mirror of https://github.com/nmvdw/HITs-Examples
B-fin => K-fin if the underlying type has decidable paths
This commit is contained in:
parent
920fdd91ab
commit
57a4535f87
|
|
@ -4,7 +4,7 @@ Require Import Sub notation variations.k_finite.
|
||||||
Require Import fsets.properties.
|
Require Import fsets.properties.
|
||||||
|
|
||||||
Section finite_hott.
|
Section finite_hott.
|
||||||
Variable A : Type.
|
Variable (A : Type).
|
||||||
Context `{Univalence} `{IsHSet A}.
|
Context `{Univalence} `{IsHSet A}.
|
||||||
|
|
||||||
(* A subobject is B-finite if its extension is B-finite as a type *)
|
(* A subobject is B-finite if its extension is B-finite as a type *)
|
||||||
|
|
@ -243,7 +243,140 @@ Section finite_hott.
|
||||||
Defined.
|
Defined.
|
||||||
|
|
||||||
End split.
|
End split.
|
||||||
|
End finite_hott.
|
||||||
|
|
||||||
|
Arguments Bfin {_} _.
|
||||||
|
|
||||||
|
Section dec_membership.
|
||||||
|
Variable (A : Type).
|
||||||
|
Context `{DecidablePaths A} `{Univalence}.
|
||||||
|
Global Instance DecidableMembership (P : Sub A) (Hfin : Bfin P) (a : A) :
|
||||||
|
Decidable (a ∈ P).
|
||||||
|
Proof.
|
||||||
|
destruct Hfin as [n Hequiv].
|
||||||
|
strip_truncations.
|
||||||
|
revert Hequiv.
|
||||||
|
revert P.
|
||||||
|
induction n.
|
||||||
|
- intros.
|
||||||
|
pose (X_empty _ P Hequiv) as p.
|
||||||
|
rewrite p.
|
||||||
|
apply _.
|
||||||
|
- intros.
|
||||||
|
pose (new_el _ P n Hequiv) as b.
|
||||||
|
destruct b as [b HX'].
|
||||||
|
destruct (split _ P n Hequiv) as [X' X'equiv]. simpl in HX'.
|
||||||
|
unfold member, sub_membership.
|
||||||
|
rewrite (HX' a).
|
||||||
|
pose (IHn X' X'equiv) as IH.
|
||||||
|
destruct IH as [IH | IH].
|
||||||
|
+ left. apply (tr (inl IH)).
|
||||||
|
+ destruct (dec (a = b)) as [Hab | Hab].
|
||||||
|
left. apply (tr (inr (tr Hab))).
|
||||||
|
right. intros α. strip_truncations.
|
||||||
|
destruct α as [β | γ]; [ | strip_truncations]; contradiction.
|
||||||
|
Defined.
|
||||||
|
End dec_membership.
|
||||||
|
|
||||||
|
Section cowd.
|
||||||
|
Variable (A : Type).
|
||||||
|
Context `{DecidablePaths A} `{Univalence}.
|
||||||
|
|
||||||
|
Definition cow := { X : Sub A | Bfin X}.
|
||||||
|
Definition empty_cow : cow.
|
||||||
|
Proof.
|
||||||
|
exists empty. apply empty_finite.
|
||||||
|
Defined.
|
||||||
|
|
||||||
|
Definition add_cow : forall a : A, cow -> cow.
|
||||||
|
Proof.
|
||||||
|
intros a [X PX].
|
||||||
|
exists (fun z => lor (X z) (merely (z = a))).
|
||||||
|
destruct (dec (a ∈ X)) as [Ha | Ha];
|
||||||
|
destruct PX as [n PX];
|
||||||
|
strip_truncations.
|
||||||
|
- (* a ∈ X *)
|
||||||
|
exists n. apply tr.
|
||||||
|
transitivity ({a : A & a ∈ X}); [ | apply PX ].
|
||||||
|
apply equiv_functor_sigma_id.
|
||||||
|
intro a'. eapply equiv_iff_hprop_uncurried ; split; cbn.
