------------------------------------------------------------------------
-- A partial order
------------------------------------------------------------------------

{-# OPTIONS --sized-types #-}

open import Prelude
open import Prelude.Size

module Delay-monad.Sized.Partial-order {a} {A : Size → Type a} where

open import Equality.Propositional
open import Logical-equivalence using (_⇔_)

open import Bijection equality-with-J using (_↔_)
open import Double-negation equality-with-J
open import Excluded-middle equality-with-J
import Extensionality equality-with-J as E
open import Function-universe equality-with-J hiding (id; _∘_)
open import H-level equality-with-J
open import H-level.Closure equality-with-J
open import Monad equality-with-J

open import Delay-monad.Sized
open import Delay-monad.Sized.Bisimilarity as B
  hiding (reflexive; symmetric; laterˡ⁻¹; laterʳ⁻¹)
open import Delay-monad.Sized.Bisimilarity.Alternative
open import Delay-monad.Sized.Bisimilarity.Negative
import Delay-monad.Sized.Termination as T

-- An ordering relation.
--
-- Capretta defines a logically equivalent relation in "General
-- Recursion via Coinductive Types".
--
-- Benton, Kennedy and Varming define a relation that is perhaps
-- logically equivalent in "Some Domain Theory and Denotational
-- Semantics in Coq".

infix  [_]_⊑_ [_]_⊑′_ _⊑_ _⊑′_

mutual

  data [_]_⊑_ (i : Size) : Delay A ∞ → Delay A ∞ → Type a where
    now    : ∀ {x} → [ i ] now x ⊑ now x
    laterʳ : ∀ {x y} → [ i ] x ⊑ force y → [ i ] x ⊑ later y
    laterˡ : ∀ {x y} → [ i ] force x ⊑′ y → [ i ] later x ⊑ y

  record [_]_⊑′_ (i : Size) (x y : Delay A ∞) : Type a where
    coinductive
    field
      force : {j : Size< i} → [ j ] x ⊑ y

open [_]_⊑′_ public

_⊑_ : Delay A ∞ → Delay A ∞ → Type a
_⊑_ = [ ∞ ]_⊑_

_⊑′_ : Delay A ∞ → Delay A ∞ → Type a
_⊑′_ = [ ∞ ]_⊑′_

-- A derived "constructor".

later-cong : ∀ {i x y} →
             [ i ] force x ⊑′ force y → [ i ] later x ⊑ later y
later-cong p = laterʳ (laterˡ p)

-- Termination predicates.

Terminates : Size → Delay A ∞ → A ∞ → Type a
Terminates i x y = [ i ] now y ⊑ x

_⇓_ : Delay A ∞ → A ∞ → Type a
_⇓_ = Terminates ∞

-- If x terminates with the values y and z, then y is equal to z.

⇓→⇓→≡ : ∀ {i x y z} → Terminates i x y → Terminates i x z → y ≡ z
⇓→⇓→≡ now        now        = refl
⇓→⇓→≡ (laterʳ p) (laterʳ q) = ⇓→⇓→≡ p q

-- If x is smaller than or equal to now y, and x terminates, then
-- x terminates with the value y.

⊑now→⇓→⇓ : ∀ {i x} {y z : A ∞} →
           x ⊑ now y → Terminates i x z → Terminates i x y
⊑now→⇓→⇓ now        now        = now
⊑now→⇓→⇓ (laterˡ p) (laterʳ q) = laterʳ (⊑now→⇓→⇓ (force p) q)

-- The notion of termination defined here is pointwise isomorphic to
-- the one defined in Delay-monad.Sized.Weak-bisimilarity.

