module BreadthFirst.Universe where
open import Data.Product using (_×_; _,_)
open import Data.Colist using (Colist; []; _∷_)
open import Relation.Binary.PropositionalEquality using (_≡_)
open import Coinduction
open import Tree using (Tree; node; leaf)
open import Stream using (Stream; _≺_)
infixr 5 _≺_ _∷_
infixr 4 _,_
infixr 2 _⊗_
data Kind : Set where
μ : Kind
ν : Kind
data U : Kind → Set1 where
tree : ∀ {k} (a : U k) → U ν
stream : ∀ {k} (a : U k) → U ν
colist : ∀ {k} (a : U k) → U ν
_⊗_ : ∀ {k₁ k₂} (a : U k₁) (b : U k₂) → U μ
⌈_⌉ : (A : Set) → U μ
El : ∀ {k} → U k → Set
El (tree a) = Tree.Tree (El a)
El (stream a) = Stream.Stream (El a)
El (colist a) = Colist (El a)
El (a ⊗ b) = El a × El b
El ⌈ A ⌉ = A
data Eq : ∀ {k} (a : U k) → El a → El a → Set1 where
leaf : ∀ {k} {a : U k} → Eq (tree a) leaf leaf
node : ∀ {k} {a : U k} {x x′ l l′ r r′}
(l≈l′ : ∞₁ (Eq (tree a) (♭ l) (♭ l′)))
(x≈x′ : Eq a x x′ )
(r≈r′ : ∞₁ (Eq (tree a) (♭ r) (♭ r′))) →
Eq (tree a) (node l x r) (node l′ x′ r′)
_≺_ : ∀ {k} {a : U k} {x x′ xs xs′}
(x≈x′ : Eq a x x′ )
(xs≈xs′ : ∞₁ (Eq (stream a) (♭ xs) (♭ xs′))) →
Eq (stream a) (x ≺ xs) (x′ ≺ xs′)
[] : ∀ {k} {a : U k} → Eq (colist a) [] []
_∷_ : ∀ {k} {a : U k} {x x′ xs xs′}
(x≈x′ : Eq a x x′ )
(xs≈xs′ : ∞₁ (Eq (colist a) (♭ xs) (♭ xs′))) →
Eq (colist a) (x ∷ xs) (x′ ∷ xs′)
_,_ : ∀ {k₁ k₂} {a : U k₁} {b : U k₂} {x x′ y y′}
(x≈x′ : Eq a x x′) (y≈y′ : Eq b y y′) →
Eq (a ⊗ b) (x , y) (x′ , y′)
⌈_⌉ : ∀ {A} {x x′} (x≡x′ : x ≡ x′) → Eq ⌈ A ⌉ x x′
data PrefixOf {k} (a : U k) : Colist (El a) → Stream (El a) → Set1 where
[] : ∀ {ys} → PrefixOf a [] ys
_∷_ : ∀ {x y xs ys}
(x≈y : Eq a x y) (p : ∞₁ (PrefixOf a (♭ xs) (♭ ys))) →
PrefixOf a (x ∷ xs) (y ≺ ys)
∞? : Kind → Set1 → Set1
∞? μ = λ A → A
∞? ν = ∞₁
♭? : ∀ k {A} → ∞? k A → A
♭? μ x = x
♭? ν x = ♭₁ x