Timings for Sigma.v

Require Import Basics.Overture Basics.Tactics.

Require Import WildCat.Core.

(** ** Indexed sum of categories *)

Section Sigma.

  Context (A : Type) (B : A -> Type)
    `{forall a, IsGraph (B a)}
    `{forall a, Is01Cat (B a)}
    `{forall a, Is0Gpd (B a)}.

#[export] Instance isgraph_sigma : IsGraph (sig B).

Proof.

  srapply Build_IsGraph.

  intros [x u] [y v].

  exact {p : x = y & p # u $-> v}.

Defined.

#[export] Instance is01cat_sigma : Is01Cat (sig B).

Proof.

  srapply Build_Is01Cat.

  +

 intros [x u].

    exists idpath.

 exact (Id u).

  +

 intros [x u] [y v] [z w] [q g] [p f].

    exists (p @ q).

    destruct p, q; cbn in *.

 
    exact (g $o f).

Defined.

#[export] Instance is0gpd_sigma : Is0Gpd (sig B).

Proof.

  constructor.

  intros [x u] [y v] [p f].

  exists (p^).

  destruct p; cbn in *.

  exact (f^$).

Defined.

End Sigma.

Instance is0functor_sigma {A : Type} (B C : A -> Type)
       `{forall a, IsGraph (B a)} `{forall a, IsGraph (C a)}
       `{forall a, Is01Cat (B a)} `{forall a, Is01Cat (C a)}
       (F : forall a, B a -> C a) {ff : forall a, Is0Functor (F a)}
  : Is0Functor (fun (x:sig B) => (x.1 ; F x.1 x.2)).

Proof.

  constructor; intros [a1 b1] [a2 b2] [p f]; cbn.

  exists p.

  destruct p; cbn in *.

  exact (fmap (F a1) f).

Defined.