Chuangjie Xu 2013 (updated in February 2015)

\begin{code}

{-# OPTIONS --without-K #-}

module UsingIrrelevantFunext.Space.DiscreteSpace where

open import Preliminaries.SetsAndFunctions hiding (_+_)
open import Preliminaries.NaturalNumber
open import Preliminaries.Boolean
open import Preliminaries.Sequence
open import Continuity.UniformContinuity
open import UsingIrrelevantFunext.Irrelevance
open import UsingIrrelevantFunext.Space.Coverage
open import UsingIrrelevantFunext.Space.Space
open import UsingIrrelevantFunext.Space.CartesianClosedness
open import UsingIrrelevantFunext.IrrelevantFunext

\end{code}

The locally constant functions ₂ℕ → X on any set X form a C-topology on X. Any
space with such a C-topology is discrete, i.e. all maps from it to any other
space is continuous.

\begin{code}

LC : {X : Set}  (₂ℕ  X)  Set
LC = locally-constant

LC-topology : (X : Set)  discrete X  probe-axioms X LC
LC-topology X dis = c₀ , c₁ , c₂ , c₃
 where
  c₀ : ∀(x : X)   α  x)  LC
  c₀ x = 0 ,  _ _ _  refl) ,  _ _  ≤-zero)

  c₁ : ∀(t : ₂ℕ  ₂ℕ)  t  C  ∀(p : ₂ℕ  X)  p  LC  (p  t)  LC
  c₁ t uct p ucp = Lemma[LM-least-modulus] _≡_ dis _⁻¹ _·_ (p  t) n prf
   where
    m : 
    m = pr₁ ucp
    n : 
    n = pr₁(uct m)
    prp : ∀(α β : ₂ℕ)  α ≡[ m ] β  p α  p β
    prp = pr₁ (pr₂ ucp)
    prt : ∀(α β : ₂ℕ)  α ≡[ n ] β  t α ≡[ m ] t β
    prt = pr₁ (pr₂ (uct m))
    prf : ∀(α β : ₂ℕ)  α ≡[ n ] β  p(t α)  p(t β)
    prf α β en = prp (t α) (t β) (prt α β en)

  c₂ : ∀(p : ₂ℕ  X)  (Σ \(n : )  ∀(s : ₂Fin n)  (p  (cons s))  LC)  p  LC
  c₂ p (n , ps) = Lemma[LM-least-modulus] _≡_ dis _⁻¹ _·_ p (n + k) prf
   where
    f : ₂Fin n  
    f s = pr₁ (ps s)
    k : 
    k = pr₁ (max-fin f)
    k-max : ∀(s : ₂Fin n)  f s  k
    k-max = pr₂ (max-fin f)
    fact : ∀(s : ₂Fin n)  ∀(α β : ₂ℕ)  α ≡[ k ] β  p(cons s α)  p(cons s β)
    fact s α β ek = pr₁ (pr₂ (ps s)) α β (Lemma[≡[]-≤] ek (k-max s))
    prf : ∀(α β : ₂ℕ)  α ≡[ n + k ] β  p α  p β
    prf α β enk = decidable-structure (dis (p α) (p β)) goal
     where
      s : ₂Fin n
      s = take n α
      en : α ≡[ n ] β
      en = Lemma[≡[]-≤] enk (Lemma[a≤a+b] n k)
      eqs : take n α  take n β
      eqs = Lemma[≡[]-take] en
      α' : ₂ℕ
      α' = drop n α
       : [ cons s α'  α ]
       = funext¹ (Lemma[cons-take-drop] n α)
           -------
      β' : ₂ℕ
      β' = drop n β
       : [ cons s β'  β ]
       = [transport]  x  cons x β'  β) (hide (eqs ⁻¹))
                       (funext¹ (Lemma[cons-take-drop] n β))
                        -------
      awk : ∀(i : )  i < k  α' i  β' i
      awk i i<k = eqα · subgoal · eqβ ⁻¹
       where
        i+n<k+n : i + n < k + n
        i+n<k+n = Lemma[a<b→a+c<b+c] i k n i<k
        i+n<n+k : i + n < n + k
        i+n<n+k = transport  m  (i + n) < m) (Lemma[n+m=m+n] k n) i+n<k+n
        subgoal : α (i + n)  β (i + n)
        subgoal = Lemma[≡[]-<] enk (i + n) i+n<n+k
        le : (n i : )  (α : ₂ℕ)  drop n α i  α (i + n)
        le 0 i α = refl
        le (succ n) i α = le n i (α  succ)
        eqα : α' i  α (i + n)
        eqα = le n i α
        eqβ : β' i  β (i + n)
        eqβ = le n i β
      ek : α' ≡[ k ] β'
      ek = Lemma[<-≡[]] awk
      goal : [ p α  p β ]
      goal = [transport]  x  p α  p x) 
                         ([transport]  x  p x  p(cons s β'))  (hide (fact s α' β' ek)))

