110 lines
4.3 KiB
Text
110 lines
4.3 KiB
Text
-- Copyright (c) 2014 Jakob von Raumer. All rights reserved.
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-- Released under Apache 2.0 license as described in the file LICENSE.
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-- Authors: Jakob von Raumer
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-- Ported from Coq HoTT
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import hott.path hott.trunc hott.equiv hott.axioms.funext
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open path truncation sigma function
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/- In hott.axioms.funext, we defined function extensionality to be the assertion
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that the map apD10 is an equivalence. We now prove that this follows
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from a couple of weaker-looking forms of function extensionality. We do
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require eta conversion, which Coq 8.4+ has judgmentally.
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This proof is originally due to Voevodsky; it has since been simplified
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by Peter Lumsdaine and Michael Shulman. -/
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-- Naive funext is the simple assertion that pointwise equal functions are equal.
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-- TODO think about universe levels
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definition naive_funext :=
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Π {A : Type} {P : A → Type} (f g : Πx, P x), (f ∼ g) → f ≈ g
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-- Weak funext says that a product of contractible types is contractible.
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definition weak_funext.{l k} :=
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Π {A : Type.{l}} (P : A → Type.{k}) [H: Πx, is_contr (P x)], is_contr (Πx, P x)
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-- The obvious implications are Funext -> NaiveFunext -> WeakFunext
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-- TODO: Get class inference to work locally
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definition naive_funext_from_funext [F : funext] : naive_funext :=
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(λ A P f g h,
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have Fefg: is_equiv (@apD10 A P f g),
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from (@funext.ap F A P f g),
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have eq1 : _, from (@is_equiv.inv _ _ (@apD10 A P f g) Fefg h),
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eq1
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)
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definition weak_funext_from_naive_funext : naive_funext → weak_funext :=
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(λ nf A P (Pc : Πx, is_contr (P x)),
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let c := λx, center (P x) in
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is_contr.mk c (λ f,
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have eq' : (λx, center (P x)) ∼ f,
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from (λx, contr (f x)),
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have eq : (λx, center (P x)) ≈ f,
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from nf A P (λx, center (P x)) f eq',
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eq
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)
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)
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/- The less obvious direction is that WeakFunext implies Funext
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(and hence all three are logically equivalent). The point is that under weak
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funext, the space of "pointwise homotopies" has the same universal property as
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the space of paths. -/
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context
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universes l k
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parameters (wf : weak_funext.{l+1 k+1}) {A : Type.{l+1}} {B : A → Type.{k+1}} (f : Π x, B x)
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protected definition idhtpy : f ∼ f := (λ x, idp)
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definition contr_basedhtpy [instance] : is_contr (Σ (g : Π x, B x), f ∼ g) :=
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is_contr.mk (dpair f idhtpy)
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(λ dp, sigma.rec_on dp
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(λ (g : Π x, B x) (h : f ∼ g),
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let r := λ (k : Π x, Σ y, f x ≈ y),
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@dpair _ (λg, f ∼ g)
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(λx, dpr1 (k x)) (λx, dpr2 (k x)) in
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let s := λ g h x, @dpair _ (λy, f x ≈ y) (g x) (h x) in
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have t1 : Πx, is_contr (Σ y, f x ≈ y),
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from (λx, !contr_basedpaths),
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have t2 : is_contr (Πx, Σ y, f x ≈ y),
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from !wf,
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have t3 : (λ x, @dpair _ (λ y, f x ≈ y) (f x) idp) ≈ s g h,
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from @path_contr (Π x, Σ y, f x ≈ y) t2 _ _,
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have t4 : r (λ x, dpair (f x) idp) ≈ r (s g h),
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from ap r t3,
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have endt : dpair f idhtpy ≈ dpair g h,
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from t4,
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endt
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)
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)
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parameters (Q : Π g (h : f ∼ g), Type) (d : Q f idhtpy)
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definition htpy_ind (g : Πx, B x) (h : f ∼ g) : Q g h :=
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@transport _ (λ gh, Q (dpr1 gh) (dpr2 gh)) (dpair f idhtpy) (dpair g h)
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(@path_contr _ contr_basedhtpy _ _) d
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definition htpy_ind_beta : htpy_ind f idhtpy ≈ d :=
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(@path2_contr _ _ _ _ !path_contr idp)⁻¹ ▹ idp
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end
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-- Now the proof is fairly easy; we can just use the same induction principle on both sides.
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universe variables l k
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theorem funext_from_weak_funext (wf : weak_funext.{l+1 k+1}) : funext.{l+1 k+1} :=
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funext.mk (λ A B f g,
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let eq_to_f := (λ g' x, f ≈ g') in
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let sim2path := htpy_ind _ f eq_to_f idp in
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have t1 : sim2path f (idhtpy f) ≈ idp,
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proof htpy_ind_beta _ f eq_to_f idp qed,
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have t2 : apD10 (sim2path f (idhtpy f)) ≈ (idhtpy f),
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proof ap apD10 t1 qed,
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have sect : apD10 ∘ (sim2path g) ∼ id,
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proof (htpy_ind _ f (λ g' x, apD10 (sim2path g' x) ≈ x) t2) g qed,
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have retr : (sim2path g) ∘ apD10 ∼ id,
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from (λ h, path.rec_on h (htpy_ind_beta _ f _ idp)),
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is_equiv.adjointify apD10 (sim2path g) sect retr)
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definition funext_from_naive_funext : naive_funext -> funext :=
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compose funext_from_weak_funext weak_funext_from_naive_funext
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