lean2/library/hott/algebra/precategory/morphism.lean

-- 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

import .basic

open path precategory sigma sigma.ops

namespace morphism
  variables {ob : Type} [C : precategory ob] include C
  variables {a b c : ob} {g : b ⟶ c} {f : a ⟶ b} {h : b ⟶ a}
  inductive is_section    [class] (f : a ⟶ b) : Type
  := mk : ∀{g}, g ∘ f ≈ id → is_section f
  inductive is_retraction [class] (f : a ⟶ b) : Type
  := mk : ∀{g}, f ∘ g ≈ id → is_retraction f
  inductive is_iso        [class] (f : a ⟶ b) : Type
  := mk : ∀{g}, g ∘ f ≈ id → f ∘ g ≈ id → is_iso f

  definition retraction_of (f : a ⟶ b) [H : is_section f] : hom b a :=
  is_section.rec (λg h, g) H
  definition section_of    (f : a ⟶ b) [H : is_retraction f] : hom b a :=
  is_retraction.rec (λg h, g) H
  definition inverse       (f : a ⟶ b) [H : is_iso f] : hom b a :=
  is_iso.rec (λg h1 h2, g) H

  postfix `⁻¹` := inverse

  theorem inverse_compose (f : a ⟶ b) [H : is_iso f] : f⁻¹ ∘ f ≈ id :=
  is_iso.rec (λg h1 h2, h1) H

  theorem compose_inverse (f : a ⟶ b) [H : is_iso f] : f ∘ f⁻¹ ≈ id :=
  is_iso.rec (λg h1 h2, h2) H

  theorem retraction_compose (f : a ⟶ b) [H : is_section f] : retraction_of f ∘ f ≈ id :=
  is_section.rec (λg h, h) H

  theorem compose_section (f : a ⟶ b) [H : is_retraction f] : f ∘ section_of f ≈ id :=
  is_retraction.rec (λg h, h) H

  theorem iso_imp_retraction [instance] (f : a ⟶ b) [H : is_iso f] : is_section f :=
  is_section.mk !inverse_compose

  theorem iso_imp_section [instance] (f : a ⟶ b) [H : is_iso f] : is_retraction f :=
  is_retraction.mk !compose_inverse

  theorem id_is_iso [instance] : is_iso (ID a) :=
  is_iso.mk !id_compose !id_compose

  theorem inverse_is_iso [instance] (f : a ⟶ b) [H : is_iso f] : is_iso (f⁻¹) :=
  is_iso.mk !compose_inverse !inverse_compose

  theorem left_inverse_eq_right_inverse {f : a ⟶ b} {g g' : hom b a}
      (Hl : g ∘ f ≈ id) (Hr : f ∘ g' ≈ id) : g ≈ g' :=
  calc
    g ≈ g ∘ id : !id_right
     ... ≈ g ∘ f ∘ g' : Hr
     ... ≈ (g ∘ f) ∘ g' : !assoc
     ... ≈ id ∘ g' : Hl
     ... ≈ g' : id_left

  theorem retraction_eq_intro [H : is_section f] (H2 : f ∘ h ≈ id) : retraction_of f ≈ h
  := left_inverse_eq_right_inverse !retraction_compose H2

  theorem section_eq_intro [H : is_retraction f] (H2 : h ∘ f ≈ id) : section_of f ≈ h
  := (left_inverse_eq_right_inverse H2 !compose_section)⁻¹

  theorem inverse_eq_intro_right [H : is_iso f] (H2 : f ∘ h ≈ id) : f⁻¹ ≈ h
  := left_inverse_eq_right_inverse !inverse_compose H2

  theorem inverse_eq_intro_left [H : is_iso f] (H2 : h ∘ f ≈ id) : f⁻¹ ≈ h
  := (left_inverse_eq_right_inverse H2 !compose_inverse)⁻¹

  theorem section_eq_retraction (f : a ⟶ b) [Hl : is_section f] [Hr : is_retraction f] :
      retraction_of f ≈ section_of f :=
  retraction_eq_intro !compose_section

  theorem section_retraction_imp_iso (f : a ⟶ b) [Hl : is_section f] [Hr : is_retraction f]
      : is_iso f :=
  is_iso.mk ((section_eq_retraction f) ▹ (retraction_compose f)) (compose_section f)

  theorem inverse_unique (H H' : is_iso f) : @inverse _ _ _ _ f H ≈ @inverse _ _ _ _ f H' :=
  inverse_eq_intro_left !inverse_compose

