Martin Escardo and Paulo Oliva 2011

\begin{code}

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

module InfinitePigeon.Logic where

\end{code}

Type of propositions denoted by Ω like in a topos.

\begin{code}

Ω = Set

data ⊥ : Ω where

⊥-elim : {A : Ω} → ⊥ → A
⊥-elim = λ ()

¬_ : Ω → Ω
¬ A = (A → ⊥)

infix   ¬_

data ⊤ : Ω where
 * : ⊤

data _∧_ (A₀ A₁ : Ω) : Ω where
 ∧-intro : A₀ → A₁ → A₀ ∧ A₁

infixr  _∧_

∧-elim₀ : {A₀ A₁ : Ω} → A₀ ∧ A₁ → A₀
∧-elim₀ (∧-intro a₀ a₁) = a₀

∧-elim₁ : {A₀ A₁ : Ω} → A₀ ∧ A₁ → A₁
∧-elim₁ (∧-intro a₀ a₁) = a₁

_↔_ : Ω → Ω → Ω
A ↔ B = (A → B) ∧ (B → A)

data _∨_ (A₀ A₁ : Ω) : Ω where
 ∨-intro₀ : A₀ → A₀ ∨ A₁
 ∨-intro₁ : A₁ → A₀ ∨ A₁

infixr  _∨_

∨-elim : {A₀ A₁ B : Ω} → (A₀ → B) → (A₁ → B) → A₀ ∨ A₁ → B
∨-elim f₀ f₁ (∨-intro₀ a₀) = f₀ a₀
∨-elim f₀ f₁ (∨-intro₁ a₁) = f₁ a₁

dependent-∨-elim : {A₀ A₁ : Ω}
                   {B : A₀ ∨ A₁ → Ω}
                 → ((a₀ : A₀) → B (∨-intro₀ a₀))
                 → ((a₁ : A₁) → B(∨-intro₁ a₁))
                 → (a : A₀ ∨ A₁)
                 → B a
dependent-∨-elim f₀ f₁ (∨-intro₀ a₀) = f₀ a₀
dependent-∨-elim f₀ f₁ (∨-intro₁ a₁) = f₁ a₁

decidable : Ω → Ω
decidable A = A ∨ ¬ A

data ∃ {X : Set} (A : X → Ω) : Ω where
 ∃-intro : (x₀ : X) → A x₀ → ∃ \(x : X) → A x

∃-witness : {X : Set} {A : X → Ω} → (∃ \(x : X) → A x) → X
∃-witness (∃-intro x a) = x

∃-elim : {X : Set}
         {A : X → Ω}
       → (proof : ∃ \(x : X) → A x) → A (∃-witness proof)
∃-elim (∃-intro x a) = a

inhabited : Set → Ω
inhabited X = ∃ \(x : X) → ⊤

\end{code}