49 lines
1.4 KiB
Text
49 lines
1.4 KiB
Text
/-
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Copyright (c) 2015 Floris van Doorn. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Authors: Floris van Doorn
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Theorems about the unit type
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-/
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open is_equiv equiv option eq pointed is_trunc function
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namespace unit
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protected definition eta : Π(u : unit), ⋆ = u
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| eta ⋆ := idp
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definition unit_equiv_option_empty [constructor] : unit ≃ option empty :=
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begin
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fapply equiv.MK,
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{ intro u, exact none},
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{ intro e, exact star},
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{ intro e, cases e, reflexivity, contradiction},
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{ intro u, cases u, reflexivity},
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end
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-- equivalences involving unit and other type constructors are in the file
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-- of the other constructor
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/- pointed and truncated unit -/
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definition punit [constructor] : Set* :=
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pSet.mk unit _ ⋆
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notation `unit*` := punit
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definition unit_arrow_eq {X : Type} (f : unit → X) : (λx, f ⋆) = f :=
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by apply eq_of_homotopy; intro u; induction u; reflexivity
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open funext
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definition unit_arrow_eq_compose {X Y : Type} (g : X → Y) (f : unit → X) :
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unit_arrow_eq (g ∘ f) = ap (λf, g ∘ f) (unit_arrow_eq f) :=
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begin
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apply eq_of_fn_eq_fn' apd10,
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refine right_inv apd10 _ ⬝ _,
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refine _ ⬝ ap apd10 (!compose_eq_of_homotopy)⁻¹,
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refine _ ⬝ (right_inv apd10 _)⁻¹,
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apply eq_of_homotopy, intro u, induction u, reflexivity
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end
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end unit
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