    ------------------------------------------------------------------------
    -- The Agda standard library
    --
    -- M-types (the dual of W-types)
    ------------------------------------------------------------------------

    {-# OPTIONS --cubical-compatible --sized-types #-}

    module Codata.Sized.M where

    open import Size using (Size; ∞)
    open import Level using (Level; _⊔_)
    open import Codata.Sized.Thunk using (Thunk; force)
    open import Data.Product.Base hiding (map)
    open import Data.Container.Core as C hiding (map)

    data M {s p} (C : Container s p) (i : Size) : Set (s ⊔ p) where
      inf : ⟦ C ⟧ (Thunk (M C) i) → M C i

    module _ {s p} {C : Container s p} where

      head : ∀ {i} → M C i → Shape C
      head (inf (x , f)) = x

      tail : (x : M C ∞) → Position C (head x) → M C ∞
      tail (inf (x , f)) = λ p → f p .force

    -- map

    module _ {s₁ s₂ p₁ p₂} {C₁ : Container s₁ p₁} {C₂ : Container s₂ p₂}
             (m : C₁ ⇒ C₂) where

      map : ∀ {i} → M C₁ i → M C₂ i
      map (inf t) = inf (⟪ m ⟫ (C.map (λ t → λ where .force → map (t .force)) t))

    -- unfold

    module _ {s p ℓ} {C : Container s p} (open Container C)
             {S : Set ℓ} (alg : S → ⟦ C ⟧ S) where

      unfold : S → ∀ {i} → M C i
      unfold seed = let (x , next) = alg seed in
                    inf (x , λ p → λ where .force → unfold (next p))