  theorem inverse_involutive (f : a ⟶ b) [H : is_iso f] : (f⁻¹)⁻¹ ≈ f :=
  inverse_eq_intro_right !inverse_compose

  theorem retraction_of_id : retraction_of (ID a) ≈ id :=
  retraction_eq_intro !id_compose

  theorem section_of_id : section_of (ID a) ≈ id :=
  section_eq_intro !id_compose

  theorem iso_of_id : ID a⁻¹ ≈ id :=
  inverse_eq_intro_left !id_compose

  theorem composition_is_section [instance] [Hf : is_section f] [Hg : is_section g]
      : is_section (g ∘ f) :=
  have aux : retraction_of g ∘ g ∘ f ≈ (retraction_of g ∘ g) ∘ f,
    from !assoc,
  is_section.mk
    (calc
      (retraction_of f ∘ retraction_of g) ∘ g ∘ f
            ≈ retraction_of f ∘ retraction_of g ∘ g ∘ f : assoc
        ... ≈ retraction_of f ∘ ((retraction_of g ∘ g) ∘ f) : aux
        ... ≈ retraction_of f ∘ id ∘ f : {retraction_compose g}
        ... ≈ retraction_of f ∘ f : id_left f
        ... ≈ id : retraction_compose f)

  theorem composition_is_retraction [instance] (Hf : is_retraction f) (Hg : is_retraction g)
      : is_retraction (g ∘ f) :=
  have aux : f ∘ section_of f ∘ section_of g ≈ (f ∘ section_of f) ∘ section_of g,
    from !assoc,
  is_retraction.mk
    (calc
      (g ∘ f) ∘ section_of f ∘ section_of g
            ≈ g ∘ f ∘ section_of f ∘ section_of g : assoc
        ... ≈ g ∘ (f ∘ section_of f) ∘ section_of g : aux
        ... ≈ g ∘ id ∘ section_of g : compose_section f
        ... ≈ g ∘ section_of g : id_left (section_of g)
        ... ≈ id : compose_section)

  theorem composition_is_inverse [instance] (Hf : is_iso f) (Hg : is_iso g) : is_iso (g ∘ f) :=
  !section_retraction_imp_iso

  structure isomorphic (a b : ob) :=
    (iso : hom a b)
    [is_iso : is_iso iso]

  infix `≅`:50 := morphism.isomorphic

  namespace isomorphic

    --open relation
    instance [persistent] is_iso

    theorem refl  (a     : ob)                           : isomorphic a a :=
    mk id

    theorem symm  ⦃a b   : ob⦄ (H  : isomorphic a b)     : isomorphic b a :=
    mk (inverse (iso H))

    theorem trans ⦃a b c : ob⦄ (H1 : isomorphic a b) (H2 : isomorphic b c) : isomorphic a c :=
    mk (iso H2 ∘ iso H1)

  --theorem is_equivalence_eq [instance] (T : Type) : is_equivalence isomorphic :=
  --is_equivalence.mk (is_reflexive.mk refl) (is_symmetric.mk symm) (is_transitive.mk trans)
  end isomorphic

  -- In a precategory, equal objects are isomorphic
  definition iso_of_path (p : a ≈ b) : isomorphic a b :=
  path.rec_on p (isomorphic.mk id)

  inductive is_mono [class] (f : a ⟶ b) : Type :=
  mk : (∀c (g h : hom c a), f ∘ g ≈ f ∘ h → g ≈ h) → is_mono f
  inductive is_epi  [class] (f : a ⟶ b) : Type :=
  mk : (∀c (g h : hom b c), g ∘ f ≈ h ∘ f → g ≈ h) → is_epi f

  theorem mono_elim [H : is_mono f] {g h : c ⟶ a} (H2 : f ∘ g ≈ f ∘ h) : g ≈ h
  := is_mono.rec (λH3, H3 c g h H2) H
  theorem epi_elim [H : is_epi f] {g h : b ⟶ c} (H2 : g ∘ f ≈ h ∘ f) : g ≈ h
  := is_epi.rec  (λH3, H3 c g h H2) H

  theorem section_is_mono [instance] (f : a ⟶ b) [H : is_section f] : is_mono f :=
  is_mono.mk
    (λ c g h H,
      calc
        g ≈ id ∘ g : id_left
      ... ≈ (retraction_of f ∘ f) ∘ g : retraction_compose f
      ... ≈ retraction_of f ∘ f ∘ g : assoc
      ... ≈ retraction_of f ∘ f ∘ h : H
      ... ≈ (retraction_of f ∘ f) ∘ h : assoc
      ... ≈ id ∘ h : retraction_compose f
      ... ≈ h : id_left)

  theorem retraction_is_epi [instance] (f : a ⟶ b) [H : is_retraction f] : is_epi f :=
  is_epi.mk
    (λ c g h H,
      calc
        g ≈ g ∘ id : id_right
      ... ≈ g ∘ f ∘ section_of f : compose_section f
      ... ≈ (g ∘ f) ∘ section_of f : assoc
      ... ≈ (h ∘ f) ∘ section_of f : H
      ... ≈ h ∘ f ∘ section_of f : assoc
      ... ≈ h ∘ id : compose_section f
      ... ≈ h : id_right)

