lean2/hott/init/ua.hlean

/-
Copyright (c) 2014 Jakob von Raumer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jakob von Raumer, Floris van Doorn

Ported from Coq HoTT
-/

prelude
import .equiv
open eq equiv is_equiv

axiom univalence (A B : Type) : is_equiv (@equiv_of_eq A B)

attribute univalence [instance]

-- This is the version of univalence axiom we will probably use most often
definition ua [reducible] {A B : Type} : A ≃ B → A = B :=
equiv_of_eq⁻¹

definition eq_equiv_equiv (A B : Type) : (A = B) ≃ (A ≃ B) :=
equiv.mk equiv_of_eq _

definition equiv_of_eq_ua [reducible] {A B : Type} (f : A ≃ B) : equiv_of_eq (ua f) = f :=
right_inv equiv_of_eq f

definition cast_ua_fn {A B : Type} (f : A ≃ B) : cast (ua f) = f :=
ap to_fun (equiv_of_eq_ua f)

definition cast_ua {A B : Type} (f : A ≃ B) (a : A) : cast (ua f) a = f a :=
ap10 (cast_ua_fn f) a

definition cast_ua_inv_fn {A B : Type} (f : A ≃ B) : cast (ua f)⁻¹ = to_inv f :=
ap to_inv (equiv_of_eq_ua f)

definition cast_ua_inv {A B : Type} (f : A ≃ B) (b : B) : cast (ua f)⁻¹ b = to_inv f b :=
ap10 (cast_ua_inv_fn f) b

definition ua_equiv_of_eq [reducible] {A B : Type} (p : A = B) : ua (equiv_of_eq p) = p :=
left_inv equiv_of_eq p

definition eq_of_equiv_lift {A B : Type} (f : A ≃ B) : A = lift B :=
ua (f ⬝e !equiv_lift)

namespace equiv

  -- One consequence of UA is that we can transport along equivalencies of types
  -- We can use this for calculation evironments
  protected definition transport_of_equiv [subst] (P : Type → Type) {A B : Type} (H : A ≃ B)
    : P A → P B :=
  eq.transport P (ua H)

  -- we can "recurse" on equivalences, by replacing them by (equiv_of_eq _)
  definition rec_on_ua [recursor] {A B : Type} {P : A ≃ B → Type}
    (f : A ≃ B) (H : Π(q : A = B), P (equiv_of_eq q)) : P f :=
  right_inv equiv_of_eq f ▸ H (ua f)

  -- a variant where we immediately recurse on the equality in the new goal
  definition rec_on_ua_idp [recursor] {A : Type} {P : Π{B}, A ≃ B → Type} {B : Type}
    (f : A ≃ B) (H : P equiv.rfl) : P f :=
  rec_on_ua f (λq, eq.rec_on q H)

  -- a variant where (equiv_of_eq (ua f)) will be replaced by f in the new goal
  definition rec_on_ua' {A B : Type} {P : A ≃ B → A = B → Type}
    (f : A ≃ B) (H : Π(q : A = B), P (equiv_of_eq q) q) : P f (ua f) :=
  right_inv equiv_of_eq f ▸ H (ua f)

  -- a variant where we do both
  definition rec_on_ua_idp' {A : Type} {P : Π{B}, A ≃ B → A = B → Type} {B : Type}
    (f : A ≃ B) (H : P equiv.rfl idp) : P f (ua f) :=
  rec_on_ua' f (λq, eq.rec_on q H)

  definition ua_refl (A : Type) : ua erfl = idpath A :=
  inj !eq_equiv_equiv (right_inv !eq_equiv_equiv erfl)

  definition ua_symm {A B : Type} (f : A ≃ B) : ua f⁻¹ᵉ = (ua f)⁻¹ :=
  begin
    apply rec_on_ua_idp f,
    refine !ua_refl ⬝ inverse2 !ua_refl⁻¹
  end

  definition ua_trans {A B C : Type} (f : A ≃ B) (g : B ≃ C) : ua (f ⬝e g) = ua f ⬝ ua g :=
  begin
    apply rec_on_ua_idp g, apply rec_on_ua_idp f,
    refine !ua_refl ⬝ concat2 !ua_refl⁻¹ !ua_refl⁻¹
  end


end equiv