/-
Copyright (c) 2015 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn

Theorems about the unit type
-/

open equiv option

namespace unit

  definition unit_equiv_option_empty : unit ≃ option empty :=
  begin
    fapply equiv.MK,
    { intro u, exact none},
    { intro e, exact star},
    { intro e, cases e, reflexivity, contradiction},
    { intro u, cases u, reflexivity},
  end

  definition unit_imp_equiv (A : Type) : (unit → A) ≃ A :=
  begin
    fapply equiv.MK,
    { intro f, exact f star},
    { intro a u, exact a},
    { intro a, reflexivity},
    { intro f, apply eq_of_homotopy, intro u, cases u, reflexivity},
  end

end unit