Martin Escardo and Paulo Oliva 2011

\begin{code}

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

module InfinitePigeon.Finite-JK-Shifts where

open import InfinitePigeon.Equality
open import InfinitePigeon.JK-Monads
open import InfinitePigeon.Logic
open import InfinitePigeon.LogicalFacts

J-∧-shift : {R A₀ A₁ : Ω} → J A₀ ∧ J A₁ → J(A₀ ∧ A₁)
J-∧-shift {R} (∧-intro ε₀ ε₁) =
 J-extend {R} (λ a₀ → J-strength (∧-intro a₀ ε₁))(ε₀)

\end{code}

This was defined in the following alternative way in Escardo's
LICS2007 and LMCS2008 papers:

\begin{code}

J-∧-shift-lics : {R A₀ A₁ : Ω}
               → J A₀ ∧ J A₁ → J(A₀ ∧ A₁)
J-∧-shift-lics {R} {A₀} {A₁} (∧-intro ε₀ ε₁) r = ∧-intro a₀ a₁
 where
  a₀ : A₀
  a₀ = ε₀(λ x₀ → J-K {R} ε₁ (λ x₁ → r (∧-intro x₀ x₁)))
  a₁ : A₁
  a₁ = ε₁ (λ x₁ → r (∧-intro a₀ x₁))

observation₁ : {R A₀ A₁ : Ω}
               (ε₀ : J {R} A₀)
               (ε₁ : J A₁)
               (r : (A₀ ∧ A₁ → R))
             → J-∧-shift (∧-intro ε₀ ε₁) r ≡ J-∧-shift-lics (∧-intro ε₀ ε₁) r
observation₁ ε₀ ε₁ r = reflexivity

K-∧-shift : {R A₀ A₁ : Ω}
          → K A₀ ∧ K A₁ → K(A₀ ∧ A₁)
K-∧-shift {R} (∧-intro φ₀ φ₁) =
 K-extend {R} (λ a₀ → K-strength (∧-intro a₀ φ₁))(φ₀)

observation₂ : {R A₀ A₁ : Ω}
               (φ₀ : K {R} A₀)
               (φ₁ : K A₁)
               (r : (A₀ ∧ A₁ → R))
             → K-∧-shift (∧-intro φ₀ φ₁) r
             ≡ φ₀(λ a₀ → φ₁(λ a₁ → r (∧-intro a₀ a₁)))
observation₂ φ₀ φ₁ r = reflexivity

observation₃ : {R A₀ A₁ : Ω}
               (ε₀ : J {R} A₀)
               (ε₁ : J A₁)
             → J-K(J-∧-shift (∧-intro ε₀ ε₁))
             ≡ K-∧-shift (∧-intro (J-K ε₀) (J-K ε₁))
observation₃ ε₀ ε₁ = reflexivity

\end{code}

Preliminary lemmas for the countable shifts (in another module).
The following definitions are the same with a parameter J/K.

\begin{code}

open import InfinitePigeon.Naturals

J-∧-shift' : {R : Ω}
             {A : ℕ → Ω}
           → J(A O) ∧ J(∀(n : ℕ) → A(succ n)) → J(∀(n : ℕ) → A n)
J-∧-shift' {R} = (J-functor {R} cons) ∘ J-∧-shift

K-∧-shift' : {R : Ω}
             {A : ℕ → Ω}
           → K(A O) ∧ K(∀(n : ℕ)
           → A(succ n)) → K(∀(n : ℕ) → A n)
K-∧-shift' {R} = (K-functor {R}  cons) ∘ K-∧-shift

\end{code}