  c₃ : ∀(p p' : ₂ℕ  X)  p  LC  (∀(α : ₂ℕ)  [ p α  p' α ])  p'  LC
  c₃ p p' (n , (prf , min)) [ex] = n , prf' , min'
   where
    ex : ∀(α : ₂ℕ)  p α  p' α
    ex α = decidable-structure (dis (p α) (p' α)) ([ex] α)
    prf' : ∀(α β : ₂ℕ)  α ≡[ n ] β  p' α  p' β
    prf' α β en = (ex α)⁻¹ · (prf α β en) · (ex β)
    min' : ∀(n' : )  (∀(α β : ₂ℕ)  α ≡[ n' ] β  p' α  p' β)  n  n'
    min' n' pr'' = min n'  α β en'  (ex α) · (pr'' α β en') · (ex β)⁻¹)


DiscreteSpace : (X : Set)  discrete X  Space
DiscreteSpace X dec = X , LC , LC-topology X dec


Lemma[discreteness] : (X : Set) (dec : discrete X) (Y : Space)
                     ∀(f : X  U Y)  continuous (DiscreteSpace X dec) Y f
Lemma[discreteness] X dec Y f p (m , prf , _) = cond₂ Y (f  p) (m , claim)
 where
  claim : ∀(s : ₂Fin m)  (f  p  (cons s))  Probe Y
  claim s = cond₃ Y  α  y) (f  p  (cons s)) (cond₀ Y y) claim₁
   where
    y : U Y
    y = f(p(cons s ))
    claim₀ : ∀(α : ₂ℕ)  p(cons s )  p(cons s α)
    claim₀ α = prf (cons s ) (cons s α) (Lemma[cons-≡[]] s  α)
    claim₁ : ∀(α : ₂ℕ)  [ y  f(p(cons s α)) ]
    claim₁ α = hide (ap f (claim₀ α))

\end{code}

All the uniformly continuous maps ₂ℕ → ₂ (and ₂ℕ → ℕ) are
locally constant. And hence they form a C-topology on ₂ (and ℕ).

The coproduct 1 + 1:

\begin{code}

₂Space : Space
₂Space = DiscreteSpace  ₂-discrete

Lemma[discrete-₂Space] : (X : Space)  ∀(f :   U X)  continuous ₂Space X f
Lemma[discrete-₂Space] X f = Lemma[discreteness]  ₂-discrete X f

continuous-if : (A : Space)  Map ₂Space (A  A  A)
continuous-if A = IF , c-IF
 where
  IF :   U (A  A  A)
  IF b = if-b , c-if-b
   where
    if-b : U A  U (A  A)
    if-b a₀ = if-b-a₀ , c-if-b-a₀
     where
      if-b-a₀ : U A  U A
      if-b-a₀ a₁ = if b a₀ a₁
      c-if-b-a₀ : continuous A A if-b-a₀
      c-if-b-a₀ p pA = lemma b
       where
        lemma : ∀(i : )   a₁  if i a₀ a₁)  p  Probe A
        lemma  = cond₀ A a₀
        lemma  = pA
    c-if-b : continuous A (A  A) if-b
    c-if-b p pA q qA t tC = lemma b
     where
      lemma : ∀(i : )   α  if i (p(t α)) (q α))  Probe A
      lemma  = cond₁ A t tC p pA
      lemma  = qA
  c-IF : continuous ₂Space (A  A  A) IF
  c-IF = Lemma[discrete-₂Space] (A  A  A) IF

