Timings for NullHomotopy.v

Require Import HoTT.Basics.

Require Import Types.Sigma.

Local Open Scope path_scope.

(** * Null homotopies of maps *)

Section NullHomotopy.

  Context `{Funext}.

  (** Geometrically, a nullhomotopy of a map [f : X -> Y] is an extension of [f] to a map [Cone X -> Y].  One might more simply call it e.g. [Constant f], but that is a little ambiguous: it could also reasonably mean e.g. a factorisation of [f] through [ Trunc -1 X ].  (Should the unique map [0 -> Y] be constant in one way, or in [Y]-many ways?) *)

 

Definition NullHomotopy {X Y : Type} (f : X -> Y)
    := {y : Y & forall x:X, f x = y}.

  Lemma istrunc_nullhomotopy {n : trunc_index}
    {X Y : Type} (f : X -> Y) `{IsTrunc n Y}
    : IsTrunc n (NullHomotopy f).

  Proof.

    apply @istrunc_sigma; auto.

    intros y.

 apply (@istrunc_forall _).

    intros x.

 apply istrunc_paths'.

  Defined.

  Definition nullhomotopy_homotopic {X Y : Type} {f g : X -> Y} (p : f == g)
    : NullHomotopy f -> NullHomotopy g.

  Proof.

    intros [y e].

    exists y.

    intros x; exact ((p x)^ @ e x).

  Defined.

  Definition nullhomotopy_composeR {X Y Z : Type} (f : X -> Y) (g : Y -> Z)
    : NullHomotopy g -> NullHomotopy (g o f).

  Proof.

    intros [z e].

    exists z.

    intros x; exact (e (f x)).

  Defined.

  Definition nullhomotopy_composeL {X Y Z : Type} (f : X -> Y) (g : Y -> Z)
    : NullHomotopy f -> NullHomotopy (g o f).

  Proof.

    intros [y e].

    exists (g y).

    intros x; exact (ap g (e x)).

  Defined.

  Definition cancelL_nullhomotopy_equiv
             {X Y Z : Type} (f : X -> Y) (g : Y -> Z) `{IsEquiv _ _ g}
    : NullHomotopy (g o f) -> NullHomotopy f.

  Proof.

    intros [z e].

    exists (g^-1 z).

    intros x; apply moveL_equiv_V, e.

  Defined.

  Definition cancelR_nullhomotopy_equiv
             {X Y Z : Type} (f : X -> Y) (g : Y -> Z) `{IsEquiv _ _ f}
    : NullHomotopy (g o f) -> NullHomotopy g.

  Proof.

    intros [z e].

    exists z.

    intros y; transitivity (g (f (f^-1 y))).

    -

 symmetry; apply ap, eisretr.

    -

 apply e.

  Defined.

  Definition nullhomotopy_ap {X Y : Type} (f : X -> Y) (x1 x2 : X)
    : NullHomotopy f -> NullHomotopy (@ap _ _ f x1 x2).

  Proof.

    intros [y e].

    unshelve eexists.

    -

 exact (e x1 @ (e x2)^).

    -

 intros p.

      apply moveL_pV.

      refine (concat_Ap e p @ _).

      refine (_ @ concat_p1 _); apply ap.

      apply ap_const.

  Defined.

End NullHomotopy.