Timings for Equalizer.v

Require Import Basics Types.Sigma Types.Paths.

(** Equalizers *)

Definition Equalizer {A B} (f g : A -> B)
  := {x : A & f x = g x}.

Definition functor_equalizer {A B A' B'} (f g : A -> B)
           (f' g' : A' -> B')
           (h : B -> B') (k : A -> A')
           (p : h o f == f' o k) (q : h o g == g' o k)
  : Equalizer f g -> Equalizer f' g'.

Proof.

  intros [x r].

  exists (k x).

  exact ((p x)^ @ (ap h r) @ (q x)).

Defined.

Definition equiv_functor_equalizer {A B A' B'} (f g : A -> B)
           (f' g' : A' -> B')
           (h : B <~> B') (k : A <~> A')
           (p : h o f == f' o k) (q : h o g == g' o k)
  : Equalizer f g <~> Equalizer f' g'.

Proof.

  unfold Equalizer.

  srapply (equiv_functor_sigma' k).

  -

 intro a; cbn.

    refine (_ oE _).

    2: rapply (equiv_ap h).

    exact (equiv_concat_r (q a) _ oE equiv_concat_l (p a)^ _).

Defined.