/-
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
-/

import algebra.group

open equiv option eq

namespace unit

  protected definition eta : Π(u : unit), ⋆ = u
  | eta ⋆ := idp

  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

open unit is_trunc

namespace algebra

  definition trivial_group [constructor] : group unit :=
  group.mk (λx y, star) _ (λx y z, idp) star (unit.rec idp) (unit.rec idp) (λx, star) (λx, idp)

  definition Trivial_group [constructor] : Group :=
  Group.mk _ trivial_group

  notation `G0` := Trivial_group

end algebra