⇓↔⇓ : ∀ {i x y} → Terminates i x y ↔ T.Terminates i x y
⇓↔⇓ = record
  { surjection = record
    { logical-equivalence = record
      { to   = to
      ; from = from
      }
    ; right-inverse-of = to∘from
    }
  ; left-inverse-of = from∘to
  }
  where
  to : ∀ {i x y} → Terminates i x y → T.Terminates i x y
  to now        = now
  to (laterʳ p) = laterʳ (to p)

  from : ∀ {i x y} → T.Terminates i x y → Terminates i x y
  from now        = now
  from (laterʳ p) = laterʳ (from p)

  from∘to : ∀ {i x y} (p : Terminates i x y) → from (to p) ≡ p
  from∘to now        = refl
  from∘to (laterʳ p) = cong laterʳ (from∘to p)

  to∘from : ∀ {i x y} (p : T.Terminates i x y) → to (from p) ≡ p
  to∘from now        = refl
  to∘from (laterʳ p) = cong laterʳ (to∘from p)

-- Terminates i is pointwise isomorphic to Terminates ∞.

Terminates↔⇓ : ∀ {i x y} → Terminates i x y ↔ x ⇓ y
Terminates↔⇓ {i} {x} {y} =
  Terminates i x y    ↝⟨ ⇓↔⇓ ⟩
  T.Terminates i x y  ↝⟨ T.Terminates↔⇓ ⟩
  x T.⇓ y             ↝⟨ inverse ⇓↔⇓ ⟩□
  x ⇓ y               □

-- The computation never is smaller than or equal to all other
-- computations.

never⊑ : ∀ {i} x → [ i ] never ⊑ x
never⊑ (now x)   = laterˡ λ { .force → never⊑ (now x) }
never⊑ (later x) = later-cong λ { .force → never⊑ (force x) }

-- The computation never does not terminate.

now⋢never : ∀ {i x} → ¬ Terminates i never x
now⋢never (laterʳ p) = now⋢never p

-- One can remove later constructors.

laterˡ⁻¹ : ∀ {i} {j : Size< i} {x y} →
           [ i ] later x ⊑ y →
           [ j ] force x ⊑ y
laterˡ⁻¹ (laterʳ p) = laterʳ (laterˡ⁻¹ p)
laterˡ⁻¹ (laterˡ p) = force p

laterʳ⁻¹ : ∀ {i x y} →
           [ i ] x ⊑ later y →
           [ i ] x ⊑ force y
laterʳ⁻¹ (laterʳ p) = p
laterʳ⁻¹ (laterˡ p) = laterˡ λ { .force → laterʳ⁻¹ (force p) }

later-cong⁻¹ : ∀ {i} {j : Size< i} {x y} →
               [ i ] later x ⊑ later y →
               [ j ] force x ⊑ force y
later-cong⁻¹ p = laterʳ⁻¹ (laterˡ⁻¹ p)

-- Weak bisimilarity is contained in the ordering relation.

≈→⊑ : ∀ {i x y} → [ i ] x ≈ y → [ i ] x ⊑ y
≈→⊑ now        = now
≈→⊑ (later  p) = later-cong λ { .force → ≈→⊑ (force p) }
≈→⊑ (laterˡ p) = laterˡ λ { .force → ≈→⊑ p }
≈→⊑ (laterʳ p) = laterʳ (≈→⊑ p)

-- The ordering relation is antisymmetric (with respect to weak
-- bisimilarity).

antisymmetric : ∀ {i x y} →
                [ i ] x ⊑ y → [ i ] y ⊑ x → [ i ] x ≈ y
antisymmetric {x = now   x} {y = now  .x} now        _          = now
antisymmetric {x = now   x} {y = later y} (laterʳ p) q          = laterʳ (_↔_.to ⇓↔⇓ p)
antisymmetric {x = later x} {y = now   y} p          (laterʳ q) = laterˡ (B.symmetric (_↔_.to ⇓↔⇓ q))
antisymmetric {x = later x} {y = later y} p          q          =
  later λ { .force → antisymmetric (later-cong⁻¹ p) (later-cong⁻¹ q) }

-- An alternative characterisation of weak bisimilarity.

