Timings for Cantor.v

Require Import HoTT.Basics HoTT.Types.

Local Open Scope nat_scope.
Local Open Scope path_scope.

(** * Cantor space 2^N *)


Definition Cantor : Type := nat -> Bool.

Definition cantor_fold : Cantor + Cantor -> Cantor.
Proof.
  intros [c|c]; intros n.
  -
 destruct n as [|n].
    +
 exact true.
    +
 exact (c n).
  -
 destruct n as [|n].
    +
 exact false.
    +
 exact (c n).
Defined.

Definition cantor_unfold : Cantor -> Cantor + Cantor.
Proof.
  intros c.
  case (c 0).
  -
 exact (inl (fun n => c n.+1)).
  -
 exact (inr (fun n => c n.+1)).
Defined.

Instance isequiv_cantor_fold `{Funext} : IsEquiv cantor_fold.
Proof.
  refine (isequiv_adjointify _ cantor_unfold _ _).
  -
 intros c; apply path_arrow; intros n.
    induction n as [|n IH].
    +
 unfold cantor_unfold.
      case (c 0); reflexivity.
    +
 unfold cantor_unfold.
      case (c 0); reflexivity.
  -
 intros [c|c]; apply path_sum; reflexivity.
Defined.

Definition equiv_cantor_fold `{Funext} : Cantor + Cantor <~> Cantor
  := Build_Equiv _ _ cantor_fold _.

Definition equiv_cantor_unfold `{Funext} : Cantor <~> Cantor + Cantor
  := Build_Equiv _ _ cantor_unfold (isequiv_inverse equiv_cantor_fold).