|
||||||
|
+ intros Ha'. strip_truncations.
|
||||||
|
destruct Ha' as [HXa' | Haa'].
|
||||||
|
* assumption.
|
||||||
|
* strip_truncations. rewrite Haa'. assumption.
|
||||||
|
+ intros HXa'. apply tr.
|
||||||
|
left. assumption.
|
||||||
|
- (* a ∉ X *)
|
||||||
|
exists (S n). apply tr.
|
||||||
|
destruct PX as [f [g Hfg Hgf adj]].
|
||||||
|
unshelve esplit.
|
||||||
|
+ intros [a' Ha']. cbn in Ha'.
|
||||||
|
destruct (dec (a' = a)) as [Haa' | Haa'].
|
||||||
|
* right. apply tt.
|
||||||
|
* assert (X a') as HXa'.
|
||||||
|
{ strip_truncations.
|
||||||
|
destruct Ha' as [Ha' | Ha']; [ assumption | ].
|
||||||
|
strip_truncations. by (contradiction (Haa' Ha')). }
|
||||||
|
apply (inl (f (a';HXa'))).
|
||||||
|
+ apply isequiv_biinv; simpl.
|
||||||
|
unshelve esplit; simpl.
|
||||||
|
* unfold Sect; simpl.
|
||||||
|
simple refine (_;_).
|
||||||
|
{ destruct 1 as [M | ?].
|
||||||
|
- destruct (g M) as [a' Ha'].
|
||||||
|
exists a'. apply tr.
|
||||||
|
by left.
|
||||||
|
- exists a. apply (tr (inr (tr idpath))). }
|
||||||
|
simpl. intros [a' Ha'].
|
||||||
|
strip_truncations.
|
||||||
|
destruct Ha' as [HXa' | Haa']; simpl;
|
||||||
|
destruct (dec (a' = a)); simpl.
|
||||||
|
** apply path_sigma' with p^. apply path_ishprop.
|
||||||
|
** rewrite Hgf; cbn. done.
|
||||||
|
** apply path_sigma' with p^. apply path_ishprop.
|
||||||
|
** rewrite Hgf; cbn. apply path_sigma' with idpath. apply path_ishprop.
|
||||||
|
* unfold Sect; simpl.
|
||||||
|
simple refine (_;_).
|
||||||
|
{ destruct 1 as [M | ?].
|
||||||
|
- destruct (g M) as [a' Ha'].
|
||||||
|
exists a'. apply tr.
|
||||||
|
by left.
|
||||||
|
- exists a. apply (tr (inr (tr idpath))). }
|
||||||
|
simpl. intros [M | [] ].
|
||||||
|
** destruct (dec (_ = a)) as [Haa' | Haa']; simpl.
|
||||||
|
{ destruct (g M) as [a' Ha']. rewrite Haa' in Ha'. by contradiction. }
|
||||||
|
{ f_ap. }
|
||||||
|
** destruct (dec (a = a)); try by contradiction.
|
||||||
|
reflexivity.
|
||||||
|
Defined.
|
||||||
|
|
||||||
|
Theorem cowy
|
||||||
|
(P : cow -> hProp)
|
||||||
|
(doge : P empty_cow)
|
||||||
|
(koeientaart : forall a c, P c -> P (add_cow a c))
|
||||||
|
:
|
||||||
|
forall X : cow, P X.
|
||||||
|
Proof.
|
||||||
|
intros [X [n FX]]. strip_truncations.
|
||||||
|
revert X FX.
|
||||||
|
induction n; intros X FX.
|
||||||
|
- pose (HX_emp:= X_empty _ X FX).
|
||||||
|
assert ((X; Build_Finite _ 0 (tr FX)) = empty_cow) as HX.
|
||||||
|
{ apply path_sigma' with HX_emp. apply path_ishprop. }
|
||||||
|
rewrite HX. assumption.
|
||||||
|
- pose (a' := new_el _ X n FX).
|
||||||
|
destruct a' as [a' Ha'].
