dan/HITs-Examples
1
0
Fork 0
mirror of https://github.com/nmvdw/HITs-Examples synced 2025-10-30 21:23:52 +01:00
Code Issues Releases Wiki Activity

Recursion rule is now defined via induction.

Browse Source
This commit is contained in:
Niels
2017-05-23 21:50:26 +02:00
parent 6bb5e8b690
commit 0bdbcdfc6c
1 changed files with 91 additions and 111 deletions
Download Patch File Download Diff File Expand all files Collapse all files

202
FiniteSets/definition.v
View File

@@ -32,116 +32,8 @@ Axiom idem : forall (x : A),
Axiom trunc : IsHSet FSet.
Fixpoint FSet_rec
(P : Type)
(H: IsHSet P)
(e : P)
(l : A -> P)
(u : P -> P -> P)
(assocP : forall (x y z : P), u x (u y z) = u (u x y) z)
(commP : forall (x y : P), u x y = u y x)
(nlP : forall (x : P), u e x = x)
(nrP : forall (x : P), u x e = x)
(idemP : forall (x : A), u (l x) (l x) = l x)
(x : FSet)
{struct x}
: P
:= (match x return _ -> _ -> _ -> _ -> _ -> _ -> P with
| E => fun _ => fun _ => fun _ => fun _ => fun _ => fun _ => e
| L a => fun _ => fun _ => fun _ => fun _ => fun _ => fun _ => l a
| U y z => fun _ => fun _ => fun _ => fun _ => fun _ => fun _ => u
(FSet_rec P H e l u assocP commP nlP nrP idemP y)
(FSet_rec P H e l u assocP commP nlP nrP idemP z)
end) assocP commP nlP nrP idemP H.
Axiom FSet_rec_beta_assoc : forall
(P : Type)
(H: IsHSet P)
(e : P)
(l : A -> P)
(u : P -> P -> P)
(assocP : forall (x y z : P), u x (u y z) = u (u x y) z)
(commP : forall (x y : P), u x y = u y x)
(nlP : forall (x : P), u e x = x)
(nrP : forall (x : P), u x e = x)
(idemP : forall (x : A), u (l x) (l x) = l x)
(x y z : FSet),
ap (FSet_rec P H e l u assocP commP nlP nrP idemP) (assoc x y z)
=
(assocP (FSet_rec P H e l u assocP commP nlP nrP idemP x)
(FSet_rec P H e l u assocP commP nlP nrP idemP y)
(FSet_rec P H e l u assocP commP nlP nrP idemP z)
).
Axiom FSet_rec_beta_comm : forall
(P : Type)
(H: IsHSet P)
(e : P)
(l : A -> P)
(u : P -> P -> P)
(assocP : forall (x y z : P), u x (u y z) = u (u x y) z)
(commP : forall (x y : P), u x y = u y x)
(nlP : forall (x : P), u e x = x)
(nrP : forall (x : P), u x e = x)
(idemP : forall (x : A), u (l x) (l x) = l x)
(x y : FSet),
ap (FSet_rec P H e l u assocP commP nlP nrP idemP) (comm x y)
=
(commP (FSet_rec P H e l u assocP commP nlP nrP idemP x)
(FSet_rec P H e l u assocP commP nlP nrP idemP y)
).
Axiom FSet_rec_beta_nl : forall
(P : Type)
(H: IsHSet P)
(e : P)
(l : A -> P)
(u : P -> P -> P)
(assocP : forall (x y z : P), u x (u y z) = u (u x y) z)
(commP : forall (x y : P), u x y = u y x)
(nlP : forall (x : P), u e x = x)
(nrP : forall (x : P), u x e = x)
(idemP : forall (x : A), u (l x) (l x) = l x)
(x : FSet),
ap (FSet_rec P H e l u assocP commP nlP nrP idemP) (nl x)
=
(nlP (FSet_rec P H e l u assocP commP nlP nrP idemP x)
).
Axiom FSet_rec_beta_nr : forall
(P : Type)
(H: IsHSet P)
(e : P)
(l : A -> P)
(u : P -> P -> P)
(assocP : forall (x y z : P), u x (u y z) = u (u x y) z)
(commP : forall (x y : P), u x y = u y x)
(nlP : forall (x : P), u e x = x)
(nrP : forall (x : P), u x e = x)
(idemP : forall (x : A), u (l x) (l x) = l x)
(x : FSet),
ap (FSet_rec P H e l u assocP commP nlP nrP idemP) (nr x)
=
(nrP (FSet_rec P H e l u assocP commP nlP nrP idemP x)
).
Axiom FSet_rec_beta_idem : forall
(P : Type)
(H: IsHSet P)
(e : P)
(l : A -> P)
(u : P -> P -> P)
(assocP : forall (x y z : P), u x (u y z) = u (u x y) z)
(commP : forall (x y : P), u x y = u y x)
(nlP : forall (x : P), u e x = x)
(nrP : forall (x : P), u x e = x)
(idemP : forall (x : A), u (l x) (l x) = l x)
(x : A),
ap (FSet_rec P H e l u assocP commP nlP nrP idemP) (idem x)
=
idemP x.
End FSet.
Section FSet_induction.
Arguments E {_}.
Arguments U {_} _ _.
@@ -201,7 +93,95 @@ 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 A x y z)
=
assocP (FSet_rec x) (FSet_rec y) (FSet_rec z).
Proof.
intros.
unfold FSet_rec.
eapply (cancelL (transport_const (assoc A 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 A x y)
=
commP (FSet_rec x) (FSet_rec y).
Proof.
intros.
unfold FSet_rec.
eapply (cancelL (transport_const (comm A 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 A x)
=
nlP (FSet_rec x).
Proof.
intros.
unfold FSet_rec.
eapply (cancelL (transport_const (nl A x) _)).
simple refine ((apD_const _ _)^ @ _).
apply FSet_ind_beta_nl.
Defined.
Definition FSet_rec_beta_nr : forall (x : FSet A),
ap FSet_rec (nr A x)
=
nrP (FSet_rec x).
Proof.
intros.
unfold FSet_rec.
eapply (cancelL (transport_const (nr A x) _)).
simple refine ((apD_const _ _)^ @ _).
apply FSet_ind_beta_nr.
Defined.
Definition FSet_rec_beta_idem : forall (a : A),
ap FSet_rec (idem A a)
=
idemP a.
Proof.
intros.
unfold FSet_rec.
eapply (cancelL (transport_const (idem A a) _)).
simple refine ((apD_const _ _)^ @ _).
apply FSet_ind_beta_idem.
Defined.
End FSet_recursion.
End definition.