≈⇔⊑×⊒ : ∀ {i x y} → [ i ] x ≈ y ⇔ ([ i ] x ⊑ y × [ i ] y ⊑ x)
≈⇔⊑×⊒ = record
  { to   = λ p → ≈→⊑ p , ≈→⊑ (B.symmetric p)
  ; from = uncurry antisymmetric
  }

-- The ordering relation is reflexive.

reflexive : ∀ {i} x → [ i ] x ⊑ x
reflexive (now x)   = now
reflexive (later x) = later-cong λ { .force → reflexive (force x) }

-- Certain instances of symmetry also hold.

symmetric : ∀ {i} {x : A ∞} {y} →
            Terminates i y x → [ i ] y ⊑ now x
symmetric now        = now
symmetric (laterʳ p) = laterˡ λ { .force → symmetric p }

-- The ordering relation is transitive.

transitive : ∀ {i} {x y z : Delay A ∞} →
             [ i ] x ⊑ y → y ⊑ z → [ i ] x ⊑ z
transitive p          now        = p
transitive p          (laterʳ q) = laterʳ (transitive p q)
transitive (laterʳ p) (laterˡ q) = transitive p (force q)
transitive (laterˡ p) q          = laterˡ λ { .force →
                                     transitive (force p) q }

-- The termination relation respects weak bisimilarity.

⇓-respects-≈ : ∀ {i x y z} → Terminates i x z → x ≈ y → Terminates i y z
⇓-respects-≈ now        q = ≈→⊑ q
⇓-respects-≈ (laterʳ p) q = ⇓-respects-≈ p (B.laterˡ⁻¹ q)

-- The ordering relation respects weak bisimilarity.

transitive-≈⊑ : ∀ {i x y z} → [ i ] x ≈ y → y ⊑ z → [ i ] x ⊑ z
transitive-≈⊑ p q = transitive (≈→⊑ p) q

transitive-⊑≈ : ∀ {i x y z} → [ i ] x ⊑ y → y ≈ z → [ i ] x ⊑ z
transitive-⊑≈ p          now        = p
transitive-⊑≈ (laterʳ p) (later  q) = laterʳ (transitive-⊑≈ p (force q))
transitive-⊑≈ (laterˡ p) q          = laterˡ λ { .force →
                                        transitive-⊑≈ (force p) q }
transitive-⊑≈ (laterʳ p) (laterˡ q) = transitive-⊑≈ p q
transitive-⊑≈ p          (laterʳ q) = laterʳ (transitive-⊑≈ p q)

-- If there is a transitivity-like function that produces an ordering
-- proof from one weak bisimilarity proof and one ordering proof, in
-- such a way that the size of the ordering proof is preserved, then
-- ∀ i → A i is uninhabited.

Transitivity-≈⊑ʳ =
  ∀ {i} {x y z : Delay A ∞} → x ≈ y → [ i ] y ⊑ z → [ i ] x ⊑ z

size-preserving-transitivity-≈⊑ʳ→uninhabited :
  Transitivity-≈⊑ʳ → ¬ (∀ i → A i)
size-preserving-transitivity-≈⊑ʳ→uninhabited =
  Transitivity-≈⊑ʳ                                                ↝⟨ (λ trans {i} x →

      [ i ] later (record { force = λ {j} → now (x j) }) ∼ never        ↝⟨ ≈→⊑ ∘ ∼→ ⟩
      [ i ] later (record { force = λ {j} → now (x j) }) ⊑ never        ↝⟨ trans (laterʳ now) ⟩
      [ i ] now (x ∞) ⊑ never                                           ↝⟨ _↔_.to ⇓↔⇓ ⟩□
      [ i ] now (x ∞) ≈ never                                           □) ⟩

  Laterˡ⁻¹-∼≈′                                                    ↝⟨ _⇔_.from size-preserving-laterˡ⁻¹-∼≈⇔size-preserving-laterˡ⁻¹-∼≈′ ⟩

  Laterˡ⁻¹-∼≈                                                     ↝⟨ size-preserving-laterˡ⁻¹-∼≈→uninhabited ⟩

  ¬ (∀ i → A i)                                                   □

-- If A ∞ is uninhabited, then there is a transitivity proof of the
-- kind discussed above (Transitivity-≈⊑ʳ).

