```------------------------------------------------------------------------
------------------------------------------------------------------------

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

open import Equality.Propositional
open import Prelude hiding (module W)

------------------------------------------------------------------------
-- Map, join and bind

mutual

-- A universe-polymorphic variant of map.

map′ : ∀ {i a b} {A : Set a} {B : Set b} →
(A → B) → Delay A i → Delay B i
map′ f (now x)   = now (f x)
map′ f (later x) = later (∞map′ f x)

∞map′ : ∀ {i a b} {A : Set a} {B : Set b} →
(A → B) → ∞Delay A i → ∞Delay B i
force (∞map′ f x) = map′ f (force x)

mutual

-- Join.

join : ∀ {i a} {A : Set a} →
Delay (Delay A i) i → Delay A i
join (now x)   = x
join (later x) = later (∞join x)

∞join : ∀ {i a} {A : Set a} →
∞Delay (Delay A i) i → ∞Delay A i
force (∞join x) = join (force x)

-- A universe-polymorphic variant of bind.

infixl 5 _>>=′_

_>>=′_ : ∀ {i a b} {A : Set a} {B : Set b} →
Delay A i → (A → Delay B i) → Delay B i
x >>=′ f = join (map′ f x)

instance

delay-raw-monad : ∀ {a i} → Raw-monad (λ (A : Set a) → Delay A i)

------------------------------------------------------------------------

left-identity′ :
∀ {a b} {A : Set a} {B : Set b} x (f : A → Delay B ∞) →
return x >>=′ f ∼ f x
left-identity′ x f = S.reflexive (f x)

mutual

right-identity′ : ∀ {a} {A : Set a} (x : Delay A ∞) →
x >>= return ∼ x
right-identity′ (now x)   = now-cong
right-identity′ (later x) =
later-cong (∞right-identity′ (force x))

∞right-identity′ : ∀ {a} {A : Set a} (x : Delay A ∞) →
x >>= return ∞∼ x
force (∞right-identity′ x) = right-identity′ x

mutual

associativity′ :
∀ {a b c} {A : Set a} {B : Set b} {C : Set c} →
(x : Delay A ∞) (f : A → Delay B ∞) (g : B → Delay C ∞) →
x >>=′ (λ x → f x >>=′ g) ∼ x >>=′ f >>=′ g
associativity′ (now x)   f g = S.reflexive (f x >>=′ g)
associativity′ (later x) f g =
later-cong (∞associativity′ (force x) f g)

∞associativity′ :
∀ {a b c} {A : Set a} {B : Set b} {C : Set c} →
(x : Delay A ∞) (f : A → Delay B ∞) (g : B → Delay C ∞) →
x >>=′ (λ x → f x >>=′ g) ∞∼ x >>=′ f >>=′ g
force (∞associativity′ x f g) = associativity′ x f g

∀ {a} → S.Extensionality a → Monad (λ (A : Set a) → Delay A ∞)
Monad.associativity  (delay-monad ext) x f g = ext (associativity′ x f g)

------------------------------------------------------------------------
-- The functions map′, join and _>>=′_ preserve strong bisimilarity

mutual

map-cong-∼ : ∀ {i a b} {A : Set a} {B : Set b}
(f : A → B) {x y : Delay A ∞} →
Strongly-bisimilar i x y →
Strongly-bisimilar i (map′ f x) (map′ f y)
map-cong-∼ f now-cong       = now-cong
map-cong-∼ f (later-cong p) = later-cong (∞map-cong-∼ f p)

∞map-cong-∼ : ∀ {i a b} {A : Set a} {B : Set b}
(f : A → B) {x y : Delay A ∞} →
∞Strongly-bisimilar i x y →
∞Strongly-bisimilar i (map′ f x) (map′ f y)
force (∞map-cong-∼ f p) = map-cong-∼ f (force p)

