89 lines
3.1 KiB
Text
89 lines
3.1 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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Authors: Jakob von Raumer, Floris van Doorn
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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
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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 cast_ua_inv_fn {A B : Type} (f : A ≃ B) : cast (ua f)⁻¹ = to_inv f :=
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ap to_inv (equiv_of_eq_ua f)
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definition cast_ua_inv {A B : Type} (f : A ≃ B) (b : B) : cast (ua f)⁻¹ b = to_inv f b :=
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ap10 (cast_ua_inv_fn f) b
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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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definition eq_of_equiv_lift {A B : Type} (f : A ≃ B) : A = lift B :=
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ua (f ⬝e !equiv_lift)
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namespace equiv
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-- One consequence of UA is that we can transport along equivalencies of types
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-- We can use this for calculation evironments
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protected definition transport_of_equiv [subst] (P : Type → Type) {A B : Type} (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 "recurse" on equivalences, by replacing them by (equiv_of_eq _)
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definition rec_on_ua [recursor] {A B : Type} {P : A ≃ B → Type}
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(f : A ≃ B) (H : Π(q : A = B), P (equiv_of_eq q)) : P f :=
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right_inv equiv_of_eq f ▸ H (ua f)
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-- a variant where we immediately recurse on the equality in the new goal
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definition rec_on_ua_idp [recursor] {A : Type} {P : Π{B}, A ≃ B → Type} {B : Type}
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(f : A ≃ B) (H : P equiv.rfl) : P f :=
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rec_on_ua f (λq, eq.rec_on q H)
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-- a variant where (equiv_of_eq (ua f)) will be replaced by f in the new goal
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definition rec_on_ua' {A B : Type} {P : A ≃ B → A = B → Type}
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(f : A ≃ B) (H : Π(q : A = B), P (equiv_of_eq q) q) : P f (ua f) :=
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right_inv equiv_of_eq f ▸ H (ua f)
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-- a variant where we do both
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definition rec_on_ua_idp' {A : Type} {P : Π{B}, A ≃ B → A = B → Type} {B : Type}
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(f : A ≃ B) (H : P equiv.rfl idp) : P f (ua f) :=
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rec_on_ua' f (λq, eq.rec_on q H)
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definition ua_refl (A : Type) : ua erfl = idpath A :=
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eq_of_fn_eq_fn !eq_equiv_equiv (right_inv !eq_equiv_equiv erfl)
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definition ua_symm {A B : Type} (f : A ≃ B) : ua f⁻¹ᵉ = (ua f)⁻¹ :=
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begin
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apply rec_on_ua_idp f,
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refine !ua_refl ⬝ inverse2 !ua_refl⁻¹
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end
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definition ua_trans {A B C : Type} (f : A ≃ B) (g : B ≃ C) : ua (f ⬝e g) = ua f ⬝ ua g :=
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begin
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apply rec_on_ua_idp g, apply rec_on_ua_idp f,
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refine !ua_refl ⬝ concat2 !ua_refl⁻¹ !ua_refl⁻¹
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end
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end equiv
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