-- An alternative definition of the partiality monad: a variant of the
-- delay monad quotiented by a notion of weak bisimilarity

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

module Partiality-monad.Coinductive.Alternative where

open import Equality.Propositional
open import H-level.Truncation.Propositional
open import Logical-equivalence using (_⇔_)
open import Prelude hiding ()
open import Quotient.HIT

open import Bijection equality-with-J using (_↔_)
open import Function-universe equality-with-J hiding (⊥↔⊥)
open import H-level equality-with-J

import Delay-monad.Alternative as A
import Delay-monad.Alternative.Equivalence as A
import Delay-monad.Alternative.Weak-bisimilarity as A
import Delay-monad.Strong-bisimilarity as Strong-bisimilarity
import Delay-monad.Weak-bisimilarity as W
import Partiality-monad.Coinductive as C

-- The partiality monad, defined as the alternative definition of the
-- delay monad quotiented by weak bisimilarity.

_⊥ :  {a}  Set a  Set a
A  = A.Delay A / λ x y  (x A.≈ y) , A.≈-propositional x y

-- The partiality monad is a set.

⊥-is-set :  {a} {A : Set a}  Is-set (A )
⊥-is-set = /-is-set

-- This definition of the partiality monad is isomorphic to the one in
-- Partiality-monad.Coinductive, for sets, assuming extensionality.

⊥↔⊥ :  {a} {A : Set a} 
      Is-set A 
      Strong-bisimilarity.Extensionality a 
      A   A C.⊥
⊥↔⊥ {A = A} A-set delay-ext = D↔D /-cong lemma
  D↔D = A.Delay↔Delay delay-ext

  lemma : (x y : A.Delay A) 
          x A.≈ y   _↔_.to D↔D x W.≈ _↔_.to D↔D y 
  lemma x y =
    x A.≈ y                            ↔⟨ inverse $ ∥∥↔ (A.≈-propositional x y) 
     x A.≈ y                         ↝⟨ ∥∥-cong-⇔ (A.≈⇔≈ A-set x y) ⟩□
     _↔_.to D↔D x W.≈ _↔_.to D↔D y