J-AC-N

Martin Escardo and Paulo Oliva 2011


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

module InfinitePigeon.J-AC-N where

open import InfinitePigeon.Addition
open import InfinitePigeon.Choice
open import InfinitePigeon.Equality
open import InfinitePigeon.J-Shift
open import InfinitePigeon.JK-LogicalFacts
open import InfinitePigeon.JK-Monads
open import InfinitePigeon.Logic
open import InfinitePigeon.LogicalFacts
open import InfinitePigeon.Naturals


------------------------------------------------------------------------
-- Proof of the J translation of the axiom of countable choice        --
-- from the proof of the axiom of choice using J-∀-shift as a lemma.  --
------------------------------------------------------------------------


J-AC-ℕ : {R : Ω}
         {X : ℕ → Set}
         {P : (n : ℕ) → X n → Ω}
       → (∀(n : ℕ) → J∃ \(x : X n) → P n x)
       → J∃ \(f : ((n : ℕ) → X n)) → ∀(n : ℕ) → P n (f n)
J-AC-ℕ {R} = (J-functor {R} AC) ∘ J-∀-shift


----------------------------------------------
--                                          --
-- Now the translations of dependent choice --
--                                          --
----------------------------------------------


J-∀-double-shift : {R : Ω}
                   {P : ℕ → ℕ → ℕ → Ω}
                 → (∀(n : ℕ) → ∀(x : ℕ) → J∃ \(y : ℕ) → P n x y)
                 → J(∀(n : ℕ) → ∀(x : ℕ) → ∃ \(y : ℕ) → P n x y)
J-∀-double-shift {R} f = J-∀-shift {R} (λ n → J-∀-shift (f n))

J-DC-ℕ : {R : Ω}
         {P : ℕ → ℕ → ℕ → Ω}
         (x₀ : ℕ)
       → (∀(n : ℕ) → ∀(x : ℕ) → J∃ \(y : ℕ) → P n x y)
       → J∃ \(α : ℕ → ℕ) → α O ≡ x₀ ∧ (∀(n : ℕ) → P n (α n) (α(n + 1)))
J-DC-ℕ {R} x₀ f = (J-functor {R} (DC x₀)) (J-∀-double-shift f)