------------------------------------------------------------------------
-- The Agda standard library
--
-- Bisimilarity for the Delay type
------------------------------------------------------------------------
{-# OPTIONS --cubical-compatible --sized-types #-}
module Codata.Sized.Delay.Bisimilarity where
open import Size using (Size; ∞)
open import Codata.Sized.Thunk using (Thunk; Thunk^R; force)
open import Codata.Sized.Delay using (Delay; now; later)
open import Level using (_⊔_)
open import Relation.Binary.Definitions
using (Reflexive; Symmetric; Transitive; Sym; Trans)
open import Relation.Binary.PropositionalEquality.Core as ≡ using (_≡_)
data Bisim {a b r} {A : Set a} {B : Set b} (R : A → B → Set r) i :
(xs : Delay A ∞) (ys : Delay B ∞) → Set (a ⊔ b ⊔ r) where
now : ∀ {x y} → R x y → Bisim R i (now x) (now y)
later : ∀ {xs ys} → Thunk^R (Bisim R) i xs ys → Bisim R i (later xs) (later ys)
module _ {a r} {A : Set a} {R : A → A → Set r} where
reflexive : Reflexive R → ∀ {i} → Reflexive (Bisim R i)
reflexive refl^R {i} {now r} = now refl^R
reflexive refl^R {i} {later rs} = later λ where .force → reflexive refl^R
module _ {a b} {A : Set a} {B : Set b}
{r} {P : A → B → Set r} {Q : B → A → Set r} where
symmetric : Sym P Q → ∀ {i} → Sym (Bisim P i) (Bisim Q i)
symmetric sym^PQ (now p) = now (sym^PQ p)
symmetric sym^PQ (later ps) = later λ where .force → symmetric sym^PQ (ps .force)
module _ {a b c} {A : Set a} {B : Set b} {C : Set c}
{r} {P : A → B → Set r} {Q : B → C → Set r} {R : A → C → Set r} where
transitive : Trans P Q R → ∀ {i} → Trans (Bisim P i) (Bisim Q i) (Bisim R i)
transitive trans^PQR (now p) (now q) = now (trans^PQR p q)
transitive trans^PQR (later ps) (later qs) =
later λ where .force → transitive trans^PQR (ps .force) (qs .force)
-- Pointwise Equality as a Bisimilarity
------------------------------------------------------------------------
module _ {ℓ} {A : Set ℓ} where
infix 1 _⊢_≈_
_⊢_≈_ : ∀ i → Delay A ∞ → Delay A ∞ → Set ℓ
_⊢_≈_ = Bisim _≡_
refl : ∀ {i} → Reflexive (i ⊢_≈_)
refl = reflexive ≡.refl
sym : ∀ {i} → Symmetric (i ⊢_≈_)
sym = symmetric ≡.sym
trans : ∀ {i} → Transitive (i ⊢_≈_)
trans = transitive ≡.trans
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