mirror of https://github.com/nmvdw/HITs-Examples
523 lines
17 KiB
Coq
523 lines
17 KiB
Coq
(* Bishop-finiteness is that "default" notion of finiteness in the HoTT library *)
|
||
Require Import HoTT HitTactics.
|
||
Require Import Sub notation variations.k_finite.
|
||
Require Import fsets.properties.
|
||
|
||
Section finite_hott.
|
||
Variable (A : Type).
|
||
Context `{Univalence} `{IsHSet A}.
|
||
|
||
(* A subobject is B-finite if its extension is B-finite as a type *)
|
||
Definition Bfin (X : Sub A) : hProp := BuildhProp (Finite {a : A & a ∈ X}).
|
||
|
||
Global Instance singleton_contr a : Contr {b : A & b ∈ {|a|}}.
|
||
Proof.
|
||
exists (a; tr idpath).
|
||
intros [b p].
|
||
simple refine (Trunc_ind (fun p => (a; tr 1%path) = (b; p)) _ p).
|
||
clear p; intro p. simpl.
|
||
apply path_sigma' with (p^).
|
||
apply path_ishprop.
|
||
Defined.
|
||
|
||
Definition singleton_fin_equiv' a : Fin 1 -> {b : A & b ∈ {|a|}}.
|
||
Proof.
|
||
intros _. apply (center {b : A & b ∈ {|a|}}).
|
||
Defined.
|
||
|
||
Global Instance singleton_fin_equiv a : IsEquiv (singleton_fin_equiv' a).
|
||
Proof. apply _. Defined.
|
||
|
||
Definition singleton : closedSingleton Bfin.
|
||
Proof.
|
||
intros a.
|
||
simple refine (Build_Finite _ 1 _).
|
||
apply tr.
|
||
symmetry.
|
||
refine (BuildEquiv _ _ (singleton_fin_equiv' a) _).
|
||
Defined.
|
||
|
||
Definition empty_finite : closedEmpty Bfin.
|
||
Proof.
|
||
simple refine (Build_Finite _ 0 _).
|
||
apply tr.
|
||
simple refine (BuildEquiv _ _ _ _).
|
||
intros [a p]; apply p.
|
||
Defined.
|
||
|
||
Definition decidable_empty_finite : hasDecidableEmpty Bfin.
|
||
Proof.
|
||
intros X Y.
|
||
destruct Y as [n Xn].
|
||
strip_truncations.
|
||
destruct Xn as [f [g fg gf adj]].
|
||
destruct n.
|
||
- refine (tr(inl _)).
|
||
apply path_forall. intro z.
|
||
apply path_iff_hprop.
|
||
* intros p.
|
||
contradiction (f (z;p)).
|
||
* contradiction.
|
||
- refine (tr(inr _)).
|
||
apply (tr(g(inr tt))).
|
||
Defined.
|
||
|
||
Lemma no_union
|
||
(f : forall (X Y : Sub A),
|
||
Bfin X -> Bfin Y -> Bfin (X ∪ Y))
|
||
(a b : A) :
|
||
hor (a = b) (a = b -> Empty).
|
||
Proof.
|
||
specialize (f {|a|} {|b|} (singleton a) (singleton b)).
|
||
unfold Bfin in f.
|
||
destruct f as [n pn].
|
||
strip_truncations.
|
||
destruct pn as [f [g fg gf _]].
|
||
destruct n as [|n].
|
||
unfold Sect in *.
|
||
- contradiction f.
|
||
exists a. apply (tr(inl(tr idpath))).
|
||
- destruct n as [|n].
|
||
+ (* If the size of the union is 1, then (a = b) *)
|
||
refine (tr (inl _)).
|
||
pose (s1 := (a;tr(inl(tr idpath)))
|
||
: {c : A & Trunc (-1) (Trunc (-1) (c = a) + Trunc (-1) (c = b))}).
|
||
pose (s2 := (b;tr(inr(tr idpath)))
|
||
: {c : A & Trunc (-1) (Trunc (-1) (c = a) + Trunc (-1) (c = b))}).
|
||
refine (ap (fun x => x.1) (gf s1)^ @ _ @ (ap (fun x => x.1) (gf s2))).
|
||
assert (fs_eq : f s1 = f s2).
|
||
{ by apply path_ishprop. }
|
||
refine (ap (fun x => (g x).1) fs_eq).
|
||
+ (* Otherwise, ¬(a = b) *)
|
||
refine (tr (inr _)).
|
||
intros p.
|
||
pose (s1 := inl (inr tt) : Fin n + Unit + Unit).
|
||
pose (s2 := inr tt : Fin n + Unit + Unit).
|
||
pose (gs1 := g s1).
|
||
pose (c := g s1).
|
||
pose (gs2 := g s2).
|
||
pose (d := g s2).
