mirror of https://github.com/nmvdw/HITs-Examples
204 lines
5.8 KiB
Coq
204 lines
5.8 KiB
Coq
(* Definitions of the Kuratowski-finite sets via a HIT *)
|
||
Require Import HoTT HitTactics.
|
||
Require Export notation.
|
||
|
||
Module Export FSet.
|
||
Section FSet.
|
||
Private Inductive FSet (A : Type) : Type :=
|
||
| E : FSet A
|
||
| L : A -> FSet A
|
||
| U : FSet A -> FSet A -> FSet A.
|
||
|
||
Global Instance fset_empty : forall A, hasEmpty (FSet A) := E.
|
||
Global Instance fset_singleton : forall A, hasSingleton (FSet A) A := L.
|
||
Global Instance fset_union : forall A, hasUnion (FSet A) := U.
|
||
|
||
Variable A : Type.
|
||
|
||
Axiom assoc : forall (x y z : FSet A),
|
||
x ∪ (y ∪ z) = (x ∪ y) ∪ z.
|
||
|
||
Axiom comm : forall (x y : FSet A),
|
||
x ∪ y = y ∪ x.
|
||
|
||
Axiom nl : forall (x : FSet A),
|
||
∅ ∪ x = x.
|
||
|
||
Axiom nr : forall (x : FSet A),
|
||
x ∪ ∅ = x.
|
||
|
||
Axiom idem : forall (x : A),
|
||
{|x|} ∪ {|x|} = {|x|}.
|
||
|
||
Axiom trunc : IsHSet (FSet A).
|
||
|
||
End FSet.
|
||
|
||
Arguments assoc {_} _ _ _.
|
||
Arguments comm {_} _ _.
|
||
Arguments nl {_} _.
|
||
Arguments nr {_} _.
|
||
Arguments idem {_} _.
|
||
|
||
Section FSet_induction.
|
||
Variable A: Type.
|
||
Variable (P : FSet A -> Type).
|
||
Variable (H : forall X : FSet A, IsHSet (P X)).
|
||
Variable (eP : P ∅).
|
||
Variable (lP : forall a: A, P {|a|}).
|
||
Variable (uP : forall (x y: FSet A), P x -> P y -> P (x ∪ y)).
|
||
Variable (assocP : forall (x y z : FSet A)
|
||
(px: P x) (py: P y) (pz: P z),
|
||
assoc x y z #
|
||
(uP x (y ∪ z) px (uP y z py pz))
|
||
=
|
||
(uP (x ∪ y) z (uP x y px py) pz)).
|
||
Variable (commP : forall (x y: FSet A) (px: P x) (py: P y),
|
||
comm x y #
|
||
uP x y px py = uP y x py px).
|
||
Variable (nlP : forall (x : FSet A) (px: P x),
|
||
nl x # uP ∅ x eP px = px).
|
||
Variable (nrP : forall (x : FSet A) (px: P x),
|
||
nr x # uP x ∅ px eP = px).
|
||
Variable (idemP : forall (x : A),
|
||
idem x # uP {|x|} {|x|} (lP x) (lP x) = lP x).
|
||
|
||
(* Induction principle *)
|
||
Fixpoint FSet_ind
|
||
(x : FSet A)
|
||
{struct x}
|
||
: P x
|
||
:= (match x return _ -> _ -> _ -> _ -> _ -> _ -> P x with
|
||
| E => fun _ _ _ _ _ _ => eP
|
||
| L a => fun _ _ _ _ _ _ => lP a
|
||
| U y z => fun _ _ _ _ _ _ => uP y z (FSet_ind y) (FSet_ind z)
|
||
end) H assocP commP nlP nrP idemP.
|
||
|
||
Axiom FSet_ind_beta_assoc : forall (x y z : FSet A),
|
||
apD FSet_ind (assoc x y z) =
|
||
(assocP x y z (FSet_ind x) (FSet_ind y) (FSet_ind z)).
|
||
|
||
Axiom FSet_ind_beta_comm : forall (x y : FSet A),
|
||
apD FSet_ind (comm x y) = (commP x y (FSet_ind x) (FSet_ind y)).
|
||
|
||
Axiom FSet_ind_beta_nl : forall (x : FSet A),
|
||
apD FSet_ind (nl x) = (nlP x (FSet_ind x)).
