is_trunc n A] : is_trunc n B :=
begin
revert A B f Hf HA,
eapply (trunc_index.rec_on n),
{ clear n, intro A B f Hf HA, cases Hf with g ε, fapply is_contr.mk,
{ exact f (center A)},
{ intro b, apply concat,
{ apply (ap f), exact (center_eq (g b))},
{ apply ε}}},
{ clear n, intro n IH A B f Hf HA, cases Hf with g ε,
apply is_trunc_succ_intro, intro b b',
fapply (IH (g b = g b')),
{ intro q, exact ((ε b)⁻¹ ⬝ ap f q ⬝ ε b')},
{ apply (is_retraction.mk (ap g)),
{ intro p, cases p, {rewrite [↑ap, con.left_inv]}}},
{ apply is_trunc_eq}}
end
definition is_embedding_to_fun (A B : Type) : is_embedding (@to_fun A B) :=
is_embedding.mk (λf f', !is_equiv_ap_to_fun)
definition is_trunc_trunctype [instance] (n : trunc_index) : is_trunc n.+1 (n-Type) :=
begin
apply is_trunc_succ_intro, intro X Y,
fapply is_trunc_equiv_closed,
{apply equiv.symm, apply trunctype_eq_equiv},
fapply is_trunc_equiv_closed,
{apply equiv.symm, apply eq_equiv_equiv},
cases n,
{apply @is_contr_of_inhabited_hprop,
{apply is_trunc_is_embedding_closed,
{apply is_embedding_to_fun} ,
{exact unit.star}},
{apply equiv_of_is_contr_of_is_contr}},
{apply is_trunc_is_embedding_closed,
{apply is_embedding_to_fun},
{exact unit.star}}
end
/- theorems about decidable equality and axiom K -/
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))
definition is_trunc_succ_of_is_trunc_loop (Hn : -1 ≤ n) (Hp : Π(a : A), is_trunc n (a = a))
: is_trunc (n.+1) A :=
begin
apply is_trunc_succ_intro, intros a a',
apply is_trunc_of_imp_is_trunc_of_leq Hn, intro p,
cases p, apply Hp
end
definition is_trunc_succ_iff_is_trunc_loop (A : Type) (Hn : -1 ≤ n) :
is_trunc (n.+1) A ↔ Π(a : A), is_trunc n (a = a) :=
iff.intro _ (is_trunc_succ_of_is_trunc_loop Hn)
-- set_option pp.implicit true
-- set_option pp.coercions true
-- set_option pp.notation false
definition is_trunc_iff_is_contr_loop_succ (n : ) (A : Type)
: is_trunc n A ↔ Π(a : A), is_contr (Ω[succ n](Pointed.mk a)) :=
begin
revert A, induction n with n IH,
{ intros, esimp [Iterated_loop_space], apply iff.intro,
{ intros H a, apply is_contr.mk idp, apply is_hprop.elim},
{ intro H, apply is_hset_of_axiom_K, intros, apply is_hprop.elim}},
{ intros, transitivity _, apply @is_trunc_succ_iff_is_trunc_loop n, constructor,
apply iff.pi_iff_pi, intros,
transitivity _, apply IH,
assert H : Πp : a = a, Ω(Pointed.mk p) = Ω(Pointed.mk (idpath a)),
{ intros, fapply Pointed_eq,
{ esimp, transitivity _,
apply eq_equiv_fn_eq_of_equiv (equiv_eq_closed_right _ p⁻¹),
esimp, apply eq_equiv_eq_closed, apply con.right_inv, apply con.right_inv},
{ esimp, apply con_right_inv2}},
transitivity _,
apply iff.pi_iff_pi, intro p,
rewrite [↑Iterated_loop_space,↓Iterated_loop_space n (Loop_space (pointed.Mk p)),H],
apply iff.refl,
apply iff.imp_iff, reflexivity}
end
definition is_trunc_iff_is_contr_loop (n : ) (A : Type)
: is_trunc (n.-2.+1) A ↔ (Π(a : A), is_contr (Ω[n](pointed.Mk a))) :=
begin
cases n with n,
{ esimp [sub_two,Iterated_loop_space], apply iff.intro,
intro H a, exact is_contr_of_inhabited_hprop a,
intro H, apply is_hprop_of_imp_is_contr, exact H},
{ apply is_trunc_iff_is_contr_loop_succ},
end
end is_trunc open is_trunc
namespace trunc
variable {A : Type}
protected definition code (n : trunc_index) (aa aa' : trunc n.+1 A) : n-Type :=
trunc.rec_on aa (λa, trunc.rec_on aa' (λa', trunctype.mk' n (trunc n (a = a'))))
protected definition encode (n : trunc_index) (aa aa' : trunc n.+1 A) : aa = aa' → trunc.code n aa aa' :=
begin
intro p, cases p, apply (trunc.rec_on aa),
intro a, esimp [trunc.code,trunc.rec_on], exact (tr idp)
end
protected definition decode (n : trunc_index) (aa aa' : trunc n.+1 A) : trunc.code n aa aa' → aa = aa' :=
begin
eapply (trunc.rec_on aa'), eapply (trunc.rec_on aa),
intro a a' x, esimp [trunc.code, trunc.rec_on] at x,
apply (trunc.rec_on x), intro p, exact (ap tr p)
end
definition trunc_eq_equiv (n : trunc_index) (aa aa' : trunc n.+1 A)
: aa = aa' ≃ trunc.code n aa aa' :=
begin
fapply equiv.MK,
{ apply trunc.encode},
{ apply trunc.decode},
{ eapply (trunc.rec_on aa'), eapply (trunc.rec_on aa),
intro a a' x, esimp [trunc.code, trunc.rec_on] at x,
apply (trunc.rec_on x), intro p, cases p, exact idp},
{ intro p, cases p, apply (trunc.rec_on aa), intro a, exact idp},
end
definition tr_eq_tr_equiv (n : trunc_index) (a a' : A)
: (tr a = tr a' :> trunc n.+1 A) ≃ trunc n (a = a') :=
!trunc_eq_equiv
definition is_trunc_trunc_of_is_trunc [instance] [priority 500] (A : Type)
(n m : trunc_index) [H : is_trunc n A] : is_trunc n (trunc m A) :=
begin
revert A m H, eapply (trunc_index.rec_on n),
{ clear n, intro A m H, apply is_contr_equiv_closed,
{ apply equiv_trunc, apply (@is_trunc_of_leq _ -2), exact unit.star} },
{ clear n, intro n IH A m H, cases m with m,
{ apply (@is_trunc_of_leq _ -2), exact unit.star},
{ apply is_trunc_succ_intro, intro aa aa',
apply (@trunc.rec_on _ _ _ aa (λy, !is_trunc_succ_of_is_hprop)),
eapply (@trunc.rec_on _ _ _ aa' (λy, !is_trunc_succ_of_is_hprop)),
intro a a', apply (is_trunc_equiv_closed_rev),
{ apply tr_eq_tr_equiv},
{ exact (IH _ _ _)}}}
end
end trunc open trunc
namespace function
variables {A B : Type}
definition is_surjective_of_is_equiv [instance] (f : A → B) [H : is_equiv f] : is_surjective f :=
is_surjective.mk (λb, !center)
definition is_equiv_equiv_is_embedding_times_is_surjective (f : A → B)
: is_equiv f ≃ (is_embedding f × is_surjective f) :=
equiv_of_is_hprop (λH, (_, _))
(λP, prod.rec_on P (λH₁ H₂, !is_equiv_of_is_surjective_of_is_embedding))
end function