  --these theorems are now proven automatically using type classes
  --should they be instances?
  theorem id_is_mono : is_mono (ID a)
  theorem id_is_epi  : is_epi  (ID a)

  theorem composition_is_mono [instance] [Hf : is_mono f] [Hg : is_mono g] : is_mono (g ∘ f) :=
  is_mono.mk
    (λ d h₁ h₂ H,
      have H2 : g ∘ (f ∘ h₁) ≈ g ∘ (f ∘ h₂),
        from calc g ∘ (f ∘ h₁) ≈ (g ∘ f) ∘ h₁ : !assoc
                          ... ≈ (g ∘ f) ∘ h₂ : H
                          ... ≈ g ∘ (f ∘ h₂) : !assoc, mono_elim (mono_elim H2))

  theorem composition_is_epi  [instance] [Hf : is_epi f] [Hg : is_epi g] : is_epi  (g ∘ f) :=
  is_epi.mk
    (λ d h₁ h₂ H,
      have H2 : (h₁ ∘ g) ∘ f ≈ (h₂ ∘ g) ∘ f,
        from calc (h₁ ∘ g) ∘ f ≈ h₁ ∘ g ∘ f : !assoc
                          ... ≈ h₂ ∘ g ∘ f : H
                          ... ≈ (h₂ ∘ g) ∘ f: !assoc, epi_elim (epi_elim H2))

end morphism
namespace morphism
--rewrite lemmas for inverses, modified from
--https://github.com/JasonGross/HoTT-categories/blob/master/theories/Categories/Category/Morphisms.v
  namespace iso
  section
  variables {ob : Type} [C : precategory ob] include C
  variables {a b c d : ob}                                         (f : b ⟶ a)
                           (r : c ⟶ d) (q : b ⟶ c) (p : a ⟶ b)
                           (g : d ⟶ c)
  variable [Hq : is_iso q] include Hq
  theorem compose_pV : q ∘ q⁻¹ ≈ id := !compose_inverse
  theorem compose_Vp : q⁻¹ ∘ q ≈ id := !inverse_compose
  theorem compose_V_pp : q⁻¹ ∘ (q ∘ p) ≈ p :=
  calc
    q⁻¹ ∘ (q ∘ p) ≈ (q⁻¹ ∘ q) ∘ p : assoc (q⁻¹) q p
      ... ≈ id ∘ p : inverse_compose q
      ... ≈ p : id_left p
  theorem compose_p_Vp : q ∘ (q⁻¹ ∘ g) ≈ g :=
  calc
     q ∘ (q⁻¹ ∘ g)  ≈ (q ∘ q⁻¹) ∘ g : assoc q (q⁻¹) g
      ... ≈ id ∘ g : compose_inverse q
      ... ≈ g : id_left g
  theorem compose_pp_V : (r ∘ q) ∘ q⁻¹ ≈ r :=
  calc
    (r ∘ q) ∘ q⁻¹ ≈ r ∘ q ∘ q⁻¹ : assoc r q (q⁻¹)⁻¹
      ... ≈ r ∘ id : compose_inverse q
      ... ≈ r : id_right r
  theorem compose_pV_p : (f ∘ q⁻¹) ∘ q ≈ f :=
  calc
    (f ∘ q⁻¹) ∘ q ≈ f ∘ q⁻¹ ∘ q : assoc f (q⁻¹) q⁻¹
      ... ≈ f ∘ id : inverse_compose q
      ... ≈ f : id_right f

