mirror of https://github.com/nmvdw/HITs-Examples
289 lines
8.6 KiB
Coq
289 lines
8.6 KiB
Coq
(* Enumerated finite sets *)
|
||
Require Import HoTT HitTactics.
|
||
Require Import disjunction Sub.
|
||
Require Import representations.cons_repr representations.definition.
|
||
Require Import variations.k_finite.
|
||
From fsets Require Import operations isomorphism properties_decidable operations_decidable.
|
||
|
||
Fixpoint listExt {A} (ls : list A) : Sub A := fun x =>
|
||
match ls with
|
||
| nil => False_hp
|
||
| cons a ls' => BuildhProp (Trunc (-1) (x = a)) ∨ listExt ls' x
|
||
end.
|
||
|
||
Fixpoint map {A B} (f : A -> B) (ls : list A) : list B :=
|
||
match ls with
|
||
| nil => nil
|
||
| cons x xs => cons (f x) (map f xs)
|
||
end.
|
||
|
||
Fixpoint filterD {A} (P : A -> Bool) (ls : list A) : list { x : A | P x = true }.
|
||
Proof.
|
||
destruct ls as [|x xs].
|
||
- exact nil.
|
||
- enough ((P x = true) + (P x = false)) as HP.
|
||
{ destruct HP as [HP | HP].
|
||
+ refine (cons (exist _ x HP) (filterD _ P xs)).
|
||
+ refine (filterD _ P xs).
|
||
}
|
||
{ destruct (P x); [left | right]; reflexivity. }
|
||
Defined.
|
||
|
||
Lemma filterD_cons {A} (P : A -> Bool) (a : A) (ls : list A) (Pa : P a = true) :
|
||
filterD P (cons a ls) = cons (a;Pa) (filterD P ls).
|
||
Proof.
|
||
simpl.
|
||
destruct (if P a as b return ((b = true) + (b = false))
|
||
then inl 1%path
|
||
else inr 1%path) as [Pa' | Pa'].
|
||
- rewrite (set_path2 Pa Pa'). reflexivity.
|
||
- rewrite Pa in Pa'. contradiction (true_ne_false Pa').
|
||
Defined.
|
||
|
||
Lemma filterD_cons_no {A} (P : A -> Bool) (a : A) (ls : list A) (Pa : P a = false) :
|
||
filterD P (cons a ls) = filterD P ls.
|
||
Proof.
|
||
simpl.
|
||
destruct (if P a as b return ((b = true) + (b = false))
|
||
then inl 1%path
|
||
else inr 1%path) as [Pa' | Pa'].
|
||
- rewrite Pa' in Pa. contradiction (true_ne_false Pa).
|
||
- reflexivity.
|
||
Defined.
|
||
|
||
Lemma filterD_lookup {A} (P : A -> Bool) (x : A) (ls : list A) (Px : P x = true) :
|
||
listExt ls x -> listExt (filterD P ls) (x;Px).
|
||
Proof.
|
||
induction ls as [| a ls].
|
||
- simpl. exact idmap.
|
||
- assert ((P a = true) + (P a = false)) as HPA.
|
||
{ destruct (P a); [left | right]; reflexivity. }
|
||
destruct HPA as [Pa | Pa].
|
||
+ rewrite (filterD_cons P a ls Pa). simpl.
|
||
simple refine (Trunc_ind _ _). intros [Hxa | HIH]; apply tr.
|
||
* left. strip_truncations.
|
||
apply tr.
|
||
apply path_sigma' with Hxa.
|
||
apply set_path2.
|
||
* right. apply (IHls HIH).
|
||
+ rewrite (filterD_cons_no P a ls Pa). simpl.
|
||
simple refine (Trunc_ind _ _). intros [Hxa | HIH].
|
||
* strip_truncations.
|
||
rewrite <- Hxa in Pa. rewrite Px in Pa.
|
||
contradiction (true_ne_false Pa).
|
||
* apply IHls. apply HIH.
|
||
Defined.
|
||
|
||
(** Definition of finite sets in an enumerated sense *)
|
||
Definition enumerated (A : Type) : hProp :=
|
||
hexists (fun ls => forall (a : A), listExt ls a).
|
||
|
||
(** Properties of enumerated sets: closed under decidable subsets *)
|
||
Lemma enumerated_comprehension (A : Type) (P : A -> Bool) :
|
||
enumerated A -> enumerated { x : A | P x = true }.
|
||
Proof.
|
||
intros HeA. strip_truncations. destruct HeA as [eA HeA].
|
||
apply tr.
|
||
exists (filterD P eA).
|
||
intros [x Px].
|
||
apply filterD_lookup.