mutual

join-cong-∼ : ∀ {i a} {A : Set a} {x y : Delay (Delay A ∞) ∞} →
Strongly-bisimilar i x y →
Strongly-bisimilar i (join x) (join y)
join-cong-∼ now-cong       = S.reflexive _
join-cong-∼ (later-cong p) = later-cong (∞join-cong-∼ p)

∞join-cong-∼ : ∀ {i a} {A : Set a} {x y : Delay (Delay A ∞) ∞} →
∞Strongly-bisimilar i x y →
∞Strongly-bisimilar i (join x) (join y)
force (∞join-cong-∼ p) = join-cong-∼ (force p)

mutual

infixl 5 _>>=-cong-∼_ _∞>>=-cong-∼_

_>>=-cong-∼_ :
∀ {i a b} {A : Set a} {B : Set b}
{x y : Delay A ∞} {f g : A → Delay B ∞} →
Strongly-bisimilar i x y →
(∀ z → Strongly-bisimilar i (f z) (g z)) →
Strongly-bisimilar i (x >>=′ f) (y >>=′ g)
now-cong     >>=-cong-∼  q = q _
later-cong p >>=-cong-∼  q = later-cong (p ∞>>=-cong-∼ q)

_∞>>=-cong-∼_ :
∀ {i a b} {A : Set a} {B : Set b}
{x y : Delay A ∞} {f g : A → Delay B ∞} →
∞Strongly-bisimilar i x y →
(∀ z → Strongly-bisimilar i (f z) (g z)) →
∞Strongly-bisimilar i (x >>=′ f) (y >>=′ g)
force (p ∞>>=-cong-∼ q) = force p >>=-cong-∼ q

------------------------------------------------------------------------
-- The functions map′, join and _>>=′_ preserve weak bisimilarity

mutual

map-cong-≈ : ∀ {i a b} {A : Set a} {B : Set b}
(f : A → B) {x y : Delay A ∞} →
Weakly-bisimilar i x y →
Weakly-bisimilar i (map′ f x) (map′ f y)
map-cong-≈ f now-cong       = now-cong
map-cong-≈ f (later-cong p) = later-cong (∞map-cong-≈ f p)
map-cong-≈ f (laterˡ p)     = laterˡ (map-cong-≈ f p)
map-cong-≈ f (laterʳ p)     = laterʳ (map-cong-≈ f p)

∞map-cong-≈ : ∀ {i a b} {A : Set a} {B : Set b}
(f : A → B) {x y : Delay A ∞} →
∞Weakly-bisimilar i x y →
∞Weakly-bisimilar i (map′ f x) (map′ f y)
force (∞map-cong-≈ f p) = map-cong-≈ f (force p)

mutual

join-cong-≈ : ∀ {i a} {A : Set a} {x y : Delay (Delay A ∞) ∞} →
Weakly-bisimilar i x y →
Weakly-bisimilar i (join x) (join y)
join-cong-≈ now-cong       = W.reflexive _
join-cong-≈ (later-cong p) = later-cong (∞join-cong-≈ p)
join-cong-≈ (laterˡ p)     = laterˡ (join-cong-≈ p)
join-cong-≈ (laterʳ p)     = laterʳ (join-cong-≈ p)

∞join-cong-≈ : ∀ {i a} {A : Set a} {x y : Delay (Delay A ∞) ∞} →
∞Weakly-bisimilar i x y →
∞Weakly-bisimilar i (join x) (join y)
force (∞join-cong-≈ p) = join-cong-≈ (force p)

mutual

infixl 5 _>>=-cong-≈_ _∞>>=-cong-≈_

_>>=-cong-≈_ :
∀ {i a b} {A : Set a} {B : Set b}
{x y : Delay A ∞} {f g : A → Delay B ∞} →
Weakly-bisimilar i x y →
(∀ z → Weakly-bisimilar i (f z) (g z)) →
Weakly-bisimilar i (x >>=′ f) (y >>=′ g)
now-cong     >>=-cong-≈  q = q _
later-cong p >>=-cong-≈  q = later-cong (p ∞>>=-cong-≈ q)
laterˡ p     >>=-cong-≈  q = laterˡ (p >>=-cong-≈ q)
laterʳ p     >>=-cong-≈  q = laterʳ (p >>=-cong-≈ q)

_∞>>=-cong-≈_ :
∀ {i a b} {A : Set a} {B : Set b}
{x y : Delay A ∞} {f g : A → Delay B ∞} →
∞Weakly-bisimilar i x y →
(∀ z → Weakly-bisimilar i (f z) (g z)) →
∞Weakly-bisimilar i (x >>=′ f) (y >>=′ g)
force (p ∞>>=-cong-≈ q) = force p >>=-cong-≈ q
```