  theorem inv_pp [H' : is_iso p] : (q ∘ p)⁻¹ ≈ p⁻¹ ∘ q⁻¹ :=
  have H1 : (p⁻¹ ∘ q⁻¹) ∘ q ∘ p ≈ p⁻¹ ∘ (q⁻¹ ∘ (q ∘ p)), from assoc (p⁻¹) (q⁻¹) (q ∘ p)⁻¹,
  have H2 : (p⁻¹) ∘ (q⁻¹ ∘ (q ∘ p)) ≈ p⁻¹ ∘ p, from ap _ (compose_V_pp q p),
  have H3 : p⁻¹ ∘ p ≈ id, from inverse_compose p,
  inverse_eq_intro_left (H1 ⬝ H2 ⬝ H3)
  --the proof using calc is hard for the unifier (needs ~90k steps)
  -- inverse_eq_intro_left
  --   (calc
  --     (p⁻¹ ∘ (q⁻¹)) ∘ q ∘ p = p⁻¹ ∘ (q⁻¹ ∘ (q ∘ p)) : assoc (p⁻¹) (q⁻¹) (q ∘ p)⁻¹
  --     ... = (p⁻¹) ∘ p : congr_arg (λx, p⁻¹ ∘ x) (compose_V_pp q p)
  --     ... = id : inverse_compose p)
  theorem inv_Vp [H' : is_iso g] : (q⁻¹ ∘ g)⁻¹ ≈ g⁻¹ ∘ q := inverse_involutive q ▹ inv_pp (q⁻¹) g
  theorem inv_pV [H' : is_iso f] : (q ∘ f⁻¹)⁻¹ ≈ f ∘ q⁻¹ := inverse_involutive f ▹ inv_pp q (f⁻¹)
  theorem inv_VV [H' : is_iso r] : (q⁻¹ ∘ r⁻¹)⁻¹ ≈ r ∘ q := inverse_involutive r ▹ inv_Vp q (r⁻¹)
  end
  section
  variables {ob : Type} {C : precategory ob} include C
  variables {d           c           b           a : ob}
                          {i : b ⟶ c} {f : b ⟶ a}
              {r : c ⟶ d} {q : b ⟶ c} {p : a ⟶ b}
              {g : d ⟶ c} {h : c ⟶ b}
                   {x : b ⟶ d} {z : a ⟶ c}
                   {y : d ⟶ b} {w : c ⟶ a}
  variable [Hq : is_iso q] include Hq

  theorem moveR_Mp (H : y ≈ q⁻¹ ∘ g) : q ∘ y ≈ g := H⁻¹ ▹ compose_p_Vp q g
  theorem moveR_pM (H : w ≈ f ∘ q⁻¹) : w ∘ q ≈ f := H⁻¹ ▹ compose_pV_p f q
  theorem moveR_Vp (H : z ≈ q ∘ p) : q⁻¹ ∘ z ≈ p := H⁻¹ ▹ compose_V_pp q p
  theorem moveR_pV (H : x ≈ r ∘ q) : x ∘ q⁻¹ ≈ r := H⁻¹ ▹ compose_pp_V r q
  theorem moveL_Mp (H : q⁻¹ ∘ g ≈ y) : g ≈ q ∘ y := moveR_Mp (H⁻¹)⁻¹
  theorem moveL_pM (H : f ∘ q⁻¹ ≈ w) : f ≈ w ∘ q := moveR_pM (H⁻¹)⁻¹
  theorem moveL_Vp (H : q ∘ p ≈ z) : p ≈ q⁻¹ ∘ z := moveR_Vp (H⁻¹)⁻¹
  theorem moveL_pV (H : r ∘ q ≈ x) : r ≈ x ∘ q⁻¹ := moveR_pV (H⁻¹)⁻¹
  theorem moveL_1V (H : h ∘ q ≈ id) : h ≈ q⁻¹ := inverse_eq_intro_left H⁻¹
  theorem moveL_V1 (H : q ∘ h ≈ id) : h ≈ q⁻¹ := inverse_eq_intro_right H⁻¹
  theorem moveL_1M (H : i ∘ q⁻¹ ≈ id) : i ≈ q := moveL_1V H ⬝ inverse_involutive q
  theorem moveL_M1 (H : q⁻¹ ∘ i ≈ id) : i ≈ q := moveL_V1 H ⬝ inverse_involutive q
  theorem moveR_1M (H : id ≈ i ∘ q⁻¹) : q ≈ i := moveL_1M (H⁻¹)⁻¹
  theorem moveR_M1 (H : id ≈ q⁻¹ ∘ i) : q ≈ i := moveL_M1 (H⁻¹)⁻¹
  theorem moveR_1V (H : id ≈ h ∘ q) : q⁻¹ ≈ h := moveL_1V (H⁻¹)⁻¹
  theorem moveR_V1 (H : id ≈ q ∘ h) : q⁻¹ ≈ h := moveL_V1 (H⁻¹)⁻¹
  end
  end iso

end morphism