lean2/hott/types/unit.hlean

/-
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 eq

namespace unit

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

  definition unit_equiv_option_empty [constructor] : 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

  -- equivalences involving unit and other type constructors are in the file
  -- of the other constructor

end unit

open unit is_trunc
feat(hott): Port remainder of §6.3 and §7.2 from the HoTT book Also prove a theorem similar to Lemma 7.3.1 There are still some sorry's in hit.suspension 2015-06-04 01:41:21 +00:00			`/-`
			`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`

refactor(types): create cubical subfolder, update markdown files 2015-06-04 17:55:02 +00:00			`Theorems about the unit type`
feat(hott): Port remainder of §6.3 and §7.2 from the HoTT book Also prove a theorem similar to Lemma 7.3.1 There are still some sorry's in hit.suspension 2015-06-04 01:41:21 +00:00			`-/`

feat(categories): add exponential laws for categories also give nicer rules to construct equalities between (pre)categories 2015-10-22 22:41:55 +00:00			`open equiv option eq`
feat(hott): Port remainder of §6.3 and §7.2 from the HoTT book Also prove a theorem similar to Lemma 7.3.1 There are still some sorry's in hit.suspension 2015-06-04 01:41:21 +00:00
			`namespace unit`

feat(categories): add exponential laws for categories also give nicer rules to construct equalities between (pre)categories 2015-10-22 22:41:55 +00:00			`protected definition eta : Π(u : unit), ⋆ = u`
			`\| eta ⋆ := idp`

feat(hott): various additions 2016-03-01 04:37:03 +00:00			`definition unit_equiv_option_empty [constructor] : unit ≃ option empty :=`
feat(hott): Port remainder of §6.3 and §7.2 from the HoTT book Also prove a theorem similar to Lemma 7.3.1 There are still some sorry's in hit.suspension 2015-06-04 01:41:21 +00:00			`begin`
			`fapply equiv.MK,`
			`{ intro u, exact none},`
			`{ intro e, exact star},`
			`{ intro e, cases e, reflexivity, contradiction},`
			`{ intro u, cases u, reflexivity},`
			`end`

feat(hott): various additions 2016-03-01 04:37:03 +00:00			`-- equivalences involving unit and other type constructors are in the file`
			`-- of the other constructor`
feat(hott): Port remainder of §6.3 and §7.2 from the HoTT book Also prove a theorem similar to Lemma 7.3.1 There are still some sorry's in hit.suspension 2015-06-04 01:41:21 +00:00
			`end unit`
feat(homotopy/circle): give all higher homotopy groups of the circle 2015-11-18 23:08:38 +00:00
			`open unit is_trunc`