Martin Escardo, 7 May 2014.

For any function f : ℕ∞ → ℕ, it is decidable whether f is non-continuous.

Π (f : ℕ∞ → ℕ). ¬ (continuous f) + ¬¬ (continuous f).

Based on the paper

Mathematical Structures in Computer Science , Volume 25,
October 2015 , pp. 1578 - 1589
DOI: https://doi.org/10.1017/S096012951300042X

The title of this paper is a bit misleading. It should have been
called "Decidability of non-continuity".

\begin{code}

{-# OPTIONS --without-K --exact-split --safe --auto-inline #-}

open import SpartanMLTT
open import UF-FunExt

module DecidabilityOfNonContinuity (fe : funext 𝓤₀ 𝓤₀) where

open import Two-Properties
open import DiscreteAndSeparated
open import GenericConvergentSequence
open import DecidableAndDetachable
open import CanonicalMapNotation

Lemma-3·1 : (q : ℕ∞ → ℕ∞ → 𝟚)
→ decidable ((m : ℕ) → ¬ ((n : ℕ) → q (ι m) (ι n) ≡ ₁))
Lemma-3·1 q = claim₄
where
A : ℕ∞ → 𝓤₀ ̇
A u = (n : ℕ) → q u (ι n) ≡ ₁

claim₀ :  (u : ℕ∞) → decidable (A u)
claim₀ u = Theorem-8·2 (q u)

p : ℕ∞ → 𝟚
p = pr₁ (indicator claim₀)

p-spec : (x : ℕ∞) → (p x ≡ ₀ → A x) × (p x ≡ ₁ → ¬ A x)
p-spec = pr₂ (indicator claim₀)

claim₁ : decidable ((n : ℕ) → p (ι n) ≡ ₁)
claim₁ = Theorem-8·2 p

claim₂ : ((n : ℕ) → ¬ A (ι n)) → (n : ℕ) → p (ι n) ≡ ₁
claim₂ φ n = different-from-₀-equal-₁ (λ v → φ n (pr₁ (p-spec (ι n)) v))

claim₃ : decidable ((n : ℕ) → p (ι n) ≡ ₁) → decidable ((n : ℕ) → ¬ A (ι n))
claim₃ (inl f) = inl (λ n → pr₂ (p-spec (ι n)) (f n))
claim₃ (inr u) = inr (contrapositive claim₂ u)

claim₄ : decidable ((n : ℕ) → ¬ (A (ι n)))
claim₄ = claim₃ claim₁

\end{code}

Omitting the inclusion function, or coercion,

ι : ℕ → ℕ∞,

a map f : ℕ∞ → ℕ is called continuous iff

∃ m. ∀ n ≥ m. f n ≡ ∞,

where m and n range over the natural numbers.

The negation of this statement is equivalent to

∀ m. ¬ ∀ n ≥ m. f n ≡ ∞.

We can implement ∀ y ≥ x. A y as ∀ x. A (max x y), so that the
continuity of f amounts to

∃ m. ∀ n. f (max m n) ≡ ∞,

and its negation to

∀ m. ¬ ∀ n. f (max m n) ≡ ∞.

\begin{code}

non-continuous : (ℕ∞ → ℕ) → 𝓤₀ ̇
non-continuous f = (m : ℕ) → ¬ ((n : ℕ) → f (max (ι m) (ι n)) ≡[ℕ] f ∞)

Theorem-3·2 : (f : ℕ∞ → ℕ) → decidable (non-continuous f)
Theorem-3·2 f = Lemma-3·1 ((λ x y → χ≡ (f (max x y)) (f ∞)))

\end{code}

(Maybe) to be continued (see the paper for the moment).

* MP gives that continuity and doubly negated continuity agree.

* WLPO is equivalent to the existence of a non-continuous function ℕ∞ → ℕ.

* ¬WLPO is equivalent to the doubly negated continuity of all functions ℕ∞ → ℕ.

* If MP and ¬WLPO then all functions ℕ∞ → ℕ are continuous.

For future use:

\begin{code}

continuous : (ℕ∞ → ℕ) → 𝓤₀ ̇
continuous f = Σ m ꞉ ℕ , ((n : ℕ) → f (max (ι m) (ι n)) ≡ f ∞)

\end{code}