|
||
assert (Hgs1 : gs1 = c) by reflexivity.
|
||
assert (Hgs2 : gs2 = d) by reflexivity.
|
||
destruct c as [x px'].
|
||
destruct d as [y py'].
|
||
simple refine (Trunc_ind _ _ px') ; intros px.
|
||
simple refine (Trunc_ind _ _ py') ; intros py.
|
||
simpl.
|
||
cut (x = y).
|
||
{
|
||
enough (s1 = s2) as X.
|
||
{
|
||
intros.
|
||
unfold s1, s2 in X.
|
||
refine (not_is_inl_and_inr' (inl(inr tt)) _ _).
|
||
+ apply tt.
|
||
+ rewrite X ; apply tt.
|
||
}
|
||
transitivity (f gs1).
|
||
{ apply (fg s1)^. }
|
||
symmetry ; transitivity (f gs2).
|
||
{ apply (fg s2)^. }
|
||
rewrite Hgs1, Hgs2.
|
||
f_ap.
|
||
simple refine (path_sigma _ _ _ _ _); [ | apply path_ishprop ]; simpl.
|
||
destruct px as [p1 | p1] ; destruct py as [p2 | p2] ; strip_truncations.
|
||
* apply (p2 @ p1^).
|
||
* refine (p2 @ _^ @ p1^). auto.
|
||
* refine (p2 @ _ @ p1^). auto.
|
||
* apply (p2 @ p1^).
|
||
}
|
||
destruct px as [px | px] ; destruct py as [py | py]; strip_truncations.
|
||
** apply (px @ py^).
|
||
** refine (px @ _ @ py^). auto.
|
||
** refine (px @ _ @ py^). symmetry. auto.
|
||
** apply (px @ py^).
|
||
Defined.
|
||
|
||
Section empty.
|
||
Variable (X : A -> hProp)
|
||
(Xequiv : {a : A & a ∈ X} <~> Fin 0).
|
||
|
||
Lemma X_empty : X = ∅.
|
||
Proof.
|
||
apply path_forall.
|
||
intro z.
|
||
apply path_iff_hprop ; try contradiction.
|
||
intro x.
|
||
destruct Xequiv as [f fequiv].
|
||
contradiction (f(z;x)).
|
||
Defined.
|
||
|
||
End empty.
|
||
|
||
Section split.
|
||
Variable (P : A -> hProp)
|
||
(n : nat)
|
||
(Xequiv : {a : A & P a } <~> Fin n + Unit).
|
||
|
||
Definition split : exists P' : Sub A, exists b : A,
|
||
({a : A & P' a} <~> Fin n) * (forall x, P x = P' x ∨ merely (x = b)).
|
||
Proof.
|
||
destruct Xequiv as [f [g fg gf adj]].
|
||
unfold Sect in *.
|
||
pose (fun x : A => sig (fun y : Fin n => x = (g (inl y)).1)) as P'.
|
||
assert (forall x, IsHProp (P' x)).
|
||
{
|
||
intros a. unfold P'.
|
||
apply hprop_allpath.
|
||
intros [x px] [y py].
|
||
simple refine (path_sigma _ _ _ _ _); [ simpl | apply path_ishprop ].
|
||
apply path_sum_inl with Unit.
|
||
cut (g (inl x) = g (inl y)).
|
||
{ intros p.
|
||
pose (ap f p) as Hp.
|
||
by rewrite !fg in Hp. }
|
||
apply path_sigma with (px^ @ py).
|
||
apply path_ishprop.
|
||
}
|
||
exists (fun a => BuildhProp (P' a)).
|
||
exists (g (inr tt)).1.
|
||
split.
|
||
{ unshelve eapply BuildEquiv.
|
||
{ refine (fun x => x.2.1). }
|
||
apply isequiv_biinv.
|
||
unshelve esplit;
|
||
exists (fun x => (((g (inl x)).1; (x; idpath)))).
|
||
- intros [a [y p]]; cbn.
|
||
eapply path_sigma with p^.
|
||
apply path_ishprop.
|
||
- intros x; cbn.
|
||
reflexivity. }
|
||
{ intros a.
|
||
unfold P'.
|
||
apply path_iff_hprop.
|
||
- intros Ha.
|
||
pose (y := f (a;Ha)).
|
||
assert (Hy : y = f (a; Ha)) by reflexivity.
|
||
destruct y as [y | []].
|
||
+ refine (tr (inl _)).
|
||
exists y.
|
||
rewrite Hy.
|
||
by rewrite gf.
|
||
+ refine (tr (inr (tr _))).
|
||
rewrite Hy.
|
||
by rewrite gf.
|
||
- intros Hstuff.
|
||
strip_truncations.
|
||
destruct Hstuff as [[y Hy] | Ha].
|
||
+ rewrite Hy.
|
||
apply (g (inl y)).2.
|
||
+ strip_truncations.
