mirror of https://github.com/nmvdw/HITs-Examples
120 lines
2.9 KiB
Coq
120 lines
2.9 KiB
Coq
Require Import HoTT HitTactics.
|
||
Require Import lattice representations.definition fsets.operations extensionality Sub fsets.properties.
|
||
|
||
Section k_finite.
|
||
|
||
Context (A : Type).
|
||
Context `{Univalence}.
|
||
|
||
Definition map (X : FSet A) : Sub A := fun a => a ∈ X.
|
||
|
||
Global Instance map_injective : IsEmbedding map.
|
||
Proof.
|
||
apply isembedding_isinj_hset. (* We use the fact that both [FSet A] and [Sub A] are hSets *)
|
||
intros X Y HXY.
|
||
apply fset_ext.
|
||
apply apD10. exact HXY.
|
||
Defined.
|
||
|
||
Definition Kf_sub_intern (B : Sub A) := exists (X : FSet A), B = map X.
|
||
|
||
Instance Kf_sub_hprop B : IsHProp (Kf_sub_intern B).
|
||
Proof.
|
||
apply hprop_allpath.
|
||
intros [X PX] [Y PY].
|
||
assert (X = Y) as HXY.
|
||
{ apply fset_ext. apply apD10.
|
||
transitivity B; [ symmetry | ]; assumption. }
|
||
apply path_sigma with HXY. simpl.
|
||
apply set_path2.
|
||
Defined.
|
||
|
||
Definition Kf_sub (B : Sub A) : hProp := BuildhProp (Kf_sub_intern B).
|
||
|
||
Definition Kf : hProp := Kf_sub (fun x => True).
|
||
|
||
Instance: IsHProp {X : FSet A & forall a : A, map X a}.
|
||
Proof.
|
||
apply hprop_allpath.
|
||
intros [X PX] [Y PY].
|
||
assert (X = Y) as HXY.
|
||
{ apply fset_ext. intros a.
|
||
unfold map in *.
|
||
apply path_hprop.
|
||
apply equiv_iff_hprop; intros.
|
||
+ apply PY.
|
||
+ apply PX. }
|
||
apply path_sigma with HXY. simpl.
|
||
apply path_forall. intro.
|
||
apply path_ishprop.
|
||
Defined.
|
||
|
||
Lemma Kf_unfold : Kf <~> (exists (X : FSet A), forall (a : A), map X a).
|
||
Proof.
|
||
apply equiv_equiv_iff_hprop. apply _. apply _.
|
||
split.
|
||
- intros [X PX]. exists X. intro a.
|
||
rewrite <- PX. done.
|
||
- intros [X PX]. exists X. apply path_forall; intro a.
|
||
apply path_hprop.
|
||
symmetry. apply if_hprop_then_equiv_Unit; [ apply _ | ].
|
||
apply PX.
|
||
Defined.
|
||
|
||
End k_finite.
|
||
|
||
Arguments map {_} {_} _.
|
||
|
||
Section structure_k_finite.
|
||
Context (A : Type).
|
||
Context `{Univalence}.
|
||
|
||
Lemma map_union : forall X Y : FSet A, map (X ∪ Y) = max_fun (map X) (map Y).
|
||
Proof.
|
||
intros.
|
||
unfold map, max_fun.
|
||
reflexivity.
|
||
Defined.
|
||
|
||
Lemma k_finite_union : closedUnion (Kf_sub A).
|
||
Proof.
|
||
unfold closedUnion, Kf_sub, Kf_sub_intern.
|
||
intros.
|
||
destruct X0 as [SX XP].
|
||
destruct X1 as [SY YP].
|
||
exists (SX ∪ SY).
|
||
rewrite map_union.
|
||
rewrite XP, YP.
|
||
reflexivity.
|
||
Defined.
|
||
|
||
Lemma k_finite_empty : closedEmpty (Kf_sub A).
|
||
Proof.
|
||
exists ∅.
|
||
reflexivity.
|
||
Defined.
|
||
|
||
Lemma k_finite_singleton : closedSingleton (Kf_sub A).
|
||
Proof.
|
||
intro.
|
||
exists {|a|}.
|
||
cbn.
|
||
apply path_forall.
|
||
intro z.
|
||
reflexivity.
|
||
Defined.
|
||
|
||
Lemma k_finite_hasDecidableEmpty : hasDecidableEmpty (Kf_sub A).
|
||
Proof.
|
||
unfold hasDecidableEmpty, closedEmpty, Kf_sub, Kf_sub_intern, map.
|
||
intros.
|
||
destruct X0 as [SX EX].
|
||
rewrite EX.
|
||
simple refine (Trunc_ind _ _ (merely_choice SX)).
|
||
intros [SXE | H1].
|
||
- rewrite SXE.
|
||
apply (tr (inl idpath)).
|
||
- apply (tr (inr H1)).
|
||
Defined.
|
||
End structure_k_finite.
|