\end{code}

The natural numbers object:

\begin{code}

ℕSpace : Space
ℕSpace = DiscreteSpace  ℕ-discrete

Lemma[discrete-ℕSpace] : (X : Space)  ∀(f :   U X)  continuous ℕSpace X f
Lemma[discrete-ℕSpace] X f = Lemma[discreteness]  ℕ-discrete X f

continuous-succ : Map ℕSpace ℕSpace
continuous-succ = succ , Lemma[discrete-ℕSpace] ℕSpace succ

continuous-rec : (A : Space)  Map A ((ℕSpace  A  A)  ℕSpace  A)
continuous-rec A = r , continuity-of-rec
 where
  ū : U(ℕSpace  A  A)    U A  U A
  ū (f , _) n x = pr₁ (f n) x
  r : U A  U((ℕSpace  A  A)  ℕSpace  A)
  r a = (g , cg)
   where
    g : U(ℕSpace  A  A)  U(ℕSpace  A)
    g f = rec a (ū f) , Lemma[discrete-ℕSpace] A (rec a (ū f))
    cg : continuous (ℕSpace  A  A) (ℕSpace  A) g
    cg p pNAA q qN t uct = cond₂ A  α  rec a (ū(p(t α))) (q α)) (n , prf)
     where
      n : 
      n = pr₁ qN
      prf : ∀(s : ₂Fin n)   α  rec a (ū(p(t(cons s α)))) (q(cons s α)))  Probe A
      prf s = cond₃ A  α  rec a (ū(p(t(cons s α)))) (q(cons s )))
                       α  rec a (ū(p(t(cons s α)))) (q(cons s α)))
                      claim₀ claim₁
       where
        ucts : uniformly-continuous-₂ℕ (t  (cons s))
        ucts = Lemma[∘-UC] t uct (cons s) (Lemma[cons-UC] s)
        lemma : ∀(k : )   α  rec a (ū(p(t(cons s α)))) k)  Probe A
        lemma 0        = cond₁ A (t  (cons s)) ucts  _  a) (cond₀ A a)
        lemma (succ k) = claim  α  rec a (ū(p(t(cons s α)))) k) (lemma k) id Lemma[id-UC]
         where
          claim :  α  pr₁ (p (t(cons s α))) k)  Probe (A  A)
          claim = pNAA  _  k) (0 ,  _ _ _  refl) ,  _ _  ≤-zero)) (t  (cons s)) ucts
        claim₀ :  α  rec a (ū(p(t(cons s α)))) (q(cons s )))  Probe A
        claim₀ = lemma (q(cons s ))
        eq : ∀(α : ₂ℕ)  q (cons s )  q (cons s α)
        eq α = pr₁ (pr₂ qN) (cons s ) (cons s α) (Lemma[cons-≡[]] s  α)
        claim₁ : ∀(α : ₂ℕ)  [ rec a (ū(p(t(cons s α)))) (q(cons s ))
                              rec a (ū(p(t(cons s α)))) (q(cons s α)) ]
        claim₁ α = hide (ap (rec a (ū(p(t(cons s α))))) (eq α))

