------------------------------------------------------------------------
-- The delay monad is a monad up to strong bisimilarity
------------------------------------------------------------------------

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

module Delay-monad.Monad where

open import Conat using (zero; suc; force)
open import Equality.Propositional
open import Prelude
open import Size

open import Monad equality-with-J

open import Delay-monad
open import Delay-monad.Bisimilarity as B

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

  -- A raw monad instance.

  delay-raw-monad :  {a i}  Raw-monad  (A : Set a)  Delay A i)
  Raw-monad.return delay-raw-monad = now
  Raw-monad._>>=_  delay-raw-monad = _>>=′_

------------------------------------------------------------------------
-- Monad laws

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 >>=′  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 }

-- The delay monad is a monad (assuming extensionality).

delay-monad :
   {a}  B.Extensionality a  Monad  (A : Set a)  Delay A )
Monad.raw-monad      (delay-monad ext)       = delay-raw-monad
Monad.left-identity  (delay-monad ext) x f   = ext (left-identity′ x f)
Monad.right-identity (delay-monad ext) x     = ext (right-identity′ x)
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-⟨$⟩ :
   {i } {A B : Set } {f : A  B}
  (x : Delay A ) 
  Conat.[ i ] steps (f ⟨$⟩ x)  steps x
steps-⟨$⟩ (now x)   = zero
steps-⟨$⟩ (later x) = suc λ { .force  steps-⟨$⟩ (x .force) }