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C-Spaces.Coverage

Chuangjie Xu 2012, ported to TypeTopology in 2025


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

module C-Spaces.Coverage where

open import MLTT.Spartan renaming (_+_ to _⊎_)
open import MLTT.Plus-Properties
open import MLTT.Two-Properties
open import Naturals.Addition
open import Naturals.Properties

open import C-Spaces.Preliminaries.Naturals.Order
open import C-Spaces.Preliminaries.Sequence
open import C-Spaces.UniformContinuity


The site we are working with is the monoid of uniformly continuous
endo-functions of the Cantor space with a coverage in which, for each
natural number n, there is a family of concatenation maps "cons s"
indexed by finite binary sequence s of length n.

The monoid of uniformly continuous ₂ℕ → ₂ℕ:


C : (₂ℕ → ₂ℕ) → Set
C = uniformly-continuous-₂ℕ

infixl 4 _∈_

_∈_ : {X : Set} → X → (X → Set) → Set
x ∈ A = A x


The coverage axiom amounts to uniform continuity of endo-functions of
the Cantor space in the following sense.


Theorem[Coverage-axiom] : ∀(m : ℕ) → ∀(t : ₂ℕ → ₂ℕ) → t ∈ C →
                   Σ \(n : ℕ) → ∀(s : ₂Fin n) →
                    Σ \(s' : ₂Fin m) → Σ \(t' : ₂ℕ → ₂ℕ) →
                     (t' ∈ C) × (∀(α : ₂ℕ) → t (cons s α) ∼ cons s' (t' α))
Theorem[Coverage-axiom] m t tC = n , prf
 where
  n : ℕ
  n = pr₁ (tC m)
  prf : ∀(s : ₂Fin n) → Σ \(s' : ₂Fin m) → Σ \(t' : ₂ℕ → ₂ℕ) →
         (t' ∈ C) × (∀(α : ₂ℕ) → t (cons s α) ∼ cons s' (t' α))
  prf s = s' ,  t' , t'C , ex
   where
    s' : ₂Fin m
    s' = take m (t (cons s 0̄))

    t' : ₂ℕ → ₂ℕ
    t' α = drop m (t (cons s α))

    t'C : t' ∈ C
    t'C k = Lemma[LM-₂ℕ-least-modulus] t' l prt'
     where
      ucts : uniformly-continuous-₂ℕ (t ∘ (cons s))
      ucts = Lemma[∘-UC] t tC (cons s) (Lemma[cons-UC] s)
      l : ℕ
      l = pr₁ (ucts (k + m))
      prts : ∀(α β : ₂ℕ) → α ＝⟦ l ⟧ β → t (cons s α) ＝⟦ k + m ⟧ t (cons s β)
      prts = pr₁ (pr₂ (ucts (k + m)))
      eq : ∀(α : ₂ℕ) → ∀(i : ℕ) → t' α i ＝ t (cons s α) (i + m)
      eq α i = Lemma[drop+] m (t (cons s α)) i
      claim₀ : ∀(α β : ₂ℕ) → α ＝⟦ l ⟧ β → t (cons s α) ＝⟦ k + m ⟧ t (cons s β) →
                ∀(i : ℕ) → i < k → t' α i ＝ t' β i
      claim₀ α β el ekm i i<k = sclaim₂ ∙ (eq β i)⁻¹
       where
        sclaim₀ : ∀(i : ℕ) → i < (k + m) → t (cons s α) i ＝ t (cons s β) i
        sclaim₀ = Lemma[＝⟦⟧-<] ekm
        sclaim₁ : t (cons s α) (i + m) ＝ t (cons s β) (i + m)
        sclaim₁ = sclaim₀ (i + m) (Lemma[a<b→a+c<b+c] i k m i<k)
        sclaim₂ : t' α i ＝ t (cons s β) (i + m)
        sclaim₂ = eq α i ∙ sclaim₁ 
      claim₁ : ∀(α β : ₂ℕ) → α ＝⟦ l ⟧ β
             → t (cons s α) ＝⟦ k + m ⟧ t (cons s β) → t' α ＝⟦ k ⟧ t' β
      claim₁ α β el ekm = Lemma[<-＝⟦⟧] (claim₀ α β el ekm)
      prt' : ∀(α β : ₂ℕ) → α ＝⟦ l ⟧ β → t' α ＝⟦ k ⟧ t' β
      prt' α β el = claim₁ α β el (prts α β el)
    ex : ∀(α : ₂ℕ) → t (cons s α) ∼ cons s' (t' α)
    ex α i = sclaim₀ ∙ sclaim₃
     where
      sclaim₀ : t (cons s α) i ＝ cons (take m (t (cons s α))) (t' α) i
      sclaim₀ = (Lemma[cons-take-drop] m (t (cons s α)) i)⁻¹
      sclaim₁ : t (cons s α) ＝⟦ m ⟧ t (cons s 0̄)
      sclaim₁ = pr₁ (pr₂ (tC m)) (cons s α) (cons s 0̄) (Lemma[cons-＝⟦⟧] s α 0̄)
      sclaim₂ : take m (t (cons s α)) ＝ s'
      sclaim₂ = Lemma[＝⟦⟧-take] sclaim₁
      sclaim₃ : cons (take m (t (cons s α))) (t' α) i ＝ cons s' (t' α) i
      sclaim₃ = ap (λ x → cons x (t' α) i) sclaim₂


A special case of Theorem[Coverage-axiom]:


Theorem[Coverage-axiom]₁ : ∀(t : ₂ℕ → ₂ℕ) → t ∈ C →
                   Σ \(n : ℕ) → ∀(s : ₂Fin n) →
                    Σ \(i : 𝟚) → Σ \(t' : ₂ℕ → ₂ℕ) →
                     (t' ∈ C) × (∀(α : ₂ℕ) → t (cons s α) ∼ cons (i ∷ ⟨⟩) (t' α))
Theorem[Coverage-axiom]₁ t tC = n , prf
 where
  n : ℕ
  n = pr₁ (Theorem[Coverage-axiom] 1 t tC)
  prf : ∀(s : ₂Fin n) → Σ \(i : 𝟚) → Σ \(t' : ₂ℕ → ₂ℕ) →
         (t' ∈ C) × (∀(α : ₂ℕ) → t (cons s α) ∼ cons (i ∷ ⟨⟩) (t' α))
  prf s = i , pr₂ (pr₂ (Theorem[Coverage-axiom] 1 t tC) s)
   where
    to₂ : ₂Fin 1 → 𝟚
    to₂ (i ∷ ⟨⟩) = i
    i : 𝟚
    i = to₂ (pr₁ (pr₂ (Theorem[Coverage-axiom] 1 t tC) s))