  continuity-of-rec : continuous A ((ℕSpace  A  A)  ℕSpace  A) r
  continuity-of-rec p pA q qNAA t uct u uN v ucv =
                    cond₂ A  α  rec (p(t(v α))) (ū(q(v α))) (u α)) (n , prf)
   where
    n : 
    n = pr₁ uN
    prf : ∀(s : ₂Fin n)
          α  rec (p(t(v(cons s α)))) (ū(q(v(cons s α)))) (u(cons s α)))  Probe A
    prf s = cond₃ A  α  rec (p(t(v(cons s α)))) (ū(q(v(cons s α)))) (u(cons s )))
                     α  rec (p(t(v(cons s α)))) (ū(q(v(cons s α)))) (u(cons s α)))
                    claim₀ claim₁
     where
      ucvs : uniformly-continuous-₂ℕ (v  (cons s))
      ucvs = Lemma[∘-UC] v ucv (cons s) (Lemma[cons-UC] s)
      uctvs : uniformly-continuous-₂ℕ (t  v  (cons s))
      uctvs = Lemma[∘-UC] t uct (v  (cons s)) ucvs
      lemma : ∀(k : )   α  rec (p(t(v(cons s α)))) (ū(q(v(cons s α)))) k)  Probe A
      lemma 0        = cond₁ A (t  v  (cons s)) uctvs p pA
      lemma (succ k) = claim  α  rec (p(t(v(cons s α)))) (ū(q(v(cons s α)))) k)
                             (lemma k) id Lemma[id-UC]
       where
        claim :  α  pr₁ (q(v(cons s α))) k)  Probe (A  A)
        claim = qNAA  _  k) (0 ,  _ _ _  refl) ,  _ _  ≤-zero)) (v  (cons s)) ucvs
      claim₀ :  α  rec (p(t(v(cons s α)))) (ū(q(v(cons s α)))) (u(cons s )))  Probe A
      claim₀ = lemma (u(cons s ))
      eq : ∀(α : ₂ℕ)  u(cons s )  u(cons s α)
      eq α = pr₁ (pr₂ uN) (cons s ) (cons s α) (Lemma[cons-≡[]] s  α)
      claim₁ : ∀(α : ₂ℕ)  [ rec (p(t(v(cons s α)))) (ū(q(v(cons s α)))) (u(cons s ))
                            rec (p(t(v(cons s α)))) (ū(q(v(cons s α)))) (u(cons s α)) ]
      claim₁ α = hide (ap (rec (p(t(v(cons s α)))) (ū(q(v(cons s α))))) (eq α))

\end{code}

When X is an hset, local constancy of ₂ℕ → X is an hprop.

\begin{code}

Lemma[LC-[hprop]] : {X : Set}  hset X  ∀(p : ₂ℕ  X)  (lc₀ lc₁ : LC p)  [ lc₀  lc₁ ]
Lemma[LC-[hprop]] hsX p (n₀ , (prf₀ , min₀)) (n₁ , (prf₁ , min₁)) = [pair⁼] e ee
 where
  e : n₀  n₁
  e = Lemma[m≤n∧n≤m→m=n] (min₀ n₁ prf₁) (min₁ n₀ prf₀)
  prf₀'min₀' :  (∀(α β : ₂ℕ)  α ≡[ n₁ ] β  p α  p β)
              × (∀(n' : )  (∀(α β : ₂ℕ)  α ≡[ n' ] β  p α  p β)  n₁  n')
  prf₀'min₀' = transport _ e (prf₀ , min₀)
  prf₀' : ∀(α β : ₂ℕ)  α ≡[ n₁ ] β  p α  p β
  prf₀' = pr₁ prf₀'min₀'
  eeprf : ∀(α β : ₂ℕ)  (en : α ≡[ n₁ ] β)  prf₀' α β en  prf₁ α β en
  eeprf α β en = hsX (prf₀' α β en) (prf₁ α β en)
  epr : [ prf₀'  prf₁ ]
  epr = funext³ eeprf
        -------
  min₀' : ∀(n' : )  (∀(α β : ₂ℕ)  α ≡[ n' ] β  p α  p β)  n₁  n'
  min₀' = pr₂ prf₀'min₀'
  eemin : ∀(n' : )  (prf' : ∀(α β : ₂ℕ)  α ≡[ n' ] β  p α  p β)  min₀' n' prf'  min₁ n' prf'
  eemin n' prf' = Lemma[≤-hprop] (min₀' n' prf') (min₁ n' prf')
  emin : [ min₀'  min₁ ]
  emin = funext² eemin
         -------
  ee : [ (prf₀' , min₀')  (prf₁ , min₁) ]
  ee = [pairˣ⁼] epr emin


