mirror of https://github.com/nmvdw/HITs-Examples
231 lines
6.7 KiB
Coq
231 lines
6.7 KiB
Coq
Require Import HoTT HitTactics.
|
||
Require Export representations.definition disjunction fsets.operations.
|
||
|
||
(* Lemmas relating operations to the membership predicate *)
|
||
Section operations_isIn.
|
||
|
||
Context {A : Type}.
|
||
Context `{Univalence}.
|
||
|
||
Lemma union_idem : forall x: FSet A, x ∪ x = x.
|
||
Proof.
|
||
hinduction ; try (intros ; apply set_path2).
|
||
- apply nl.
|
||
- apply idem.
|
||
- intros x y P Q.
|
||
rewrite assoc.
|
||
rewrite (comm x y).
|
||
rewrite <- (assoc y x x).
|
||
rewrite P.
|
||
rewrite (comm y x).
|
||
rewrite <- (assoc x y y).
|
||
f_ap.
|
||
Defined.
|
||
|
||
(** ** Properties about subset relation. *)
|
||
Lemma subset_union (X Y : FSet A) :
|
||
X ⊆ Y -> X ∪ Y = Y.
|
||
Proof.
|
||
hinduction X ; try (intros; apply path_forall; intro; apply set_path2).
|
||
- intros. apply nl.
|
||
- intros a.
|
||
hinduction Y ; try (intros; apply path_forall; intro; apply set_path2).
|
||
+ intro.
|
||
contradiction.
|
||
+ intro a0.
|
||
simple refine (Trunc_ind _ _).
|
||
intro p ; simpl.
|
||
rewrite p; apply idem.
|
||
+ intros X1 X2 IH1 IH2.
|
||
simple refine (Trunc_ind _ _).
|
||
intros [e1 | e2].
|
||
++ rewrite assoc.
|
||
rewrite (IH1 e1).
|
||
reflexivity.
|
||
++ rewrite comm.
|
||
rewrite <- assoc.
|
||
rewrite (comm X2).
|
||
rewrite (IH2 e2).
|
||
reflexivity.
|
||
- intros X1 X2 IH1 IH2 [G1 G2].
|
||
rewrite <- assoc.
|
||
rewrite (IH2 G2).
|
||
apply (IH1 G1).
|
||
Defined.
|
||
|
||
Lemma subset_union_l (X : FSet A) :
|
||
forall Y, X ⊆ X ∪ Y.
|
||
Proof.
|
||
hinduction X ; try (repeat (intro; intros; apply path_forall);
|
||
intro ; apply equiv_hprop_allpath ; apply _).
|
||
- apply (fun _ => tt).
|
||
- intros a Y.
|
||
apply (tr(inl(tr idpath))).
|
||
- intros X1 X2 HX1 HX2 Y.
|
||
split.
|
||
* rewrite <- assoc. apply HX1.
|
||
* rewrite (comm X1 X2). rewrite <- assoc. apply HX2.
|
||
Defined.
|
||
|
||
(* simplify it using extensionality *)
|
||
Lemma comprehension_or : forall ϕ ψ (x: FSet A),
|
||
comprehension (fun a => orb (ϕ a) (ψ a)) x
|
||
= (comprehension ϕ x) ∪ (comprehension ψ x).
|
||
Proof.
|
||
intros ϕ ψ.
|
||
hinduction ; try (intros; apply set_path2).
|
||
- apply (union_idem _)^.
|
||
- intros.
|
||
destruct (ϕ a) ; destruct (ψ a) ; symmetry.
|
||
* apply union_idem.
|
||
* apply nr.
|
||
* apply nl.
|
||
* apply union_idem.
|
||
- simpl. intros x y P Q.
|
||
rewrite P.
|
||
rewrite Q.
|
||
rewrite <- assoc.
|
||
rewrite (assoc (comprehension ψ x)).
|
||
rewrite (comm (comprehension ψ x) (comprehension ϕ y)).
|
||
rewrite <- assoc.
|
||
rewrite <- assoc.
|
||
reflexivity.
|
||
Defined.
|
||
|
||
End operations_isIn.
|
||
|
||
(* Other properties *)
|
||
Section properties.
|
||
|
||
Context {A : Type}.
|
||
Context `{Univalence}.
|
||
|
||
(** isIn properties *)
|
||
Definition empty_isIn (a: A) : a ∈ E -> Empty := idmap.
|
||
|
||
Definition singleton_isIn (a b: A) : a ∈ {|b|} -> Trunc (-1) (a = b) := idmap.
|
||
|
||
Definition union_isIn (X Y : FSet A) (a : A)
|
||
: a ∈ X ∪ Y = a ∈ X ∨ a ∈ Y := idpath.
|
||
|
||
Lemma comprehension_isIn (ϕ : A -> Bool) (a : A) : forall X : FSet A,
|
||
a ∈ (comprehension ϕ X) = if ϕ a then a ∈ X else False_hp.
|
||
Proof.
|
||
hinduction ; try (intros ; apply set_path2) ; cbn.
|
||
- destruct (ϕ a) ; reflexivity.
|
||
- intros b.
