Hedberg's theorem
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{-# OPTIONS --cubical #-} | |
module hedberg where | |
open import Cubical.Core.Everything | |
open import Cubical.Foundations.Prelude | |
open import Cubical.Foundations.GroupoidLaws | |
data Empty : Set where | |
absurd : {A : Set} → Empty → A | |
absurd () | |
_∘_ : {A B C : Set} → (B → C) → (A → B) → A → C | |
(f ∘ g) x = f (g x) | |
id : {A : Set} → A → A | |
id x = x | |
pcong : {A B : Set} → {x y : A} → {f g : A → B} → (f ≡ g) → (f x ≡ f y) → (g x ≡ g y) | |
pcong {x = x} {y = y} f≡g = transp (λ i → f≡g i x ≡ f≡g i y) i0 | |
data ∥_∥ (A : Set) : Set where | |
∣_∣ : A → ∥ A ∥ | |
squash : (x y : ∥ A ∥) → x ≡ y | |
-- ∥ X ∥ → X iff constant endomap is available | |
hprop : Set → Set | |
hprop X = (x y : X) → x ≡ y | |
h-set : Set → Set | |
h-set X = (x y : X) → hprop (x ≡ y) | |
data decidable (A : Set) : Set where | |
yes : A → decidable A | |
no : (A → Empty) → decidable A | |
discrete : Set → Set | |
discrete X = (x y : X) → decidable (x ≡ y) | |
constant : {X Y : Set} → (X → Y) → Set | |
constant {X} f = (x y : X) → f x ≡ f y | |
collapsible : Set → Set | |
collapsible X = Σ[ f ∈ (X → X) ] constant f | |
path-collapsible : Set → Set | |
path-collapsible X = (x y : X) → collapsible (x ≡ y) | |
discrete→path-collapsible : {X : Set} | |
→ discrete X → path-collapsible X | |
discrete→path-collapsible {X} d x y | |
= map , cm | |
where map : x ≡ y → x ≡ y | |
map x≡y with d x y | |
map x≡y | yes x≡y' = x≡y' | |
map x≡y | no f = absurd (f x≡y) | |
cm : constant map | |
cm p q with d x y | |
cm p q | yes x = (λ i → x) | |
cm p q | no x = absurd (x p) | |
path-collapsible→h-set : {X : Set} | |
→ path-collapsible X → h-set X | |
path-collapsible→h-set {X} pc x y p q = pcong lif (cong g (snd (pc x y) p q)) | |
where g : x ≡ y → x ≡ y | |
g x≡y = x≡y ∙ sym (fst (pc y y) refl) | |
lif : g ∘ (fst (pc x y)) ≡ id | |
lif i x≡y = J (λ y x≡y → fst (pc x y) x≡y ∙ sym (fst (pc y y) refl) ≡ x≡y) | |
(rCancel (fst (pc x x) refl)) | |
x≡y i | |
-- Hedberg's theorem | |
discrete→h-set : {X : Set} → discrete X → h-set X | |
discrete→h-set d = path-collapsible→h-set (discrete→path-collapsible d) | |
stable : Set → Set | |
stable X = ((X → Empty) → Empty) → X | |
separated : Set → Set | |
separated X = (x y : X) → stable (x ≡ y) | |
separated→h-set : {X : Set} → separated X → h-set X | |
separated→h-set {X} sep = path-collapsible→h-set λ x y → (sep x y ∘ obvious x y) , c x y | |
where obvious : (x y : X) → (x ≡ y) → ((x ≡ y → Empty) → Empty) | |
obvious x y x≡y f = f x≡y | |
lemma : {X : Set} (p q : X → Empty) → (x : X) → p x ≡ q x | |
lemma p q x = absurd (p x) | |
lemma₂ : {X : Set} (p q : X → Empty) → p ≡ q | |
lemma₂ p q i x = lemma p q x i | |
c : (x y : X) → constant (sep x y ∘ obvious x y) | |
c x y x≡y₁ x≡y₂ i j = sep x y (lemma₂ (obvious x y x≡y₁) (obvious x y x≡y₂) i) j | |
cong-preserves-constant : {A B : Set} → {f : A → B} {x y : A} | |
→ constant f → constant (cong {x = x} {y = y} f) | |
cong-preserves-constant {f = f} {x = x} {y = y} cf p q = lemma p ∙ sym (lemma q) | |
where lemma : ∀ p → cong {x = x} {y = y} f p ≡ sym (cf x x) ∙ cf x y | |
lemma = J (λ y x≡y → cong {x = x} {y = y} f x≡y ≡ sym (cf x x) ∙ cf x y) | |
(sym (lCancel (cf x x))) | |
cong-constant : {A : Set} {B : Set} {f : A → B} {x : A} {x≡x : x ≡ x} | |
→ constant f | |
→ cong f x≡x ≡ refl | |
cong-constant {f = f} {x} {x≡x} c = cong-preserves-constant c _ _ | |
fixed-point-lemma : {X : Set} (f : X → X) → constant f | |
→ hprop (Σ[ x ∈ X ] x ≡ f x) | |
fixed-point-lemma {X} f cf (x , x≡fx) (y , y≡fy) = cong₂ (_,_) x≡fx eq1 | |
∙ cong₂ (_,_) (cf x y) eq2 | |
∙ cong₂ (_,_) (sym y≡fy) eq3 | |
where transport' : {X Y : Set} → (h k : X → Y) | |
→ {x y : X} → (t : x ≡ y) → (p : h x ≡ k x) → h y ≡ k y | |
transport' h k t p = sym (cong h t) ∙ p ∙ cong k t | |
x≡y : x ≡ y | |
x≡y = x≡fx ∙ cf x y ∙ sym y≡fy | |
x≡x : x ≡ x | |
x≡x = x≡fx ∙ sym (subst (λ z → z ≡ f z) (sym x≡y) y≡fy) | |
fx≡ffx : f x ≡ f (f x) | |
fx≡ffx = cong f x≡fx | |
fy≡ffy : f y ≡ f (f y) | |
fy≡ffy = cong f y≡fy | |
fx≡ffy : f x ≡ f (f y) | |
fx≡ffy = cong f (x≡fx ∙ cf x y) | |
eq1 : PathP (λ i → x≡fx i ≡ f (x≡fx i)) x≡fx fx≡ffx | |
eq1 i j = hfill (λ k → λ{(i = i0) → x≡fx j | |
;(i = i1) → fx≡ffx k | |
;(j = i1) → f (x≡fx (i ∧ k))}) (inS (x≡fx (i ∨ j))) j | |
eq2 : PathP (λ i → cf x y i ≡ f (cf x y i)) fx≡ffx fy≡ffy | |
eq2 i j = {!!} | |
eq3 : PathP (λ i → y≡fy (~ i) ≡ f (y≡fy (~ i))) fy≡ffy y≡fy | |
eq3 i j = hfill (λ k → λ{(i = i0) → fy≡ffy (j ∧ k) | |
;(i = i1) → y≡fy j | |
;(j = i1) → f (y≡fy (~ i ∧ k))}) (inS (y≡fy (j ∨ (~ i)))) j |
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