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

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

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.Bisimilarity as B
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 →
B.Extensionality a →
A ⊥ ↔ A C.⊥
⊥↔⊥ {A = A} A-set delay-ext = D↔D /-cong lemma
where
D↔D = A.Delay↔Delay delay-ext

lemma : (x y : A.Delay A) →
x A.≈ y ⇔ ∥ _↔_.to D↔D x B.≈ _↔_.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 B.≈ _↔_.to D↔D y ∥  □
```