|
||
apply (HeA x).
|
||
Defined.
|
||
|
||
Lemma map_listExt {A B} (f : A -> B) (ls : list A) (y : A) :
|
||
listExt ls y -> listExt (map f ls) (f y).
|
||
Proof.
|
||
induction ls.
|
||
- simpl. apply idmap.
|
||
- simpl. simple refine (Trunc_ind _ _). intros [Hxa | HIH]; apply tr.
|
||
+ left. strip_truncations. apply tr. f_ap.
|
||
+ right. apply IHls. apply HIH.
|
||
Defined.
|
||
|
||
(** Properties of enumerated sets: closed under surjections *)
|
||
Lemma enumerated_surj (A B : Type) (f : A -> B) :
|
||
IsSurjection f -> enumerated A -> enumerated B.
|
||
Proof.
|
||
intros Hsurj HeA. strip_truncations; apply tr.
|
||
destruct HeA as [eA HeA].
|
||
exists (map f eA).
|
||
intros x. specialize (Hsurj x).
|
||
pose (t := center (merely (hfiber f x))).
|
||
simple refine (@Trunc_rec (-1) (hfiber f x) (listExt (map f eA) x) _ _ t).
|
||
intros [y Hfy].
|
||
specialize (HeA y). rewrite <- Hfy.
|
||
apply map_listExt. apply HeA.
|
||
Defined.
|
||
|
||
Lemma listExt_app_r {A} (ls ls' : list A) (x : A) :
|
||
listExt ls x -> listExt (ls ++ ls') x.
|
||
Proof.
|
||
induction ls; simpl.
|
||
- exact Empty_rec.
|
||
- simple refine (Trunc_ind _ _). intros [Hxa | HIH]; apply tr.
|
||
+ left. apply Hxa.
|
||
+ right. apply IHls. apply HIH.
|
||
Defined.
|
||
|
||
Lemma listExt_app_l {A} (ls ls' : list A) (x : A) :
|
||
listExt ls x -> listExt (ls' ++ ls) x.
|
||
Proof.
|
||
induction ls'; simpl.
|
||
- apply idmap.
|
||
- intros Hls.
|
||
apply tr.
|
||
right. apply IHls'. apply Hls.
|
||
Defined.
|
||
|
||
(** Properties of enumerated sets: closed under sums *)
|
||
Lemma enumerated_sum (A B : Type) :
|
||
enumerated A -> enumerated B -> enumerated (A + B).
|
||
Proof.
|
||
intros HeA HeB.
|
||
strip_truncations; apply tr.
|
||
destruct HeA as [eA HeA], HeB as [eB HeB].
|
||
exists (app (map inl eA) (map inr eB)).
|
||
intros [x | x].
|
||
- apply listExt_app_r. apply map_listExt. apply HeA.
|
||
- apply listExt_app_l. apply map_listExt. apply HeB.
|
||
Defined.
|
||
|
||
Fixpoint listProd_sing {A B} (x : A) (ys : list B) : list (A * B).
|
||
Proof.
|
||
destruct ys as [|y ys].
|
||
- exact nil.
|
||
- refine (cons (x,y) _).
|
||
apply (listProd_sing _ _ x ys).
|
||
Defined.
|
||
|
||
Fixpoint listProd {A B} (xs : list A) (ys : list B) : list (A * B).
|
||
Proof.
|
||
destruct xs as [|x xs].
|
||
- exact nil.
|
||
- refine (app _ _).
|
||
+ exact (listProd_sing x ys).
|
||
+ exact (listProd _ _ xs ys).
|
||
Defined.
|
||
|
||
Lemma listExt_prod_sing {A B} (x : A) (y : B) (ys : list B) :
|
||
listExt ys y -> listExt (listProd_sing x ys) (x, y).
|
||
Proof.
|
||
induction ys; simpl.
|
||
- exact idmap.
|
||
- simple refine (Trunc_ind _ _). intros [Hxy | HIH]; simpl; apply tr.
|
||
+ left. strip_truncations. apply tr. f_ap.
|
||
+ right. apply IHys. apply HIH.
|
||
Defined.
|
||
|
||
Lemma listExt_prod `{Funext} {A B} (xs : list A) (ys : list B) : forall (x : A) (y : B),
|
||
listExt xs x -> listExt ys y -> listExt (listProd xs ys) (x,y).
|
||
Proof.
|
||
induction xs as [| x' xs]; intros x y.
|
||
- simpl. contradiction.
|
||
- simpl. simple refine (Trunc_ind _ _). intros Htx. simpl.
|
||
induction ys as [| y' ys].
|
||
+ simpl. contradiction.