|
||
assert (forall c d, ϕ a = c -> ϕ b = d ->
|
||
a ∈ (if ϕ b then {|b|} else ∅)
|
||
=
|
||
(if ϕ a then BuildhProp (Trunc (-1) (a = b)) else False_hp)) as X.
|
||
{
|
||
intros c d Hc Hd.
|
||
destruct c ; destruct d ; rewrite Hc, Hd ; try reflexivity
|
||
; apply path_iff_hprop ; try contradiction ; intros ; strip_truncations
|
||
; apply (false_ne_true).
|
||
* apply (Hd^ @ ap ϕ X^ @ Hc).
|
||
* apply (Hc^ @ ap ϕ X @ Hd).
|
||
}
|
||
apply (X (ϕ a) (ϕ b) idpath idpath).
|
||
- intros X Y H1 H2.
|
||
rewrite H1, H2.
|
||
destruct (ϕ a).
|
||
* reflexivity.
|
||
* apply path_iff_hprop.
|
||
** intros Z ; strip_truncations.
|
||
destruct Z ; assumption.
|
||
** intros ; apply tr ; right ; assumption.
|
||
Defined.
|
||
|
||
Context {B : Type}.
|
||
|
||
Definition isIn_product : forall (a : A) (b : B) (X : FSet A) (Y : FSet B),
|
||
isIn a X -> isIn b Y -> isIn (a,b) (product X Y).
|
||
Proof.
|
||
intros a b X Y.
|
||
hinduction X ; try (intros ; apply path_forall ; intro ; apply path_ishprop).
|
||
- contradiction.
|
||
- intros c Hc.
|
||
hinduction Y ; try (intros ; apply path_forall ; intro ; apply path_ishprop).
|
||
* contradiction.
|
||
* intros d Hd.
|
||
strip_truncations.
|
||
apply tr.
|
||
rewrite Hc, Hd.
|
||
reflexivity.
|
||
* intros Y1 Y2 HY1 HY2 HOr.
|
||
strip_truncations.
|
||
apply tr.
|
||
destruct HOr as [H1 | H2].
|
||
** apply (inl (HY1 H1)).
|
||
** apply (inr (HY2 H2)).
|
||
- intros X1 X2 HX1 HX2 Hor HY.
|
||
strip_truncations.
|
||
apply tr.
|
||
destruct Hor as [H1 | H2].
|
||
* apply (inl(HX1 H1 HY)).
|
||
* apply (inr (HX2 H2 HY)).
|
||
Defined.
|
||
|
||
(* The proof can be simplified using extensionality *)
|
||
(** comprehension properties *)
|
||
Lemma comprehension_false Y : comprehension (fun (_ : A) => false) Y = ∅.
|
||
Proof.
|
||
hrecursion Y; try (intros; apply set_path2).
|
||
- reflexivity.
|
||
- reflexivity.
|
||
- intros x y IHa IHb.
|
||
rewrite IHa, IHb.
|
||
apply union_idem.
|
||
Defined.
|
||
|
||
(* Can be simplified using extensionality *)
|
||
Lemma comprehension_subset : forall ϕ (X : FSet A),
|
||
(comprehension ϕ X) ∪ X = X.
|
||
Proof.
|
||
intros ϕ.
|
||
hrecursion; try (intros ; apply set_path2) ; cbn.
|
||
- apply union_idem.
|
||
- intro a.
|
||
destruct (ϕ a).
|
||
* apply union_idem.
|
||
* apply nl.
|
||
- intros X Y P Q.
|
||
rewrite assoc.
|
||
rewrite <- (assoc (comprehension ϕ X) (comprehension ϕ Y) X).
|
||
rewrite (comm (comprehension ϕ Y) X).
|
||
rewrite assoc.
|
||
rewrite P.
|
||
rewrite <- assoc.
|
||
rewrite Q.
|
||
reflexivity.
|
||
Defined.
|
||
|
||
Lemma merely_choice : forall X : FSet A, hor (X = ∅) (hexists (fun a => a ∈ X)).
|
||
Proof.
|
||
hinduction; try (intros; apply equiv_hprop_allpath ; apply _).
|
||
- apply (tr (inl idpath)).
|
||
- intro a.
|
||
refine (tr (inr (tr (a ; tr idpath)))).
|
||
- intros X Y TX TY.
|
||
strip_truncations.
|
||
destruct TX as [XE | HX] ; destruct TY as [YE | HY] ; try(strip_truncations ; apply tr).
|
||
* refine (tr (inl _)).
|
||
rewrite XE, YE.
|
||
apply (union_idem E).
|
||
* destruct HY as [a Ya].
|
||
refine (inr (tr _)).
|
||
exists a.
|
||
apply (tr (inr Ya)).
|
||
* destruct HX as [a Xa].
|
||
refine (inr (tr _)).
|
||
exists a.
|
||
apply (tr (inl Xa)).
|
||
* destruct (HX, HY) as [[a Xa] [b Yb]].
|
||
refine (inr (tr _)).
|
||
exists a.
|
||
apply (tr (inl Xa)).
|
||
Defined.
|
||
|
||
End properties.
|