By Martin Escardo, 2 September 2011.

Modified in December 2011 assuming the axiom of extensionality (which
is not used directly in this module, but instead in
GenericConvergentSequence).

We prove that the generic convergent sequence ℕ∞ is searchable, which
amounts to Theorem-3·6 of the paper

   http://www.cs.bham.ac.uk/~mhe/papers/omniscient.pdf, 
   http://www.cs.bham.ac.uk/~mhe/.talks/dagstuhl2011/omniscient.pdf

and conclude as a corollary that it is searchable and satisfies the
principle of omniscience.

\begin{code}

module ConvergentSequenceSearchable where

\end{code}


(Continuity axioms and the fan principle are not assumed.)

\begin{code}

open import CurryHoward
open import Equality
open import Naturals
open import Two 
open import Cantor
open import GenericConvergentSequence
open import SetsAndFunctions
open import Searchable

\end{code}

This is the main theorem proved in this module:

\begin{code}

ℕ∞-is-searchable : searchable ℕ∞
ℕ∞-is-searchable p = ∃-intro a Lemma
 where 
  α : ₂ℕ 
  α        = p(under )
  α(succ n) = min₂ (α n) (p(under(succ n)))

  a : ℕ∞
  a = (α , λ i → Lemma[minab≤a])

  Dagger₀ : ∀(n : ℕ) → a ≡ under n → p(under n) ≡ ₀
  Dagger₀  r =  cong (λ w → incl w ) r
  Dagger₀ (succ n) r = trans w t
   where 
    s : α n ≡ ₁
    s = trans (cong (λ w → incl w n) r) (under-diagonal₁ n)

    t : α(succ n) ≡ ₀
    t = trans (cong (λ w → incl w(succ n)) r) (under-diagonal₀ n)

    w : p(under(succ n)) ≡ α(succ n)
    w = sym(cong(λ b → min₂ b (p(under(succ n)))) s)

  Dagger₁ : a ≡ ∞ → ∀(n : ℕ) → p(under n) ≡ ₁
  Dagger₁ r  = cong (λ w → incl w ) r
  Dagger₁ r (succ n) = trans w t
   where 
    s : α n ≡ ₁
    s = cong (λ w → incl w n) r

    t : α(succ n) ≡ ₁
    t = cong (λ w → incl w (succ n)) r

    w : p(under(succ n)) ≡ α(succ n)
    w = sym(cong(λ b → min₂ b (p(under(succ n)))) s)

  Claim₀ : p a ≡ ₁ → ∀(n : ℕ) → a ≠ under n
  Claim₀ r n s = Lemma[b≡₁→b≠₀] r (Lemma s)
   where 
    Lemma : a ≡ under n → p a ≡ ₀
    Lemma t = trans (cong p t) (Dagger₀ n t)

  Claim₁ : p a ≡ ₁ → a ≡ ∞
  Claim₁ r = not-ℕ-is-∞ (Claim₀ r) 

  Claim₂ : p a ≡ ₁ → ∀(n : ℕ) → p(under n) ≡ ₁
  Claim₂ r = Dagger₁(Claim₁ r)

  Claim₃ : p a ≡ ₁ → p ∞ ≡ ₁
  Claim₃ r = Lemma[x≡y→x≡z→z≡y] r (cong p (Claim₁ r))

  Lemma : p a ≡ ₁ → ∀(v : ℕ∞) → p v ≡ ₁
  Lemma r = ℕ∞-density (Claim₂ r) (Claim₃ r)

\end{code}

Corollary:

\begin{code}

open import Omniscience

ℕ∞-is-omniscient : omniscient ℕ∞
ℕ∞-is-omniscient = searchable-implies-omniscient ℕ∞-is-searchable 

\end{code}