------------------------------------------------------------------------
-- The Agda standard library
--
-- Coinductive "natural" numbers
------------------------------------------------------------------------

{-# OPTIONS --safe --cubical-compatible --guardedness #-}

module Codata.Musical.Conat where

open import Codata.Musical.Notation
open import Data.Nat.Base using (; zero; suc)
open import Function.Base using (_∋_)
open import Relation.Binary.Bundles using (Setoid)
open import Relation.Binary.Definitions
  using (Reflexive; Symmetric; Transitive)
open import Relation.Binary.PropositionalEquality.Core as P using (_≡_)

------------------------------------------------------------------------
-- Re-exporting the type and basic operations

open import Codata.Musical.Conat.Base public

------------------------------------------------------------------------
-- Some properties

module Coℕ-injective where

 suc-injective :  {m n}  (Coℕ  suc m)  suc n  m  n
 suc-injective P.refl = P.refl

fromℕ-injective :  {m n}  fromℕ m  fromℕ n  m  n
fromℕ-injective {zero}  {zero}  eq = P.refl
fromℕ-injective {suc m} {suc n} eq = P.cong suc (fromℕ-injective (P.cong pred eq))

------------------------------------------------------------------------
-- Equality

infix 4 _≈_

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))