mirror of https://github.com/nmvdw/HITs-Examples
Recursion rule is now defined via induction.
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Niels
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@ -32,116 +32,8 @@ Axiom idem : forall (x : A),
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Axiom trunc : IsHSet FSet.
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Fixpoint FSet_rec
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(P : Type)
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(H: IsHSet P)
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(e : P)
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(l : A -> P)
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(u : P -> P -> P)
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(assocP : forall (x y z : P), u x (u y z) = u (u x y) z)
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(commP : forall (x y : P), u x y = u y x)
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(nlP : forall (x : P), u e x = x)
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(nrP : forall (x : P), u x e = x)
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(idemP : forall (x : A), u (l x) (l x) = l x)
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(x : FSet)
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{struct x}
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: P
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:= (match x return _ -> _ -> _ -> _ -> _ -> _ -> P with
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| E => fun _ => fun _ => fun _ => fun _ => fun _ => fun _ => e
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| L a => fun _ => fun _ => fun _ => fun _ => fun _ => fun _ => l a
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| U y z => fun _ => fun _ => fun _ => fun _ => fun _ => fun _ => u
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(FSet_rec P H e l u assocP commP nlP nrP idemP y)
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(FSet_rec P H e l u assocP commP nlP nrP idemP z)
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end) assocP commP nlP nrP idemP H.
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Axiom FSet_rec_beta_assoc : forall
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(P : Type)
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(H: IsHSet P)
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(e : P)
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(l : A -> P)
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(u : P -> P -> P)
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(assocP : forall (x y z : P), u x (u y z) = u (u x y) z)
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(commP : forall (x y : P), u x y = u y x)
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(nlP : forall (x : P), u e x = x)
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(nrP : forall (x : P), u x e = x)
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(idemP : forall (x : A), u (l x) (l x) = l x)
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(x y z : FSet),
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ap (FSet_rec P H e l u assocP commP nlP nrP idemP) (assoc x y z)
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=
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(assocP (FSet_rec P H e l u assocP commP nlP nrP idemP x)
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(FSet_rec P H e l u assocP commP nlP nrP idemP y)
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(FSet_rec P H e l u assocP commP nlP nrP idemP z)
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).
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Axiom FSet_rec_beta_comm : forall
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(P : Type)
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(H: IsHSet P)
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(e : P)
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(l : A -> P)
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(u : P -> P -> P)
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(assocP : forall (x y z : P), u x (u y z) = u (u x y) z)
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(commP : forall (x y : P), u x y = u y x)
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(nlP : forall (x : P), u e x = x)
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(nrP : forall (x : P), u x e = x)
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(idemP : forall (x : A), u (l x) (l x) = l x)
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(x y : FSet),
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ap (FSet_rec P H e l u assocP commP nlP nrP idemP) (comm x y)
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=
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(commP (FSet_rec P H e l u assocP commP nlP nrP idemP x)
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(FSet_rec P H e l u assocP commP nlP nrP idemP y)
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).
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Axiom FSet_rec_beta_nl : forall
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(P : Type)
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(H: IsHSet P)
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(e : P)
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(l : A -> P)
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(u : P -> P -> P)
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(assocP : forall (x y z : P), u x (u y z) = u (u x y) z)
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(commP : forall (x y : P), u x y = u y x)
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(nlP : forall (x : P), u e x = x)
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(nrP : forall (x : P), u x e = x)
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(idemP : forall (x : A), u (l x) (l x) = l x)
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(x : FSet),
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ap (FSet_rec P H e l u assocP commP nlP nrP idemP) (nl x)
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=
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(nlP (FSet_rec P H e l u assocP commP nlP nrP idemP x)
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).
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Axiom FSet_rec_beta_nr : forall
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(P : Type)
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(H: IsHSet P)
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(e : P)
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(l : A -> P)
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(u : P -> P -> P)
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(assocP : forall (x y z : P), u x (u y z) = u (u x y) z)
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(commP : forall (x y : P), u x y = u y x)
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(nlP : forall (x : P), u e x = x)
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(nrP : forall (x : P), u x e = x)
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(idemP : forall (x : A), u (l x) (l x) = l x)
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(x : FSet),
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ap (FSet_rec P H e l u assocP commP nlP nrP idemP) (nr x)
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=
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(nrP (FSet_rec P H e l u assocP commP nlP nrP idemP x)
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).