|
||||||
|
destruct (split _ X n FX) as [X' FX'].
|
||||||
|
pose (X'cow := (X'; Build_Finite _ n (tr FX')) : cow).
|
||||||
|
assert ((X; Build_Finite _ (n.+1) (tr FX)) = add_cow a' X'cow) as ℵ.
|
||||||
|
{ simple refine (path_sigma' _ _ _); [ | apply path_ishprop ].
|
||||||
|
apply path_forall. intros a.
|
||||||
|
unfold X'cow.
|
||||||
|
specialize (Ha' a). rewrite Ha'. simpl. reflexivity. }
|
||||||
|
rewrite ℵ.
|
||||||
|
apply koeientaart.
|
||||||
|
apply IHn.
|
||||||
|
Defined.
|
||||||
|
|
||||||
Definition bfin_to_kfin : forall (X : Sub A), Bfin X -> Kf_sub _ X.
|
Definition bfin_to_kfin : forall (X : Sub A), Bfin X -> Kf_sub _ X.
|
||||||
Proof.
|
Proof.
|
||||||
|
|
@ -263,8 +396,8 @@ Section finite_hott.
|
||||||
apply (fun Xz => f(z;Xz)).
|
apply (fun Xz => f(z;Xz)).
|
||||||
- intros.
|
- intros.
|
||||||
simpl in *.
|
simpl in *.
|
||||||
destruct (new_el X n iso) as [a HXX'].
|
destruct (new_el _ X n iso) as [a HXX'].
|
||||||
destruct (split X n iso) as [X' X'equiv].
|
destruct (split _ X n iso) as [X' X'equiv].
|
||||||
destruct (IHn X' X'equiv) as [Y HY].
|
destruct (IHn X' X'equiv) as [Y HY].
|
||||||
exists (Y ∪ {|a|}).
|
exists (Y ∪ {|a|}).
|
||||||
unfold map in *.
|
unfold map in *.
|
||||||
|
|
@ -277,111 +410,111 @@ Section finite_hott.
|
||||||
reflexivity.
|
reflexivity.
|
||||||
Defined.
|
Defined.
|
||||||
|
|
||||||
Context `{A_deceq : DecidablePaths A}.
|
Lemma kfin_is_bfin : @closedUnion A Bfin.
|
||||||
|
|
||||||
(*
|
|
||||||
Lemma kfin_is_bfin : closedUnion Bfin.
|
|
||||||
Proof.
|
Proof.
|
||||||
intros X Y HX HY.
|
intros X Y HX HY.
|
||||||
unfold Bfin in *.
|
pose (Xcow := (X; HX) : cow).
|
||||||
destruct HX as [n Xequiv].
|
pose (Ycow := (Y; HY) : cow).
|
||||||
revert X Xequiv.
|
simple refine (cowy (fun C => Bfin (C.1 ∪ Y)) _ _ Xcow); simpl.
|
||||||
induction n.
|
- assert ((fun a => Trunc (-1) (Empty + Y a)) = (fun a => Y a)) as Help.
|
||||||
- intros.
|
{ apply path_forall. intros z; simpl.
|
||||||
strip_truncations.
|
apply path_iff_ishprop.
|
||||||
rewrite (X_empty X Xequiv).
|
+ intros; strip_truncations; auto.
|
||||||
assert(∅ ∪ Y = Y).
|
destruct X0; auto. destruct e.
|
||||||
{ apply path_forall ; intro z.
|
+ intros ?. apply tr. right; assumption.
|
||||||
compute-[lor].
|
(* TODO FIX THIS with sum_empty_l *)
|
||||||
eauto with lattice_hints typeclass_instances.
|
|
||||||
}
|
}
|
||||||
rewrite X0.
|
rewrite Help. apply HY.
|
||||||
apply HY.
|
- intros a [X' HX'] [n FX'Y]. strip_truncations.
|
||||||
- simpl in *.
|
destruct (dec(a ∈ X')) as [HaX' | HaX'].
|
||||||
intros.
|
* exists n. apply tr.
|
||||||
destruct HY as [m Yequiv].
