lean2/hott/types/pointed.hlean

/-
Copyright (c) 2014 Jakob von Raumer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.

Module: types.pointed
Author: Jakob von Raumer

Ported from Coq HoTT
-/

open eq prod is_trunc sigma

structure is_pointed [class] (A : Type) :=
  (point : A)

namespace is_pointed
  variables {A B : Type} (f : A → B)

  -- Any contractible type is pointed
  protected definition contr [instance] [H : is_contr A] : is_pointed A :=
    is_pointed.mk !center

  -- A pi type with a pointed target is pointed
  protected definition pi [instance] {P : A → Type} [H : Πx, is_pointed (P x)]
      : is_pointed (Πx, P x) :=
    is_pointed.mk (λx, point (P x))

  -- A sigma type of pointed components is pointed
  protected definition sigma [instance] {P : A → Type} [G : is_pointed A]
      [H : is_pointed (P !point)] : is_pointed (Σx, P x) :=
    is_pointed.mk ⟨!point,!point⟩

  protected definition prod [H1 : is_pointed A] [H2 : is_pointed B]
      : is_pointed (A × B) :=
    is_pointed.mk (!point,!point)

  protected definition loop_space (a : A) : is_pointed (a = a) :=
    is_pointed.mk idp

end is_pointed