Martin Escardo and Paulo Oliva 2011

\begin{code}

{-# OPTIONS --without-K --no-termination-check #-}

\end{code}

Termination is proved externally, using bar induction and
continuity in the case of this module (Escardo-Oliva 2010).

\begin{code}

module InfinitePigeon.J-Shift-Selection where

open import InfinitePigeon.Finite-JK-Shifts
open import InfinitePigeon.JK-Monads
open import InfinitePigeon.Logic
open import InfinitePigeon.Naturals

\end{code}

Definition given by Escardo and Oliva MSCS2010
(infinite iteration of the J-∧-shift'):

\begin{code}

J-∀-shift-selection : {R : Ω}
                      {A : ℕ → Ω}
                    → (∀(n : ℕ) → J {R} (A n))
                    → J {R} (∀(n : ℕ) → A n)
J-∀-shift-selection εs =
 J-∧-shift'(∧-intro (head εs) (J-∀-shift-selection(tail εs)))

\end{code}

That's it. What follows is for information only.

This is one of definitions given in both LICS2007,
https://doi.org/10.1109/LICS.2007.25, and LMCS2008,
https://doi.org/10.2168/LMCS-4(3:3)2008 (Section 8.1, called Π):

\begin{code}

prod : {R : Ω}
       {A : ℕ → Ω}
     → (∀ (n : ℕ) → J {R} (A n))
     → J {R} (∀ (n : ℕ) → A n)
prod {R} {A} εs p = cons (∧-intro a₀ ((prod {R} (tail εs)) (prefix a₀ p)))
 where
  a₀ : A O
  a₀ = head εs (λ a → prefix a p ((prod {R} (tail εs))(prefix a p)))

\end{code}

It is equivalent after we expand the definitions (both satisfy the
same recursive equation in normal form).