lean2/hott/types/arrow.hlean

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

Module: types.arrow
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