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

{-# OPTIONS --without-K --safe --sized-types #-}

open import Equality.Propositional
open import Prelude
open import Prelude.Size

open import Conat equality-with-J as Conat using (zero; suc; force)

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

-- 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 λ { .force → map′ f (force x) }

-- Join.

join : ∀ {i a} {A : Set a} →
Delay (Delay A i) i → Delay A i
join (now   x) = x
join (later x) = later λ { .force → 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 = reflexive (f x)

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

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

∀ {a} → B.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 and weak
-- bisimilarity and expansion

map-cong : ∀ {k i a b} {A : Set a} {B : Set b}
(f : A → B) {x y : Delay A ∞} →
[ i ] x ⟨ k ⟩ y → [ i ] map′ f x ⟨ k ⟩ map′ f y
map-cong f now        = now
map-cong f (later  p) = later λ { .force → map-cong f (force p) }
map-cong f (laterˡ p) = laterˡ (map-cong f p)
map-cong f (laterʳ p) = laterʳ (map-cong f p)

join-cong : ∀ {k i a} {A : Set a} {x y : Delay (Delay A ∞) ∞} →
[ i ] x ⟨ k ⟩ y → [ i ] join x ⟨ k ⟩ join y
join-cong now        = reflexive _
join-cong (later  p) = later λ { .force → join-cong (force p) }
join-cong (laterˡ p) = laterˡ (join-cong p)
join-cong (laterʳ p) = laterʳ (join-cong p)

infixl 5 _>>=-cong_

_>>=-cong_ :
∀ {k i a b} {A : Set a} {B : Set b}
{x y : Delay A ∞} {f g : A → Delay B ∞} →
[ i ] x ⟨ k ⟩ y → (∀ z → [ i ] f z ⟨ k ⟩ g z) →
[ i ] x >>=′ f ⟨ k ⟩ y >>=′ g
now      >>=-cong  q = q _
later  p >>=-cong  q = later λ { .force → force p >>=-cong q }
laterˡ p >>=-cong  q = laterˡ (p >>=-cong q)
laterʳ p >>=-cong  q = laterʳ (p >>=-cong q)

------------------------------------------------------------------------
-- Some lemmas relating monadic combinators to steps

-- Use of map′ does not affect the number of steps in the computation.

steps-map′ :
∀ {i a b} {A : Set a} {B : Set b} {f : A → B}
(x : Delay A ∞) →
Conat.[ i ] steps (map′ f x) ∼ steps x
steps-map′ (now x)   = zero
steps-map′ (later x) = suc λ { .force → steps-map′ (x .force) }

-- Use of _⟨\$⟩_ does not affect the number of steps in the
-- computation.

steps-⟨\$⟩ :
∀ {i ℓ} {A B : Set ℓ} {f : A → B}
(x : Delay A ∞) →
Conat.[ i ] steps (f ⟨\$⟩ x) ∼ steps x
steps-⟨\$⟩ (now x)   = zero
steps-⟨\$⟩ (later x) = suc λ { .force → steps-⟨\$⟩ (x .force) }
```