Lemma[Map-discrete] : (X : Space)(Y : Set)(d : discrete Y)(h : hset Y) 
                      (f g : Map X (DiscreteSpace Y d)) 
                      (∀(x : U X)  pr₁ f x  pr₁ g x)  [ f  g ]
Lemma[Map-discrete] X Y d h (f , cf) (g , cg) ex = extension claim [e₀]
 where
  [e₀] : [ f  g ]
  [e₀] = funext¹ ex
         -------
  claim : f  g  [ (f , cf)  (g , cg) ]
  claim e₀ = [pair⁼] e₀ [e₁]
   where
    W : ((p : ₂ℕ  U X)  p  Probe X  )  Set
    W φ = (∀(p : ₂ℕ  U X) (pX : p  Probe X) 
           ∀(α β : ₂ℕ)  α ≡[ φ p pX ] β  g(p α)  g(p β))
        × (∀(p : ₂ℕ  U X)  ∀(pX : p  Probe X) 
           ∀(m : )  (∀(α β : ₂ℕ)  α ≡[ m ] β  g(p α)  g(p β))  φ p pX  m)

    cf' : ∀(p : ₂ℕ  U X)  p  Probe X 
          Σ \(n : ) 
            (∀(α β : ₂ℕ)  α ≡[ n ] β  g(p α)  g(p β))
          × (∀(m : )  (∀(α β : ₂ℕ)  α ≡[ m ] β  g(p α)  g(p β))  n  m)
    cf' = transport _ e₀ cf
    φf : (p : ₂ℕ  U X)  p  Probe X  
    φf p pX = pr₁ (cf' p pX)
    Φf₀ : ∀(p : ₂ℕ  U X) (pX : p  Probe X)
         ∀(α β : ₂ℕ)  α ≡[ φf p pX ] β  g(p α)  g(p β)
    Φf₀ p pX = pr₁ (pr₂ (cf' p pX))
    Φf₁ : ∀(p : ₂ℕ  U X) (pX : p  Probe X)
          ∀(m : )  (∀(α β : ₂ℕ)  α ≡[ m ] β  g(p α)  g(p β))  φf p pX  m
    Φf₁ p pX = pr₂ (pr₂ (cf' p pX))
    CF : Σ \(φ : (p : ₂ℕ  U X)  p  Probe X  )  W φ
    CF = (φf , Φf₀ , Φf₁)

    φg : (p : ₂ℕ  U X)  p  Probe X  
    φg p pX = pr₁ (cg p pX)
    Φg₀ : ∀(p : ₂ℕ  U X) (pX : p  Probe X)
         ∀(α β : ₂ℕ)  α ≡[ φg p pX ] β  g(p α)  g(p β)
    Φg₀ p pX = pr₁ (pr₂ (cg p pX))
    Φg₁ : ∀(p : ₂ℕ  U X) (pX : p  Probe X)
         ∀(m : )  (∀(α β : ₂ℕ)  α ≡[ m ] β  g(p α)  g(p β))  φg p pX  m
    Φg₁ p pX = pr₂ (pr₂ (cg p pX))
    CG : Σ \(φ : (p : ₂ℕ  U X)  p  Probe X  )  W φ
    CG = (φg , Φg₀ , Φg₁)

    [eφ] : [ φf  φg ]
    [eφ] = funext² epx
           -------
     where
      epx : (p : ₂ℕ  U X)  (pX : p  Probe X)  φf p pX  φg p pX
      epx p pX = Lemma[m≤n∧n≤m→m=n] claim₀ claim₁
       where
        claim₀ : φf p pX  φg p pX
        claim₀ = Φf₁ p pX (φg p pX) (Φg₀ p pX)
        claim₁ : φg p pX  φf p pX
        claim₁ = Φg₁ p pX (φf p pX) (Φf₀ p pX)

