{-# OPTIONS --without-K --safe --sized-types #-}
module Codata.Colist.Bisimilarity where
open import Level using (_⊔_)
open import Size
open import Codata.Thunk
open import Codata.Colist
open import Relation.Binary
open import Relation.Binary.PropositionalEquality as Eq using (_≡_)
data Bisim {a b r} {A : Set a} {B : Set b} (R : A → B → Set r) (i : Size) :
(xs : Colist A ∞) (ys : Colist B ∞) → Set (r ⊔ a ⊔ b) where
[] : Bisim R i [] []
_∷_ : ∀ {x y xs ys} → R x y → Thunk^R (Bisim R) i xs ys → Bisim R i (x ∷ xs) (y ∷ 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} {[]} = []
reflexive refl^R {i} {r ∷ rs} = refl^R ∷ 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 [] = []
symmetric sym^PQ (p ∷ ps) = sym^PQ p ∷ 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 [] [] = []
transitive trans^PQR (p ∷ ps) (q ∷ qs) =
trans^PQR p q ∷ λ where .force → transitive trans^PQR (ps .force) (qs .force)
module _ {ℓ} {A : Set ℓ} where
infix 1 _⊢_≈_
_⊢_≈_ : ∀ i → Colist A ∞ → Colist A ∞ → Set ℓ
_⊢_≈_ = Bisim _≡_
refl : ∀ {i} → Reflexive (i ⊢_≈_)
refl = reflexive Eq.refl
sym : ∀ {i} → Symmetric (i ⊢_≈_)
sym = symmetric Eq.sym
trans : ∀ {i} → Transitive (i ⊢_≈_)
trans = transitive Eq.trans