(** ** Uniqueness of W-types
W-types are unique up to propositional equality.
Author: Langston Barrett (@siddharthist)
*)
Require Import UniMath.Foundations.PartD.
Require Import UniMath.MoreFoundations.Univalence.
Require Import UniMath.CategoryTheory.Core.Categories.
Require Import UniMath.CategoryTheory.Core.Functors.
Require Import UniMath.Induction.FunctorAlgebras_legacy.
Require Import UniMath.Induction.PolynomialFunctors.
Require Import UniMath.Induction.W.Core.
Section Uniqueness.
Local Open Scope functions.
Local Open Scope cat.
Context (A : UU).
Context (B : A → UU).
(** Following the paper, we have [X ⇒ Y] = Hom(X, Y) *)
Local Notation F := (polynomial_functor A B). (* can be coerced to a function *)
Local Notation "F*" := (polynomial_functor_arr A B).
Local Notation "X ⇒ Y" := (algebra_mor F X Y).
(** Since we can't use the standard categorical proof, we must re-prove that
initial algebras are unique up to isomorphism.
*)
(** We prove that their carriers (first projections) are isomorphic, and hence
equal (by univalence).
This is standard categorical reasoning: each has exactly one arrow to the
other, which, composing, gives an endormorphism. However, each has exactly
one endomorphism, the identity map. Therefore, they are isomorphic. *)
Lemma W_carriers_iso : ∏ X Y : W B, (alg_carrier F X) ≃
(alg_carrier F Y).
Proof.
intros X Y.
(** Get the algebra morphisms X → Y and Y → X via initiality *)
pose (X_mor_Y := iscontrpr1 (pr2 X Y)).
pose (Y_mor_X := iscontrpr1 (pr2 Y X)).
apply (weq_iso
(mor_from_algebra_mor _ _ _ X_mor_Y)
(mor_from_algebra_mor _ _ _ Y_mor_X)).
- intro x.
apply (toforallpaths _ (Y_mor_X ∘ X_mor_Y) (idfun _)).
refine (base_paths (algebra_mor_comp _ _ _ _ X_mor_Y Y_mor_X)
(algebra_mor_id F X) _).
apply (proofirrelevancecontr (pr2 X X)).
- intro y.
apply (toforallpaths _ (X_mor_Y ∘ Y_mor_X) (idfun _)).
refine (base_paths (algebra_mor_comp _ _ _ _ Y_mor_X X_mor_Y)
(algebra_mor_id F Y) _).
apply (proofirrelevancecontr (pr2 Y Y)).
Defined.
(** Note the crucial use of univalence *)
Lemma W_carriers_eq : ∏ X Y : W B, (alg_carrier _ X) = (alg_carrier _ Y).
Proof.
exact (fun X Y => weqtopaths (W_carriers_iso X Y)).
Defined.
(** Now we must prove that the algebra morphisms, when transported along
the path W_carriers_eq, will be equal. *)
Lemma W_alg_eq : ∏ X Y : W B, W_algebra B X = W_algebra B Y.
Proof.
intros X Y.
pose (f := pr1 ((pr2 X) (W_algebra B Y))).
pose (pr1eq := (W_carriers_eq X Y)).
apply (total2_paths_f pr1eq).
(** Some shorthands for items we'll use *)
pose (is_final_X := pr2 X).
pose (is_final_Y := pr2 Y).
pose (θ := pr2 (pr1 X)).
pose (ψ := pr2 (pr1 Y)).
(** substⁱ-lemma in HoTT/W-types *)
assert (trans_fun : forall {X Y : UU} {F : UU → UU} {f : F X → X} {g : F Y → Y}
(p : X = Y),
(forall (x : F X),
g (transportf F p x) = (transportf (idfun UU) p (f x))) →
(* (forall (x : X), *)
(* transportf F p (f x) = g (transportf (idfun UU) p x)) → *)
transportf (λ X, F X → X) p f = g).
{
intros ? ? ? ? ? p H.
induction p.
apply funextfun.
intro x.
exact (!H x).
}
apply trans_fun.
intro x.
assert (arr_transport :
forall {X Y : UU} (p : X = Y), F* (transportf (idfun _) p) = transportf F p).
{
intros ? ? p.
induction p.
reflexivity.
}
assert (lemma1 : forall x : pr1 (pr1 X),
(pr1 f) x = transportf (idfun UU) pr1eq x ).
{
intro.
refine (toforallpaths _ _ _ _ x0).
refine (_ @ !(weqpath_transport (W_carriers_iso X Y))).
reflexivity.
}
assert (lemma2 : transportf F pr1eq = F* (pr1 f)).
{
refine (!(arr_transport _ _ pr1eq) @ _).
apply maponpaths.
unfold pr1eq, W_carriers_eq.
refine ((weqpath_transport (W_carriers_iso X Y)) @ _).
reflexivity.
}
refine (_ @ lemma1 (θ x)).
refine (maponpaths ψ (toforallpaths _ _ _ lemma2 x) @ _).
(** Now our goal is simply the condition that f is a algebra morphism *)
apply (toforallpaths _ (λ x, ψ (F* (pr1 f) x)) (pr1 f ∘ θ)).
exact (!pr2 f).
Defined.
Lemma isaprop_W : isaprop (W B).
apply invproofirrelevance.
intros X Y.
apply subtypePath.
- exact isaprop_is_initial.
- exact (W_alg_eq X Y).
Defined.
End Uniqueness.