    sclaim : φf  φg  [ CF  CG ]
    sclaim  = [pair⁼]  [eΦ]
     where
      Φf' : W φg
      Φf' = transport W  (Φf₀ , Φf₁)
      Φf₀' : ∀(p : ₂ℕ  U X) (pX : p  Probe X)
            ∀(α β : ₂ℕ)  α ≡[ φg p pX ] β  g(p α)  g(p β)
      Φf₀' = pr₁ Φf'
      Φf₁' : ∀(p : ₂ℕ  U X) (pX : p  Probe X)
            ∀(m : )  (∀(α β : ₂ℕ)  α ≡[ m ] β  g(p α)  g(p β))  φg p pX  m
      Φf₁' = pr₂ Φf'
      [eΦ]₀ : [ Φf₀'  Φg₀ ]
      [eΦ]₀ = funext⁵ fact
              -------
       where
        fact : (p : ₂ℕ  U X)  (pX : p  Probe X)  ∀(α β : ₂ℕ)  (en : α ≡[ φg p pX ] β) 
               Φf₀' p pX α β en  Φg₀ p pX α β en
        fact p pX α β en = h (Φf₀' p pX α β en) (Φg₀ p pX α β en)
      [eΦ]₁ : [ Φf₁'  Φg₁ ]
      [eΦ]₁ = funext⁴ fact
              -------
       where
        fact : (p : ₂ℕ  U X)  (pX : p  Probe X)  (m : ) 
               (pr : ∀(α β : ₂ℕ)  α ≡[ m ] β  g(p α)  g(p β)) 
               Φf₁' p pX m pr  Φg₁ p pX m pr
        fact p pX m pr = Lemma[≤-hprop] (Φf₁' p pX m pr) (Φg₁ p pX m pr)
      [eΦ] : [ (Φf₀' , Φf₁')  (Φg₀ , Φg₁) ]
      [eΦ] = [pairˣ⁼] [eΦ]₀ [eΦ]₁

    [E] : [ CF  CG ]
    [E] = extension sclaim [eφ]

    [e₁] : [ cf'  cg ]
    [e₁] = functor (ap  w p pX  (pr₁ w p pX , pr₁(pr₂ w) p pX , pr₂(pr₂ w) p pX))) [E]

Lemma[Map-₂-[≡]] : (X : Space)  (f g : Map X ₂Space) 
                   (∀(x : U X)  pr₁ f x  pr₁ g x)  [ f  g ]
Lemma[Map-₂-[≡]] X = Lemma[Map-discrete] X  ₂-discrete ₂-hset

Lemma[Map-ℕ-[≡]] : (X : Space)  (f g : Map X ℕSpace) 
                   (∀(x : U X)  pr₁ f x  pr₁ g x)  [ f  g ]
Lemma[Map-ℕ-[≡]] X = Lemma[Map-discrete] X  ℕ-discrete ℕ-hset

\end{code}

Assuming funext, local constancy is an hproposition.

\begin{code}

Lemma[funext→LC-hprop] : Funext  {X : Set}  hset X  ∀(p : ₂ℕ  X)  hprop (p  LC)
Lemma[funext→LC-hprop] funext hsX p (n₀ , (prf₀ , min₀)) (n₁ , (prf₁ , min₁)) = pair⁼ e ee
 where
  e : n₀  n₁
  e = Lemma[m≤n∧n≤m→m=n] (min₀ n₁ prf₁) (min₁ n₀ prf₀)
  prf₀'min₀' :  (∀(α β : ₂ℕ)  α ≡[ n₁ ] β  p α  p β)
             × (∀(n' : )  (∀(α β : ₂ℕ)  α ≡[ n' ] β  p α  p β)  n₁  n')
  prf₀'min₀' = transport _ e (prf₀ , min₀)
  prf₀' : ∀(α β : ₂ℕ)  α ≡[ n₁ ] β  p α  p β
  prf₀' α β en = pr₁ prf₀'min₀' α β en
  eepr : ∀(α β : ₂ℕ)  (en : α ≡[ n₁ ] β)  prf₀' α β en  prf₁ α β en
  eepr α β en = hsX (prf₀' α β en) (prf₁ α β en)
  epr : prf₀'  prf₁
  epr = funext  α  funext  β  funext  en  eepr α β en)))
       --------
  min₀' : ∀(n' : )  (∀(α β : ₂ℕ)  α ≡[ n' ] β  p α  p β)  n₁  n'
  min₀' = pr₂ prf₀'min₀'
  eemin : ∀(n' : )  (pr' : ∀(α β : ₂ℕ)  α ≡[ n' ] β  p α  p β)  min₀' n' pr'  min₁ n' pr'
  eemin n' pr' = Lemma[≤-hprop] (min₀' n' pr') (min₁ n' pr')
  emin : min₀'  min₁
  emin = funext  n'  funext  pr'  eemin n' pr'))
        --------
  ee : (prf₀' , min₀')  (prf₁ , min₁)
  ee = pairˣ⁼ epr emin

\end{code}