|
transitivity ({a : A & Trunc (-1) (X' a + Y a)}); [| assumption ].
|
||||||
strip_truncations.
|
apply equiv_functor_sigma_id. intro a'.
|
||||||
destruct (new_el X n Xequiv) as [a HXX'].
|
apply equiv_iff_hprop.
|
||||||
destruct (split X n Xequiv) as [X' X'equiv].
|
{ intros Q. strip_truncations.
|
||||||
destruct (IHn X' (tr X'equiv)) as [k Hk].
|
destruct Q as [Q | Q].
|
||||||
strip_truncations.
|
- strip_truncations.
|
||||||
cbn in *.
|
apply tr. left.
|
||||||
rewrite (path_forall _ _ HXX').
|
destruct Q ; auto.
|
||||||
assert
|
strip_truncations. rewrite t; assumption.
|
||||||
(forall a0,
|
- apply (tr (inr Q)). }
|
||||||
BuildhProp (Trunc (-1) (X' a0 ∨ merely (a0 = a) + Y a0))
|
{ intros Q. strip_truncations.
|
||||||
=
|
destruct Q as [Q | Q]; apply tr.
|
||||||
BuildhProp (hor (Trunc (-1) (X' a0 + Y a0)) (merely (a0 = a)))
|
- left. apply tr. left. done.
|
||||||
).
|
- right. done. }
|
||||||
{
|
* destruct (dec (a ∈ Y)) as [HaY | HaY ].
|
||||||
intros.
|
** exists n. apply tr.
|
||||||
apply path_iff_hprop.
|
transitivity ({a : A & Trunc (-1) (X' a + Y a)}); [| assumption ].
|
||||||
* intros X0.
|
apply equiv_functor_sigma_id. intro a'.
|
||||||
strip_truncations.
|
apply equiv_iff_hprop.
|
||||||
destruct X0 as [X0 | X0].
|
{ intros Q. strip_truncations.
|
||||||
** strip_truncations.
|
destruct Q as [Q | Q].
|
||||||
destruct X0 as [X0 | X0].
|
- strip_truncations.
|
||||||
*** refine (tr(inl(tr _))).
|
apply tr.
|
||||||
apply (inl X0).
|
destruct Q.
|
||||||
*** refine (tr(inr X0)).
|
left. auto.
|
||||||
** refine (tr(inl(tr _))).
|
right. strip_truncations. rewrite t; assumption.
|
||||||
apply (inr X0).
|
- apply (tr (inr Q)). }
|
||||||
* intros X0.
|
{ intros Q. strip_truncations.
|
||||||
strip_truncations.
|
destruct Q as [Q | Q]; apply tr.
|
||||||
destruct X0 as [X0 | X0].
|
- left. apply tr. left. done.
|
||||||
** strip_truncations.
|
- right. done. }
|
||||||
destruct X0 as [X0 | X0].
|
** exists (n.+1). apply tr.
|
||||||
*** refine (tr(inl(tr(inl X0)))).
|
destruct FX'Y as [f [g Hfg Hgf adj]].
|
||||||
*** refine (tr(inr X0)).
|
unshelve esplit.
|
||||||
** refine (tr(inl(tr(inr X0)))).
|
{ intros [a' Ha']. cbn in Ha'.
|
||||||
}
|
destruct (dec (BuildhProp (a' = a))) as [Ha'a | Ha'a].
|
||||||
(* rewrite (path_forall _ _ X0). *)
|
- right. apply tt.
|
||||||
assert
|
- left. refine (f (a';_)).
|
||||||
(
|
strip_truncations.
|
||||||
{a0 : A & hor (Trunc (-1) (X' a0 + Y a0)) (merely (a0 = a))}
|
destruct Ha' as [Ha' | Ha'].
|
||||||
=
|
+ strip_truncations.
|
||||||
{a0 : A & Trunc (-1) (X' a0 + Y a0)}
|
destruct Ha' as [Ha' | Ha'].
|
||||||
+
|
* apply (tr (inl Ha')).
