Chuangjie Xu 2012 (updated in February 2015)

\begin{code}

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

module UsingSetoid.Space.Coverage where

open import Preliminaries.SetsAndFunctions hiding (_+_)
open import Preliminaries.NaturalNumber
open import Preliminaries.Sequence
open import Continuity.UniformContinuity
open import UsingSetoid.Setoid

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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:

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C : Subset E-map-₂ℕ-₂ℕ
C (t , _) = uniformly-continuous-₂ℕ t

Lemma[◎-UC] : ∀ t → t ∈ C → ∀ t' → t' ∈ C → (t ◎ t') ∈ C
Lemma[◎-UC] (t , _) tC (t' , _) t'C = Lemma[∘-UC] t tC t' t'C

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

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Theorem[Coverage-axiom] : ∀(m : ℕ) → ∀(t : E-map-₂ℕ-₂ℕ) → t ∈ C →
Σ \(n : ℕ) → ∀(s : ₂Fin n) →
Σ \(s' : ₂Fin m) → Σ \(t' : E-map-₂ℕ-₂ℕ) →
(t' ∈ C) × (∀ α → pr₁ t (cons s α) ≣ cons s' (pr₁ t' α))
Theorem[Coverage-axiom] m (t , et) tC = n , prf
where
n : ℕ
n = pr₁ (tC m)
prf : ∀(s : ₂Fin n) → Σ \(s' : ₂Fin m) → Σ \(t' : E-map-₂ℕ-₂ℕ) →
(t' ∈ C) × (∀ α → t (cons s α) ≣ cons s' (pr₁ t' α))
prf s = s' ,  (t' , et') , 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))
et' : ∀ α α' → α ≣ α' → t' α ≣ t' α'
et' α α' e = Lemma[≣-drop] m _ _ (et _ _ (cons-E-map s _ _ e))
t'C : (t' , et') ∈ 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) → ∀(i : ℕ) → t (cons s α) i ≡ cons s' (t' α) i
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

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