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{-# OPTIONS --without-K --safe #-}
open import Relation.Binary
module Cfe.Language.Base
{a ℓ} (setoid : Setoid a ℓ)
where
open Setoid setoid renaming (Carrier to A)
open import Data.Empty.Polymorphic
open import Data.List
open import Data.List.Relation.Binary.Equality.Setoid setoid
open import Data.Nat hiding (_≤_; _⊔_)
open import Data.Product
open import Data.Sum
open import Function
open import Level renaming (suc to lsuc)
Language : Set (lsuc a ⊔ lsuc ℓ)
Language = List A → Set (a ⊔ ℓ)
∅ : Language
∅ = const ⊥
｛ε｝ : Language
｛ε｝ = [] ≋_
｛_｝ : A → Language
｛ a ｝ = [ a ] ≋_
infix 4 _∪_
infix 4 _∙_
_∪_ : Language → Language → Language
(A ∪ B) l = A l ⊎ B l
_∙_ : Language → Language → Language
(A ∙ B) l = ∃[ l₁ ] ∃[ l₂ ] (A l₁ × B l₂ × l₁ ++ l₂ ≋ l)
iterate : (Language → Language) → ℕ → Language → Language
iterate f ℕ.zero = id
iterate f (suc n) = f ∘ iterate f n
fix : (Language → Language) → Language
fix f l = ∃[ n ] iterate f n ∅ l
_≤_ : Language → Language → Set (a ⊔ ℓ)
A ≤ B = ∀ l → A l → B l
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