lean2/hott/init/axioms/funext_of_ua.hlean

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

Module: init.axioms.funext_of_ua
Author: Jakob von Raumer

Ported from Coq HoTT
-/

prelude
import ..equiv ..datatypes ..types.prod
import .funext_varieties .ua

open eq function prod is_trunc sigma equiv is_equiv unit

section
  universe variables l

  private theorem ua_isequiv_postcompose {A B : Type.{l}} {C : Type}
      {w : A → B} [H0 : is_equiv w] : is_equiv (@compose C A B w) :=
    let w' := equiv.mk w H0 in
    let eqinv : A = B := ((@is_equiv.inv _ _ _ (univalence A B)) w') in
    let eq' := equiv_of_eq eqinv in
    is_equiv.adjointify (@compose C A B w)
      (@compose C B A (is_equiv.inv w))
      (λ (x : C → B),
        have eqretr : eq' = w',
          from (@retr _ _ (@equiv_of_eq A B) (univalence A B) w'),
        have invs_eq : (to_fun eq')⁻¹ = (to_fun w')⁻¹,
          from inv_eq eq' w' eqretr,
        have eqfin : (to_fun eq') ∘ ((to_fun eq')⁻¹ ∘ x) = x,
          from (λ p,
            (@eq.rec_on Type.{l} A
              (λ B' p', Π (x' : C → B'), (to_fun (equiv_of_eq p'))
                ∘ ((to_fun (equiv_of_eq p'))⁻¹ ∘ x') = x')
              B p (λ x', idp))
            ) eqinv x,
        have eqfin' : (to_fun w') ∘ ((to_fun eq')⁻¹ ∘ x) = x,
          from eqretr ▹ eqfin,
        have eqfin'' : (to_fun w') ∘ ((to_fun w')⁻¹ ∘ x) = x,
          from invs_eq ▹ eqfin',
        eqfin''
      )
      (λ (x : C → A),
        have eqretr : eq' = w',
          from (@retr _ _ (@equiv_of_eq A B) (univalence A B) w'),
        have invs_eq : (to_fun eq')⁻¹ = (to_fun w')⁻¹,
          from inv_eq eq' w' eqretr,
        have eqfin : (to_fun eq')⁻¹ ∘ ((to_fun eq') ∘ x) = x,
          from (λ p, eq.rec_on p idp) eqinv,
        have eqfin' : (to_fun eq')⁻¹ ∘ ((to_fun w') ∘ x) = x,
          from eqretr ▹ eqfin,
        have eqfin'' : (to_fun w')⁻¹ ∘ ((to_fun w') ∘ x) = x,
          from invs_eq ▹ eqfin',
        eqfin''
      )

  -- We are ready to prove functional extensionality,
  -- starting with the naive non-dependent version.
  private definition diagonal [reducible] (B : Type) : Type
    := Σ xy : B × B, pr₁ xy = pr₂ xy

  private definition isequiv_src_compose {A B : Type}
      : @is_equiv (A → diagonal B)
                 (A → B)
                 (compose (pr₁ ∘ pr1)) :=
    @ua_isequiv_postcompose _ _ _ (pr₁ ∘ pr1)
        (is_equiv.adjointify (pr₁ ∘ pr1)
          (λ x, sigma.mk (x , x) idp) (λx, idp)
          (λ x, sigma.rec_on x
            (λ xy, prod.rec_on xy
              (λ b c p, eq.rec_on p idp))))

  private definition isequiv_tgt_compose {A B : Type}
      : @is_equiv (A → diagonal B)
                 (A → B)
                 (compose (pr₂ ∘ pr1)) :=
    @ua_isequiv_postcompose _ _ _ (pr2 ∘ pr1)
        (is_equiv.adjointify (pr2 ∘ pr1)
          (λ x, sigma.mk (x , x) idp) (λx, idp)
          (λ x, sigma.rec_on x
            (λ xy, prod.rec_on xy
              (λ b c p, eq.rec_on p idp))))

  set_option class.conservative false
  theorem nondep_funext_from_ua {A : Type} {B : Type}
      : Π {f g : A → B}, f ∼ g → f = g :=
    (λ (f g : A → B) (p : f ∼ g),
        let d := λ (x : A), sigma.mk (f x , f x) idp in
        let e := λ (x : A), sigma.mk (f x , g x) (p x) in
        let precomp1 :=  compose (pr₁ ∘ pr1) in
        have equiv1 [visible] : is_equiv precomp1,
          from @isequiv_src_compose A B,
        have equiv2 [visible] : Π x y, is_equiv (ap precomp1),
          from is_equiv.is_equiv_ap precomp1,
        have H' : Π (x y : A → diagonal B),
            pr₁ ∘ pr1 ∘ x = pr₁ ∘ pr1 ∘ y → x = y,
          from (λ x y, is_equiv.inv (ap precomp1)),
        have eq2 : pr₁ ∘ pr1 ∘ d = pr₁ ∘ pr1 ∘ e,
          from idp,
        have eq0 : d = e,
          from H' d e eq2,
        have eq1 : (pr₂ ∘ pr1) ∘ d = (pr₂ ∘ pr1) ∘ e,
          from ap _ eq0,
        eq1
    )

end

-- Now we use this to prove weak funext, which as we know
-- implies (with dependent eta) also the strong dependent funext.
theorem weak_funext_of_ua : weak_funext :=
  (λ (A : Type) (P : A → Type) allcontr,
    let U := (λ (x : A), unit) in
  have pequiv : Π (x : A), P x ≃ U x,
    from (λ x, @equiv_unit_of_is_contr (P x) (allcontr x)),
  have psim : Π (x : A), P x = U x,
    from (λ x, @is_equiv.inv _ _
      equiv_of_eq (univalence _ _) (pequiv x)),
  have p : P = U,
    from @nondep_funext_from_ua A Type P U psim,
  have tU' : is_contr (A → unit),
    from is_contr.mk (λ x, ⋆)
      (λ f, @nondep_funext_from_ua A unit (λ x, ⋆) f
        (λ x, unit.rec_on (f x) idp)),
  have tU : is_contr (Π x, U x),
    from tU',
  have tlast : is_contr (Πx, P x),
    from eq.transport _ p⁻¹ tU,
  tlast
)

-- In the following we will proof function extensionality using the univalence axiom
definition funext_of_ua : funext :=
  funext_of_weak_funext (@weak_funext_of_ua)

variables {A : Type} {P : A → Type} {f g : Π x, P x}

namespace funext
  definition is_equiv_apD [instance] (f g : Π x, P x) : is_equiv (@apD10 A P f g) :=
  funext_of_ua f g
end funext

open funext

definition eq_equiv_homotopy : (f = g) ≃ (f ∼ g) :=
equiv.mk apD10 _

definition eq_of_homotopy [reducible] : f ∼ g → f = g :=
(@apD10 A P f g)⁻¹

--TODO: rename to eq_of_homotopy_idp
definition eq_of_homotopy_id (f : Π x, P x) : eq_of_homotopy (λx : A, idpath (f x)) = idpath f :=
is_equiv.sect apD10 idp

definition naive_funext_of_ua : naive_funext :=
λ A P f g h, eq_of_homotopy h

protected definition homotopy.rec_on {Q : (f ∼ g) → Type} (p : f ∼ g)
  (H : Π(q : f = g), Q (apD10 q)) : Q p :=
retr apD10 p ▹ H (eq_of_homotopy p)