uninhabited→size-preserving-transitivity-≈⊑ʳ : ¬ A ∞ → Transitivity-≈⊑ʳ
uninhabited→size-preserving-transitivity-≈⊑ʳ =
  ¬ A ∞             ↝⟨ uninhabited→trivial ⟩
  (∀ x y → x ≈ y)   ↝⟨ (λ trivial _ _ → ≈→⊑ (trivial _ _)) ⟩
  (∀ x y → x ⊑ y)   ↝⟨ (λ trivial {_ _ _ _} _ _ → trivial _ _) ⟩□
  Transitivity-≈⊑ʳ  □

-- If transitivity can be made size-preserving in the second argument,
-- then ∀ i → A i is uninhabited.

Transitivityʳ =
  ∀ {i} {x y z : Delay A ∞} → x ⊑ y → [ i ] y ⊑ z → [ i ] x ⊑ z

size-preserving-transitivityʳ→uninhabited :
  Transitivityʳ → ¬ (∀ i → A i)
size-preserving-transitivityʳ→uninhabited =
  Transitivityʳ     ↝⟨ _∘ ≈→⊑ ⟩
  Transitivity-≈⊑ʳ  ↝⟨ size-preserving-transitivity-≈⊑ʳ→uninhabited ⟩□
  ¬ (∀ i → A i)     □

-- If A ∞ is uninhabited, then transitivity can be made
-- size-preserving in the second argument.

uninhabited→size-preserving-transitivityʳ : ¬ A ∞ → Transitivityʳ
uninhabited→size-preserving-transitivityʳ =
  ¬ A ∞            ↝⟨ uninhabited→trivial ⟩
  (∀ x y → x ≈ y)  ↝⟨ (λ trivial _ _ → ≈→⊑ (trivial _ _)) ⟩
  (∀ x y → x ⊑ y)  ↝⟨ (λ trivial {_ _ _ _} _ _ → trivial _ _) ⟩□
  Transitivityʳ    □

-- If there is a transitivity-like function that produces an ordering
-- proof from one ordering proof and one weak bisimilarity proof, in
-- such a way that the size of the weak bisimilarity proof is
-- preserved, then ∀ i → A i is uninhabited.

Transitivity-⊑≈ʳ =
  ∀ {i} {x y z : Delay A ∞} → x ⊑ y → [ i ] y ≈ z → [ i ] x ⊑ z

size-preserving-transitivity-⊑≈ʳ→uninhabited :
  Transitivity-⊑≈ʳ → ¬ (∀ i → A i)
size-preserving-transitivity-⊑≈ʳ→uninhabited =
  Transitivity-⊑≈ʳ                                                ↝⟨ (λ trans {i} x →

      [ i ] later (record { force = λ {j} → now (x j) }) ∼ never        ↝⟨ ∼→ ⟩
      [ i ] later (record { force = λ {j} → now (x j) }) ≈ never        ↝⟨ trans (laterʳ now) ⟩
      [ i ] now (x ∞) ⊑ never                                           ↝⟨ _↔_.to ⇓↔⇓ ⟩□
      [ i ] now (x ∞) ≈ never                                           □) ⟩

  Laterˡ⁻¹-∼≈′                                                    ↝⟨ _⇔_.from size-preserving-laterˡ⁻¹-∼≈⇔size-preserving-laterˡ⁻¹-∼≈′ ⟩

  Laterˡ⁻¹-∼≈                                                     ↝⟨ size-preserving-laterˡ⁻¹-∼≈→uninhabited ⟩

  ¬ (∀ i → A i)                                                   □

-- If A ∞ is uninhabited, then there is a transitivity-like function of the
-- kind discussed above (Transitivity-⊑≈ʳ).

