Timings for Notnot.v

(* -*- mode: coq; mode: visual-line -*- *)

Require Import HoTT.Basics HoTT.Types.

Require Import Modality.

Local Open Scope path_scope.

(** * The double negation modality *)

(** This is Exercise 7.12 in the book.  Note that it is (apparently) *not* accessible unless we assume propositional resizing. *)

Definition NotNot `{Funext} : Modality.

Proof.

  snrapply easy_modality.

  -

 intros X; exact (~ (~ X)).

  -

 intros T x nx; exact (nx x).

  -

 intros A B f z nBz.

    apply z; intros a.

    exact (f a (transport (fun x => ~ (B x))
                          (path_ishprop _ _)
                          nBz)).

  -

 intros A B f a.

    apply path_ishprop.

  -

 intros A z z'.

    refine (isequiv_iff_hprop _ _).

    intros; apply path_ishprop.

Defined.