{- Alternative definition of M as a record type, together with some of its properties

-}

{-# OPTIONS --safe --guardedness #-}

module Cubical.Codata.M.MRecord where

open import Cubical.Foundations.Prelude

private
  variable
    ℓ ℓ' ℓ'' : Level

record M (S : Type ℓ) (P : S → Type ℓ') : Type (ℓ-max ℓ ℓ') where
  constructor sup-M
  coinductive
  field
    shape : S
    pos : P shape → M S P
open M

-- Lifting M to relations
record M-R {S : Type ℓ} {Q : S → Type ℓ'} (R : M S Q → M S Q → Type ℓ'')
           (m₀ m₁ : M S Q) : Type (ℓ-max ℓ (ℓ-max ℓ' ℓ'')) where
  field
   s-eq : shape m₀ ≡ shape m₁
   p-eq : (q₀ : Q (shape m₀)) (q₁ : Q (shape m₁))
          (q-eq : PathP (λ i → Q (s-eq i)) q₀ q₁) → R (pos m₀ q₀) (pos m₁ q₁)
open M-R

-- (Propositional) η-equality for M
ηEqM : {S' : Type ℓ} {Q' : S' → Type ℓ'} (m : M S' Q') → sup-M (shape m) (pos m) ≡ m
shape (ηEqM m i) = shape m
pos (ηEqM m i) = pos m