lean2/hott/types/arrow.hlean

56 lines
1.6 KiB
Text
Raw Normal View History

/-
Copyright (c) 2014 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Author: Floris van Doorn
Ported from Coq HoTT
Theorems about arrow types (function spaces)
-/
import types.pi
open eq equiv is_equiv funext pi equiv.ops
namespace arrow
variables {A A' : Type} {B B' : Type} {C : A → B → Type}
{a a' a'' : A} {b b' b'' : B} {f g : A → B}
-- all lemmas here are special cases of the ones for pi-types
/- Functorial action -/
variables (f0 : A' → A) (f1 : B → B')
definition arrow_functor : (A → B) → (A' → B') := pi_functor f0 (λa, f1)
/- Equivalences -/
definition is_equiv_arrow_functor
[H0 : is_equiv f0] [H1 : is_equiv f1] : is_equiv (arrow_functor f0 f1) :=
is_equiv_pi_functor f0 (λa, f1)
definition arrow_equiv_arrow_rev (f0 : A' ≃ A) (f1 : B ≃ B') : (A → B) ≃ (A' → B') :=
equiv.mk _ (is_equiv_arrow_functor f0 f1)
definition arrow_equiv_arrow (f0 : A ≃ A') (f1 : B ≃ B') : (A → B) ≃ (A' → B') :=
arrow_equiv_arrow_rev (equiv.symm f0) f1
definition arrow_equiv_arrow_right (f1 : B ≃ B') : (A → B) ≃ (A → B') :=
arrow_equiv_arrow_rev equiv.refl f1
definition arrow_equiv_arrow_left_rev (f0 : A' ≃ A) : (A → B) ≃ (A' → B) :=
arrow_equiv_arrow_rev f0 equiv.refl
definition arrow_equiv_arrow_left (f0 : A ≃ A') : (A → B) ≃ (A' → B) :=
arrow_equiv_arrow f0 equiv.refl
/- Transport -/
definition arrow_transport {B C : A → Type} (p : a = a') (f : B a → C a)
: (transport (λa, B a → C a) p f) (λb, p ▸ f (p⁻¹ ▸ b)) :=
eq.rec_on p (λx, idp)
end arrow