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Axiom FSet_rec_beta_idem : forall
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(P : Type)
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(H: IsHSet P)
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(e : P)
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(l : A -> P)
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(u : P -> P -> P)
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(assocP : forall (x y z : P), u x (u y z) = u (u x y) z)
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(commP : forall (x y : P), u x y = u y x)
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(nlP : forall (x : P), u e x = x)
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(nrP : forall (x : P), u x e = x)
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(idemP : forall (x : A), u (l x) (l x) = l x)
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(x : A),
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ap (FSet_rec P H e l u assocP commP nlP nrP idemP) (idem x)
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=
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idemP x.
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End FSet.
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Section FSet_induction.
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Arguments E {_}.
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Arguments U {_} _ _.
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@ -201,7 +93,95 @@ Axiom FSet_ind_beta_nr : forall (x : FSet A),
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apD FSet_ind (nr x) = (nrP x (FSet_ind x)).
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Axiom FSet_ind_beta_idem : forall (x : A), apD FSet_ind (idem x) = idemP x.
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End FSet_induction.
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Section FSet_recursion.
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Variable A : Type.
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Variable P : Type.
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Variable H: IsHSet P.
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Variable e : P.
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Variable l : A -> P.
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Variable u : P -> P -> P.
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Variable assocP : forall (x y z : P), u x (u y z) = u (u x y) z.
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Variable commP : forall (x y : P), u x y = u y x.
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Variable nlP : forall (x : P), u e x = x.
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Variable nrP : forall (x : P), u x e = x.
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Variable idemP : forall (x : A), u (l x) (l x) = l x.
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Definition FSet_rec : FSet A -> P.
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Proof.
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simple refine (FSet_ind A _ _ _ _ _ _ _ _ _ _) ; try (intros ; simple refine ((transport_const _ _) @ _)) ; cbn.
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- apply e.
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- apply l.
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- intros x y ; apply u.
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- apply assocP.
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- apply commP.
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- apply nlP.
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- apply nrP.
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- apply idemP.
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Defined.
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Definition FSet_rec_beta_assoc : forall (x y z : FSet A),
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ap FSet_rec (assoc A x y z)
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=
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assocP (FSet_rec x) (FSet_rec y) (FSet_rec z).
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Proof.
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intros.
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unfold FSet_rec.
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eapply (cancelL (transport_const (assoc A x y z) _)).
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simple refine ((apD_const _ _)^ @ _).
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apply FSet_ind_beta_assoc.
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Defined.
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Definition FSet_rec_beta_comm : forall (x y : FSet A),
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ap FSet_rec (comm A x y)
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=
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commP (FSet_rec x) (FSet_rec y).
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Proof.
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intros.
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unfold FSet_rec.
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eapply (cancelL (transport_const (comm A x y) _)).
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simple refine ((apD_const _ _)^ @ _).
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apply FSet_ind_beta_comm.
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Defined.
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Definition FSet_rec_beta_nl : forall (x : FSet A),
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ap FSet_rec (nl A x)
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=
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nlP (FSet_rec x).
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Proof.
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intros.
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unfold FSet_rec.
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eapply (cancelL (transport_const (nl A x) _)).
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simple refine ((apD_const _ _)^ @ _).
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apply FSet_ind_beta_nl.
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Defined.
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Definition FSet_rec_beta_nr : forall (x : FSet A),
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ap FSet_rec (nr A x)
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=
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nrP (FSet_rec x).
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Proof.
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intros.
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unfold FSet_rec.
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eapply (cancelL (transport_const (nr A x) _)).
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simple refine ((apD_const _ _)^ @ _).
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apply FSet_ind_beta_nr.
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Defined.
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Definition FSet_rec_beta_idem : forall (a : A),
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ap FSet_rec (idem A a)
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=
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idemP a.
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Proof.
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intros.
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unfold FSet_rec.
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eapply (cancelL (transport_const (idem A a) _)).
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simple refine ((apD_const _ _)^ @ _).
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apply FSet_ind_beta_idem.
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Defined.
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End FSet_recursion.
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End definition.
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