66 lines
1.8 KiB
Text
66 lines
1.8 KiB
Text
/-
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Copyright (c) 2014 Jakob von Raumer. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Module: init.ua
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Author: Jakob von Raumer
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Ported from Coq HoTT
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-/
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prelude
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import .equiv
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open eq equiv is_equiv equiv.ops
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--Ensure that the types compared are in the same universe
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section
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universe variable l
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variables {A B : Type.{l}}
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definition is_equiv_cast_of_eq (H : A = B) : is_equiv (cast H) :=
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(@is_equiv_tr Type (λX, X) A B H)
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definition equiv_of_eq (H : A = B) : A ≃ B :=
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equiv.mk _ (is_equiv_cast_of_eq H)
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end
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axiom univalence (A B : Type) : is_equiv (@equiv_of_eq A B)
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attribute univalence [instance]
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-- This is the version of univalence axiom we will probably use most often
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definition ua [reducible] {A B : Type} : A ≃ B → A = B :=
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equiv_of_eq⁻¹
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definition eq_equiv_equiv (A B : Type) : (A = B) ≃ (A ≃ B) :=
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equiv.mk equiv_of_eq _
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definition equiv_of_eq_ua [reducible] {A B : Type} (f : A ≃ B) : equiv_of_eq (ua f) = f :=
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right_inv equiv_of_eq f
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definition cast_ua_fn {A B : Type} (f : A ≃ B) : cast (ua f) = f :=
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ap to_fun (equiv_of_eq_ua f)
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definition cast_ua {A B : Type} (f : A ≃ B) (a : A) : cast (ua f) a = f a :=
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ap10 (cast_ua_fn f) a
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definition ua_equiv_of_eq [reducible] {A B : Type} (p : A = B) : ua (equiv_of_eq p) = p :=
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left_inv equiv_of_eq p
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-- One consequence of UA is that we can transport along equivalencies of types
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namespace equiv
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universe variable l
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protected definition transport_of_equiv [subst] (P : Type → Type) {A B : Type.{l}} (H : A ≃ B)
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: P A → P B :=
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eq.transport P (ua H)
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-- We can use this for calculation evironments
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definition rec_on_of_equiv_of_eq {A B : Type} {P : (A ≃ B) → Type}
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(p : A ≃ B) (H : Π(q : A = B), P (equiv_of_eq q)) : P p :=
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right_inv equiv_of_eq p ▸ H (ua p)
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end equiv
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