|
||
rewrite Ha.
|
||
apply (g (inr tt)).2. }
|
||
Defined.
|
||
|
||
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.
|
||
destruct (split _ P n Hequiv) as
|
||
(P' & b & HP' & HP).
|
||
unfold member, sub_membership.
|
||
rewrite (HP a).
|
||
destruct (IHn P' HP') 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.
|
||
- destruct (split _ X n FX) as
|
||
(X' & b & FX' & HX).
|
||
pose (X'cow := (X'; Build_Finite _ n (tr FX')) : cow).
|
||
assert ((X; Build_Finite _ (n.+1) (tr FX)) = add_cow b X'cow) as ℵ.
|
||
{ simple refine (path_sigma' _ _ _); [ | apply path_ishprop ].
|
||
apply path_forall. intros a.
|
||
unfold X'cow.
|
||
rewrite (HX a). simpl. reflexivity. }
|
||
rewrite ℵ.
|
||
apply koeientaart.
|
||
apply IHn.
|
||
Defined.
|
||
|
||
Definition bfin_to_kfin : forall (P : Sub A), Bfin P -> Kf_sub _ P.
|
||
Proof.
|
||
intros P [n f].
|
||
strip_truncations.
|
||
unfold Kf_sub, Kf_sub_intern.
|
||
revert f. revert P.
|
||
induction n; intros P f.
|
||
- exists ∅.
|
||
apply path_forall; intro a; simpl.
|
||
apply path_iff_hprop; [ | contradiction ].
|
||
intros p.
|
||
apply (f (a;p)).
|
||
- destruct (split _ P n f) as
|
||
(P' & b & HP' & HP).
|
||
destruct (IHn P' HP') as [Y HY].
|
||
exists (Y ∪ {|b|}).
|
||
apply path_forall; intro a. simpl.
|
||
rewrite <- HY.
|
||
apply HP.
|
||
Defined.
|
||
|
||
Lemma kfin_is_bfin : @closedUnion A Bfin.
|
||
Proof.
|
||
intros X Y HX HY.
|
||
pose (Xcow := (X; HX) : cow).
|
||
pose (Ycow := (Y; HY) : cow).
|
||
simple refine (cowy (fun C => Bfin (C.1 ∪ Y)) _ _ Xcow); simpl.
|
||
- assert ((fun a => Trunc (-1) (Empty + Y a)) = (fun a => Y a)) as Help.
|
||
{ apply path_forall. intros z; simpl.
|
||
apply path_iff_ishprop.
|
||
+ intros; strip_truncations; auto.
|
||
destruct X0; auto. destruct e.
|
||
+ intros ?. apply tr. right; assumption.
|
||
(* TODO FIX THIS with sum_empty_l *)
|
||
}
|
||
rewrite Help. apply HY.
|
||
- intros a [X' HX'] [n FX'Y]. strip_truncations.
|
||
destruct (dec(a ∈ X')) as [HaX' | HaX'].
|
||
* exists n. apply tr.
|
||
transitivity ({a : A & Trunc (-1) (X' a + Y a)}); [| assumption ].
|
||
apply equiv_functor_sigma_id. intro a'.
|
||
apply equiv_iff_hprop.
|
||
{ intros Q. strip_truncations.
|
||
destruct Q as [Q | Q].
|
||
- strip_truncations.
|
||
apply tr. left.
|
||
destruct Q ; auto.
|
||
strip_truncations. rewrite t; assumption.
|
||
- apply (tr (inr Q)). }
|
||
{ intros Q. strip_truncations.
|
||
destruct Q as [Q | Q]; apply tr.
|
||
- left. apply tr. left. done.
|
||
- right. done. }
|
||
* destruct (dec (a ∈ Y)) as [HaY | HaY ].
|
||
** exists n. apply tr.
|
||
transitivity ({a : A & Trunc (-1) (X' a + Y a)}); [| assumption ].
|
||
apply equiv_functor_sigma_id. intro a'.
|
||
apply equiv_iff_hprop.
|
||
{ intros Q. strip_truncations.
|
||
destruct Q as [Q | Q].
|
||
- strip_truncations.
|
||
apply tr.
|
||
destruct Q.
|
||
left. auto.
|
||
right. strip_truncations. rewrite t; assumption.
|
||
- apply (tr (inr Q)). }
|
||
{ intros Q. strip_truncations.
|
||
destruct Q as [Q | Q]; apply tr.
|
||
- left. apply tr. left. done.
|
||
- right. done. }
|
||
** exists (n.+1). apply tr.
|
||
destruct FX'Y as [f [g Hfg Hgf adj]].
|
||
unshelve esplit.
|
||
{ intros [a' Ha']. cbn in Ha'.
|
||
destruct (dec (BuildhProp (a' = a))) as [Ha'a | Ha'a].