uninhabited→size-preserving-transitivity-⊑≈ʳ : ¬ A ∞ → Transitivity-⊑≈ʳ
uninhabited→size-preserving-transitivity-⊑≈ʳ =
  ¬ A ∞             ↝⟨ uninhabited→trivial ⟩
  (∀ x y → x ≈ y)   ↝⟨ (λ trivial _ _ → ≈→⊑ (trivial _ _)) ⟩
  (∀ x y → x ⊑ y)   ↝⟨ (λ trivial {_ _ _ _} _ _ → trivial _ _) ⟩□
  Transitivity-⊑≈ʳ  □

-- An alternative characterisation of the ordering relation.
--
-- Capretta proves a similar result in "General Recursion via
-- Coinductive Types".
--
-- One might wonder if the equivalence can be made size-preserving in
-- some way. However, note that x ⇓ y is in bijective correspondence
-- with Terminates i x y for any size i (see Terminates↔⇓).

⊑⇔⇓→⇓ : ∀ {x y} → x ⊑ y ⇔ (∀ z → x ⇓ z → y ⇓ z)
⊑⇔⇓→⇓ = record
  { to   = to
  ; from = from _
  }
  where
  to : ∀ {x y} → x ⊑ y → ∀ z → x ⇓ z → y ⇓ z
  to p          z now   = p
  to (laterʳ p) z q          = laterʳ (to p z q)
  to (laterˡ p) z (laterʳ q) = to (force p) z q

  from : ∀ {i} x {y} → (∀ z → x ⇓ z → y ⇓ z) → [ i ] x ⊑ y
  from (now x)   p = p x now
  from (later x) p = laterˡ λ { .force →
                       from (force x) (λ z q → p z (laterʳ q)) }

-- An alternative characterisation of weak bisimilarity.
--
-- Capretta proves a similar result in "General Recursion via
-- Coinductive Types".

≈⇔≈₂ : {x y : Delay A ∞} → x ≈ y ⇔ x ≈₂ y
≈⇔≈₂ {x} {y} =
  x ≈ y                                                  ↝⟨ ≈⇔⊑×⊒ ⟩
  x ⊑ y × y ⊑ x                                          ↝⟨ ⊑⇔⇓→⇓ ×-cong ⊑⇔⇓→⇓ ⟩
  (∀ z → x ⇓ z → y ⇓ z) × (∀ z → y ⇓ z → x ⇓ z)          ↝⟨ ∀-cong _ (λ _ → →-cong _ (from-bijection ⇓↔⇓) (from-bijection ⇓↔⇓))
                                                              ×-cong
                                                            ∀-cong _ (λ _ → →-cong _ (from-bijection ⇓↔⇓) (from-bijection ⇓↔⇓)) ⟩
  (∀ z → x T.⇓ z → y T.⇓ z) × (∀ z → y T.⇓ z → x T.⇓ z)  ↝⟨ record { to   = uncurry λ to from z → record { to = to z; from = from z }
                                                                   ; from = λ hyp → _⇔_.to ∘ hyp , _⇔_.from ∘ hyp
                                                                   } ⟩□
  (∀ z → x T.⇓ z ⇔ y T.⇓ z)                              □

-- If A ∞ is a set, then every computation is weakly bisimilar to
-- never or now something (assuming excluded middle and
-- extensionality).

⇑⊎⇓ :
  Excluded-middle a → E.Extensionality a a →
  Is-set (A ∞) → (x : Delay A ∞) → never ≈ x ⊎ ∃ λ y → x T.⇓ y
⇑⊎⇓ em ext A-set x =
  ⊎-map (_⇔_.from ≈⇔≈₂) id $
  Excluded-middle→Double-negation-elimination em
    (⊎-closure-propositional
       (λ { x⇑ (y , x⇓y) →
            now≉never (now y  ≈⟨ x⇓y ⟩
                       x      ≈⟨ B.symmetric (_⇔_.from ≈⇔≈₂ x⇑) ⟩∎
                       never  ∎) })
       (≈₂-propositional ext A-set)
       (T.∃-Terminates-propositional A-set))
    (⊎-map (_⇔_.to ≈⇔≈₂) id ⟨$⟩ T.¬¬[⇑⊎⇓] x)