2015-06-04 01:41:21 +00:00
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/-
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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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2015-06-04 17:55:02 +00:00
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Theorems about the unit type
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2015-06-04 01:41:21 +00:00
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-/
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2015-11-18 23:08:38 +00:00
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import algebra.group
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2015-10-22 22:41:55 +00:00
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open equiv option eq
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2015-06-04 01:41:21 +00:00
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namespace unit
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2015-10-22 22:41:55 +00:00
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protected definition eta : Π(u : unit), ⋆ = u
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| eta ⋆ := idp
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2015-06-04 01:41:21 +00:00
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definition unit_equiv_option_empty : 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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definition unit_imp_equiv (A : Type) : (unit → A) ≃ A :=
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begin
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fapply equiv.MK,
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{ intro f, exact f star},
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{ intro a u, exact a},
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{ intro a, reflexivity},
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{ intro f, apply eq_of_homotopy, intro u, cases u, reflexivity},
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end
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end unit
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2015-11-18 23:08:38 +00:00
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open unit is_trunc
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namespace algebra
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definition trivial_group [constructor] : group unit :=
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group.mk (λx y, star) _ (λx y z, idp) star (unit.rec idp) (unit.rec idp) (λx, star) (λx, idp)
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definition Trivial_group [constructor] : Group :=
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Group.mk _ trivial_group
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notation `G0` := Trivial_group
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end algebra
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