|
||
|
||
Axiom FSet_ind_beta_nr : forall (x : FSet A),
|
||
apD FSet_ind (nr x) = (nrP x (FSet_ind x)).
|
||
|
||
Axiom FSet_ind_beta_idem : forall (x : A), apD FSet_ind (idem x) = idemP x.
|
||
End FSet_induction.
|
||
|
||
Section FSet_recursion.
|
||
|
||
Variable A : Type.
|
||
Variable P : Type.
|
||
Variable H: IsHSet P.
|
||
Variable e : P.
|
||
Variable l : A -> P.
|
||
Variable u : P -> P -> P.
|
||
Variable assocP : forall (x y z : P), u x (u y z) = u (u x y) z.
|
||
Variable commP : forall (x y : P), u x y = u y x.
|
||
Variable nlP : forall (x : P), u e x = x.
|
||
Variable nrP : forall (x : P), u x e = x.
|
||
Variable idemP : forall (x : A), u (l x) (l x) = l x.
|
||
|
||
Definition FSet_rec : FSet A -> P.
|
||
Proof.
|
||
simple refine (FSet_ind A _ _ _ _ _ _ _ _ _ _)
|
||
; try (intros ; simple refine ((transport_const _ _) @ _)) ; cbn.
|
||
- apply e.
|
||
- apply l.
|
||
- intros x y ; apply u.
|
||
- apply assocP.
|
||
- apply commP.
|
||
- apply nlP.
|
||
- apply nrP.
|
||
- apply idemP.
|
||
Defined.
|
||
|
||
Definition FSet_rec_beta_assoc : forall (x y z : FSet A),
|
||
ap FSet_rec (assoc x y z)
|
||
=
|
||
assocP (FSet_rec x) (FSet_rec y) (FSet_rec z).
|
||
Proof.
|
||
intros.
|
||
unfold FSet_rec.
|
||
eapply (cancelL (transport_const (assoc x y z) _)).
|
||
simple refine ((apD_const _ _)^ @ _).
|
||
apply FSet_ind_beta_assoc.
|
||
Defined.
|
||
|
||
Definition FSet_rec_beta_comm : forall (x y : FSet A),
|
||
ap FSet_rec (comm x y)
|
||
=
|
||
commP (FSet_rec x) (FSet_rec y).
|
||
Proof.
|
||
intros.
|
||
unfold FSet_rec.
|
||
eapply (cancelL (transport_const (comm x y) _)).
|
||
simple refine ((apD_const _ _)^ @ _).
|
||
apply FSet_ind_beta_comm.
|
||
Defined.
|
||
|
||
Definition FSet_rec_beta_nl : forall (x : FSet A),
|
||
ap FSet_rec (nl x)
|
||
=
|
||
nlP (FSet_rec x).
|
||
Proof.
|
||
intros.
|
||
unfold FSet_rec.
|
||
eapply (cancelL (transport_const (nl x) _)).
|
||
simple refine ((apD_const _ _)^ @ _).
|
||
apply FSet_ind_beta_nl.
|
||
Defined.
|
||
|
||
Definition FSet_rec_beta_nr : forall (x : FSet A),
|
||
ap FSet_rec (nr x)
|
||
=
|
||
nrP (FSet_rec x).
|
||
Proof.
|
||
intros.
|
||
unfold FSet_rec.
|
||
eapply (cancelL (transport_const (nr x) _)).
|
||
simple refine ((apD_const _ _)^ @ _).
|
||
apply FSet_ind_beta_nr.
|
||
Defined.
|
||
|
||
Definition FSet_rec_beta_idem : forall (a : A),
|
||
ap FSet_rec (idem a)
|
||
=
|
||
idemP a.
|
||
Proof.
|
||
intros.
|
||
unfold FSet_rec.
|
||
eapply (cancelL (transport_const (idem a) _)).
|
||
simple refine ((apD_const _ _)^ @ _).
|
||
apply FSet_ind_beta_idem.
|
||
Defined.
|
||
|
||
End FSet_recursion.
|
||
|
||
Instance FSet_recursion A : HitRecursion (FSet A) :=
|
||
{
|
||
indTy := _; recTy := _;
|
||
H_inductor := FSet_ind A; H_recursor := FSet_rec A
|
||
}.
|
||
|
||
End FSet.
|
||
|
||
Lemma union_idem {A : Type} : 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. |