module Data.Conat where
open import Coinduction
open import Data.Nat.Base using (ℕ; zero; suc)
open import Function
open import Relation.Binary
open import Relation.Binary.PropositionalEquality as P using (_≡_)
data Coℕ : Set where
zero : Coℕ
suc : (n : ∞ Coℕ) → Coℕ
module Coℕ-injective where
suc-injective : ∀ {m n} → (Coℕ ∋ suc m) ≡ suc n → m ≡ n
suc-injective P.refl = P.refl
pred : Coℕ → Coℕ
pred zero = zero
pred (suc n) = ♭ n
fromℕ : ℕ → Coℕ
fromℕ zero = zero
fromℕ (suc n) = suc (♯ fromℕ n)
fromℕ-injective : ∀ {m n} → fromℕ m ≡ fromℕ n → m ≡ n
fromℕ-injective {zero} {zero} eq = P.refl
fromℕ-injective {zero} {suc n} ()
fromℕ-injective {suc m} {zero} ()
fromℕ-injective {suc m} {suc n} eq = P.cong suc (fromℕ-injective (P.cong pred eq))
∞ℕ : Coℕ
∞ℕ = suc (♯ ∞ℕ)
infixl 6 _+_
_+_ : Coℕ → Coℕ → Coℕ
zero + n = n
suc m + n = suc (♯ (♭ m + n))
data _≈_ : Coℕ → Coℕ → Set where
zero : zero ≈ zero
suc : ∀ {m n} (m≈n : ∞ (♭ m ≈ ♭ n)) → suc m ≈ suc n
module ≈-injective where
suc-injective : ∀ {m n p q} → (suc m ≈ suc n ∋ suc p) ≡ suc q → p ≡ q
suc-injective P.refl = P.refl
setoid : Setoid _ _
setoid = record
{ Carrier = Coℕ
; _≈_ = _≈_
; isEquivalence = record
{ refl = refl
; sym = sym
; trans = trans
}
}
where
refl : Reflexive _≈_
refl {zero} = zero
refl {suc n} = suc (♯ refl)
sym : Symmetric _≈_
sym zero = zero
sym (suc m≈n) = suc (♯ sym (♭ m≈n))
trans : Transitive _≈_
trans zero zero = zero
trans (suc m≈n) (suc n≈k) = suc (♯ trans (♭ m≈n) (♭ n≈k))