|
||
- right. apply tt.
|
||
- left. refine (f (a';_)).
|
||
strip_truncations.
|
||
destruct Ha' as [Ha' | Ha'].
|
||
+ strip_truncations.
|
||
destruct Ha' as [Ha' | Ha'].
|
||
* apply (tr (inl Ha')).
|
||
* strip_truncations. contradiction.
|
||
+ apply (tr (inr Ha')). }
|
||
{ apply isequiv_biinv; simpl.
|
||
unshelve esplit; simpl.
|
||
- unfold Sect; simpl.
|
||
simple refine (_;_).
|
||
{ destruct 1 as [M | ?].
|
||
- destruct (g M) as [a' Ha'].
|
||
exists a'.
|
||
strip_truncations; apply tr.
|
||
destruct Ha' as [Ha' | Ha'].
|
||
+ left. apply (tr (inl Ha')).
|
||
+ right. done.
|
||
- exists a. apply (tr (inl (tr (inr (tr idpath))))). }
|
||
{ intros [a' Ha']; simpl.
|
||
strip_truncations.
|
||
destruct Ha' as [HXa' | Haa']; simpl;
|
||
destruct (dec (a' = a)); simpl.
|
||
** apply path_sigma' with p^. apply path_ishprop.
|
||
** rewrite Hgf; cbn. apply path_sigma' with idpath. apply path_ishprop.
|
||
** apply path_sigma' with p^. apply path_ishprop.
|
||
** rewrite Hgf; cbn. done. }
|
||
- unfold Sect; simpl.
|
||
simple refine (_;_).
|
||
{ destruct 1 as [M | ?].
|
||
- (* destruct (g M) as [a' Ha']. *)
|
||
exists (g M).1.
|
||
simple refine (Trunc_rec _ (g M).2).
|
||
intros Ha'.
|
||
apply tr.
|
||
(* strip_truncations; apply tr. *)
|
||
destruct Ha' as [Ha' | Ha'].
|
||
+ left. apply (tr (inl Ha')).
|
||
+ right. done.
|
||
- exists a. apply (tr (inl (tr (inr (tr idpath))))). }
|
||
simpl. intros [M | [] ].
|
||
** destruct (dec (_ = a)) as [Haa' | Haa']; simpl.
|
||
{ destruct (g M) as [a' Ha']. simpl in Haa'.
|
||
strip_truncations.
|
||
rewrite Haa' in Ha'. destruct Ha'; by contradiction. }
|
||
{ f_ap. transitivity (f (g M)); [ | apply Hfg].
|
||
f_ap. apply path_sigma' with idpath.
|
||
apply path_ishprop. }
|
||
** destruct (dec (a = a)); try by contradiction.
|
||
reflexivity. }
|
||
Defined.
|
||
|
||
End cowd.
|
||
|
||
Section Kf_to_Bf.
|
||
Context `{Univalence}.
|
||
|
||
Definition FSet_to_Bfin (A : Type) `{DecidablePaths A} : forall (X : FSet A), Bfin (map X).
|
||
Proof.
|
||
hinduction; try (intros; apply path_ishprop).
|
||
- exists 0. apply tr. simpl.
|
||
simple refine (BuildEquiv _ _ _ _).
|
||
destruct 1 as [? []].
|
||
- intros a.
|
||
exists 1. apply tr. simpl.
|
||
transitivity Unit; [ | symmetry; apply sum_empty_l ].
|
||
unshelve esplit.
|
||
+ exact (fun _ => tt).
|
||
+ apply isequiv_biinv. split.
|
||
* exists (fun _ => (a; tr(idpath))).
|
||
intros [b Hb]. strip_truncations.
|
||
apply path_sigma' with Hb^.
|
||
apply path_ishprop.
|
||
* exists (fun _ => (a; tr(idpath))).
|
||
intros []. reflexivity.
|
||
- intros Y1 Y2 HY1 HY2.
|
||
apply kfin_is_bfin; auto.
|
||
Defined.
|
||
|
||
Instance Kf_to_Bf (X : Type) (Hfin : Kf X) `{DecidablePaths X} : Finite X.
|
||
Proof.
|
||
apply Kf_unfold in Hfin.
|
||
destruct Hfin as [Y HY].
|
||
pose (X' := FSet_to_Bfin _ Y).
|
||
unfold Bfin in X'.
|
||
simple refine (finite_equiv' _ _ X').
|
||
unshelve esplit.
|
||
- intros [a ?]. apply a.
|
||
- apply isequiv_biinv. split.
|
||
* exists (fun a => (a;HY a)).
|
||
intros [b Hb].
|
||
apply path_sigma' with idpath.
|
||
apply path_ishprop.
|
||
* exists (fun a => (a;HY a)).
|
||
intros b. reflexivity.
|
||
Defined.
|
||
|
||
End Kf_to_Bf.
|