|
||
+ simpl. simple refine (Trunc_ind _ _). intros Hty. simpl. apply tr.
|
||
destruct Htx as [Hxx' | Hxs], Hty as [Hyy' | Hys].
|
||
* left. strip_truncations. apply tr. f_ap.
|
||
* right. strip_truncations. rewrite <- Hxx'. clear Hxx'.
|
||
apply listExt_app_r.
|
||
apply listExt_prod_sing. assumption.
|
||
* right. strip_truncations. rewrite <- Hyy'.
|
||
rewrite <- Hyy' in IHxs.
|
||
apply listExt_app_l. apply IHxs. assumption.
|
||
simpl. apply tr. left. apply tr. reflexivity.
|
||
* right.
|
||
apply listExt_app_l.
|
||
apply IHxs. assumption.
|
||
simpl. apply tr. right. assumption.
|
||
Defined.
|
||
|
||
(** Properties of enumerated sets: closed under products *)
|
||
Lemma enumerated_prod (A B : Type) `{Funext} :
|
||
enumerated A -> enumerated B -> enumerated (A * B).
|
||
Proof.
|
||
intros HeA HeB.
|
||
strip_truncations; apply tr.
|
||
destruct HeA as [eA HeA], HeB as [eB HeB].
|
||
exists (listProd eA eB).
|
||
intros [x y].
|
||
apply listExt_prod; [ apply HeA | apply HeB ].
|
||
Defined.
|
||
|
||
(** If a set is enumerated is it Kuratowski-finite *)
|
||
Section enumerated_fset.
|
||
Variable A : Type.
|
||
Context `{Univalence}.
|
||
|
||
Fixpoint list_to_fset (ls : list A) : FSet A :=
|
||
match ls with
|
||
| nil => ∅
|
||
| cons x xs => {|x|} ∪ (list_to_fset xs)
|
||
end.
|
||
|
||
Lemma list_to_fset_ext (ls : list A) (a : A):
|
||
listExt ls a -> isIn a (list_to_fset ls).
|
||
Proof.
|
||
induction ls as [|x xs]; simpl.
|
||
- apply idmap.
|
||
- intros Hin.
|
||
strip_truncations. apply tr.
|
||
destruct Hin as [Hax | Hin].
|
||
+ left. exact Hax.
|
||
+ right. by apply IHxs.
|
||
Defined.
|
||
|
||
Lemma enumerated_Kf : enumerated A -> Kf A.
|
||
Proof.
|
||
intros Hls.
|
||
strip_truncations.
|
||
destruct Hls as [ls Hls].
|
||
exists (list_to_fset ls).
|
||
apply path_forall. intro a.
|
||
symmetry. apply path_hprop.
|
||
apply if_hprop_then_equiv_Unit. apply _.
|
||
by apply list_to_fset_ext.
|
||
Defined.
|
||
End enumerated_fset.
|
||
|
||
Section fset_dec_enumerated.
|
||
Variable A : Type.
|
||
Context `{Univalence}.
|
||
|
||
Definition Kf_fsetc :
|
||
Kf A -> exists (X : FSetC A), forall (a : A), k_finite.map (FSetC_to_FSet X) a.
|
||
Proof.
|
||
intros [X HX].
|
||
exists (FSet_to_FSetC X).
|
||
rewrite repr_iso_id_l.
|
||
by rewrite <- HX.
|
||
Defined.
|
||
|
||
Definition merely_enumeration_FSetC :
|
||
forall (X : FSetC A),
|
||
hexists (fun (ls : list A) => forall a, a ∈ (FSetC_to_FSet X) = listExt ls a).
|
||
Proof.
|
||
hinduction.
|
||
- apply tr. exists nil. simpl. done.
|
||
- intros a X Hls.
|
||
strip_truncations. apply tr.
|
||
destruct Hls as [ls Hls].
|
||
exists (cons a ls). intros b. simpl.
|
||
f_ap.
|
||
- intros. apply path_ishprop.
|
||
- intros. apply path_ishprop.
|
||
Defined.
|
||
|
||
Definition Kf_enumerated : Kf A -> enumerated A.
|
||
Proof.
|
||
intros HKf. apply Kf_fsetc in HKf.
|
||
destruct HKf as [X HX].
|
||
pose (ls' := (merely_enumeration_FSetC X)).
|
||
simple refine (@Trunc_rec _ _ _ _ _ ls'). clear ls'.
|
||
intros [ls Hls].
|
||
apply tr. exists ls.
|
||
intros a. rewrite <- Hls. apply (HX a).
|
||
Defined.
|
||
End fset_dec_enumerated.
|