|
||||||
{a0 : A & (merely (a0 = a))}
|
* strip_truncations. contradiction.
|
||||||
).
|
+ apply (tr (inr Ha')). }
|
||||||
{
|
{ apply isequiv_biinv; simpl.
|
||||||
assert ({a0 : A & Trunc (-1) (X' a0 + Y a0)} + {a0 : A & merely (a0 = a)} ->
|
unshelve esplit; simpl.
|
||||||
{a0 : A & hor (Trunc (-1) (X' a0 + Y a0)) (merely (a0 = a))}).
|
- unfold Sect; simpl.
|
||||||
{
|
simple refine (_;_).
|
||||||
intros.
|
{ destruct 1 as [M | ?].
|
||||||
destruct X1.
|
- destruct (g M) as [a' Ha'].
|
||||||
* destruct s as [c p].
|
exists a'.
|
||||||
exists c.
|
strip_truncations; apply tr.
|
||||||
apply tr.
|
destruct Ha' as [Ha' | Ha'].
|
||||||
left.
|
+ left. apply (tr (inl Ha')).
|
||||||
apply p.
|
+ right. done.
|
||||||
* destruct s as [c p].
|
- exists a. apply (tr (inl (tr (inr (tr idpath))))). }
|
||||||
exists c.
|
{ intros [a' Ha']; simpl.
|
||||||
apply tr.
|
strip_truncations.
|
||||||
right. apply p.
|
destruct Ha' as [HXa' | Haa']; simpl;
|
||||||
|
destruct (dec (a' = a)); simpl.
|
||||||
simple refine (path_universe _).
|
** apply path_sigma' with p^. apply path_ishprop.
|
||||||
* intros [a0 p].
|
** rewrite Hgf; cbn. apply path_sigma' with idpath. apply path_ishprop.
|
||||||
destruct (dec (a0 = a)).
|
** apply path_sigma' with p^. apply path_ishprop.
|
||||||
** right. exists a0. apply (tr p0).
|
** rewrite Hgf; cbn. done. }
|
||||||
** left.
|
- unfold Sect; simpl.
|
||||||
exists a0.
|
simple refine (_;_).
|
||||||
strip_truncations.
|
{ destruct 1 as [M | ?].
|
||||||
destruct p ; strip_truncations.
|
- (* destruct (g M) as [a' Ha']. *)
|
||||||
*** apply (tr t).
|
exists (g M).1.
|
||||||
*** contradiction (n0 t).
|
simple refine (Trunc_rec _ (g M).2).
|
||||||
* apply isequiv_biinv.
|
intros Ha'.
|
||||||
unfold BiInv.
|
apply tr.
|
||||||
split.
|
(* strip_truncations; apply tr. *)
|
||||||
**
|
destruct Ha' as [Ha' | Ha'].
|
||||||
|
+ left. apply (tr (inl Ha')).
|
||||||
exists a0
|
+ right. done.
|
||||||
}
|
- exists a. apply (tr (inl (tr (inr (tr idpath))))). }
|
||||||
rewrite X1.
|
simpl. intros [M | [] ].
|
||||||
apply finite_sum.
|
** destruct (dec (_ = a)) as [Haa' | Haa']; simpl.
|
||||||
* simple refine (Build_Finite _ k (tr Hk)).
|
{ destruct (g M) as [a' Ha']. simpl in Haa'.
|
||||||
* apply singleton.
|
strip_truncations.
|
||||||
Admitted.
|
rewrite Haa' in Ha'. destruct Ha'; by contradiction. }
|
||||||
*)
|
{ f_ap. transitivity (f (g M)); [ | apply Hfg].
|
||||||
|
f_ap. apply path_sigma' with idpath.
|
||||||
End finite_hott.
|
apply path_ishprop. }
|
||||||
|
** destruct (dec (a = a)); try by contradiction.
|
||||||
|
reflexivity. }
|
||||||
|
Defined.
|
||||||
|
End cowd.
|
||||||
|
|
|
||||||
Loading…
Reference in New Issue