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{-# OPTIONS --without-K --safe #-}
open import Relation.Binary as B using (Setoid)
module Cfe.Language.Base
{c ℓ} (over : Setoid c ℓ)
where
open Setoid over using () renaming (Carrier to C)
open import Algebra
open import Data.Empty
open import Data.List hiding (null)
open import Data.Product
open import Data.Unit using (⊤; tt)
open import Function hiding (Injection; Surjection; Inverse)
import Function.Equality as Equality using (setoid)
open import Level as L hiding (Lift)
open import Relation.Binary.Indexed.Heterogeneous.Construct.Trivial as Trivial
open import Relation.Binary.PropositionalEquality as ≡ using (_≡_)
open import Relation.Binary.Indexed.Heterogeneous
infix 4 _∈_
Language : ∀ a aℓ → Set (suc c ⊔ suc a ⊔ suc aℓ)
Language a aℓ = IndexedSetoid (List C) a aℓ
∅ : Language 0ℓ 0ℓ
∅ = Trivial.indexedSetoid (≡.setoid ⊥)
｛ε｝ : Language c 0ℓ
｛ε｝ = record
{ Carrier = [] ≡_
; _≈_ = λ _ _ → ⊤
; isEquivalence = record
{ refl = tt
; sym = λ _ → tt
; trans = λ _ _ → tt
}
}
Lift : ∀ {a aℓ} → (b bℓ : Level) → Language a aℓ → Language (a ⊔ b) (aℓ ⊔ bℓ)
Lift b bℓ A = record
{ Carrier = L.Lift b ∘ A.Carrier
; _≈_ = λ (lift x) (lift y) → L.Lift bℓ (x A.≈ y)
; isEquivalence = record
{ refl = lift A.refl
; sym = λ (lift x) → lift (A.sym x)
; trans = λ (lift x) (lift y) → lift (A.trans x y)
}
}
where
module A = IndexedSetoid A
𝕃 : ∀ {a aℓ} → Language a aℓ → List C → Set a
𝕃 = IndexedSetoid.Carrier
_∈_ : ∀ {a aℓ} → List C → Language a aℓ → Set a
_∈_ = flip 𝕃
∈-cong : ∀ {a aℓ} → (A : Language a aℓ) → ∀ {l₁ l₂} → l₁ ≡ l₂ → l₁ ∈ A → l₂ ∈ A
∈-cong A ≡.refl l∈A = l∈A
≈ᴸ : ∀ {a aℓ} → (A : Language a aℓ) → ∀ {l₁ l₂} → 𝕃 A l₁ → 𝕃 A l₂ → Set aℓ
≈ᴸ = IndexedSetoid._≈_
≈ᴸ-refl : ∀ {a aℓ} → (A : Language a aℓ) → Reflexive (𝕃 A) (≈ᴸ A)
≈ᴸ-refl = IsIndexedEquivalence.refl ∘ IndexedSetoid.isEquivalence
≈ᴸ-sym : ∀ {a aℓ} → (A : Language a aℓ) → Symmetric (𝕃 A) (≈ᴸ A)
≈ᴸ-sym = IsIndexedEquivalence.sym ∘ IndexedSetoid.isEquivalence
≈ᴸ-trans : ∀ {a aℓ} → (A : Language a aℓ) → Transitive (𝕃 A) (≈ᴸ A)
≈ᴸ-trans = IsIndexedEquivalence.trans ∘ IndexedSetoid.isEquivalence
≈ᴸ-cong : ∀ {a aℓ} → (A : Language a aℓ) → ∀ {l₁ l₂ l₃ l₄} →
(l₁≡l₂ : l₁ ≡ l₂) → (l₃≡l₄ : l₃ ≡ l₄) →
(l₁∈A : l₁ ∈ A) → (l₃∈A : l₃ ∈ A) →
≈ᴸ A l₁∈A l₃∈A →
≈ᴸ A (∈-cong A l₁≡l₂ l₁∈A) (∈-cong A l₃≡l₄ l₃∈A)
≈ᴸ-cong A ≡.refl ≡.refl l₁∈A l₃∈A eq = eq
record _≤_ {a aℓ b bℓ} (A : Language a aℓ) (B : Language b bℓ) : Set (c ⊔ a ⊔ aℓ ⊔ b ⊔ bℓ) where
field
f : ∀ {l} → l ∈ A → l ∈ B
cong : ∀ {l₁ l₂ l₁∈A l₂∈A} → ≈ᴸ A {l₁} {l₂} l₁∈A l₂∈A → ≈ᴸ B (f l₁∈A) (f l₂∈A)
record _≈_ {a aℓ b bℓ} (A : Language a aℓ) (B : Language b bℓ) : Set (c ⊔ ℓ ⊔ a ⊔ aℓ ⊔ b ⊔ bℓ) where
field
f : ∀ {l} → l ∈ A → l ∈ B
f⁻¹ : ∀ {l} → l ∈ B → l ∈ A
cong₁ : ∀ {l₁ l₂ l₁∈A l₂∈A} → ≈ᴸ A {l₁} {l₂} l₁∈A l₂∈A → ≈ᴸ B (f l₁∈A) (f l₂∈A)
cong₂ : ∀ {l₁ l₂ l₁∈B l₂∈B} → ≈ᴸ B {l₁} {l₂} l₁∈B l₂∈B → ≈ᴸ A (f⁻¹ l₁∈B) (f⁻¹ l₂∈B)
≈-refl : ∀ {a aℓ} → B.Reflexive (_≈_ {a} {aℓ})
≈-refl {x = A} = record
{ f = id
; f⁻¹ = id
; cong₁ = id
; cong₂ = id
}
≈-sym : ∀ {a aℓ b bℓ} → B.Sym (_≈_ {a} {aℓ} {b} {bℓ}) _≈_
≈-sym A≈B = record
{ f = A≈B.f⁻¹
; f⁻¹ = A≈B.f
; cong₁ = A≈B.cong₂
; cong₂ = A≈B.cong₁
}
where
module A≈B = _≈_ A≈B
≈-trans : ∀ {a aℓ b bℓ c cℓ} → B.Trans (_≈_ {a} {aℓ}) (_≈_ {b} {bℓ} {c} {cℓ}) _≈_
≈-trans {i = A} {B} {C} A≈B B≈C = record
{ f = B≈C.f ∘ A≈B.f
; f⁻¹ = A≈B.f⁻¹ ∘ B≈C.f⁻¹
; cong₁ = B≈C.cong₁ ∘ A≈B.cong₁
; cong₂ = A≈B.cong₂ ∘ B≈C.cong₂
}
where
module A≈B = _≈_ A≈B
module B≈C = _≈_ B≈C
≈-isEquivalence : ∀ {a aℓ} → B.IsEquivalence (_≈_ {a} {aℓ} {a} {aℓ})
≈-isEquivalence = record
{ refl = ≈-refl
; sym = ≈-sym
; trans = ≈-trans
}
setoid : ∀ a aℓ → B.Setoid (suc (c ⊔ a ⊔ aℓ)) (c ⊔ ℓ ⊔ a ⊔ aℓ)
setoid a aℓ = record { isEquivalence = ≈-isEquivalence {a} {aℓ} }
≤-refl : ∀ {a aℓ} → B.Reflexive (_≤_ {a} {aℓ})
≤-refl = record
{ f = id
; cong = id
}
≤-reflexive : ∀ {a aℓ b bℓ} → _≈_ {a} {aℓ} {b} {bℓ} B.⇒ _≤_
≤-reflexive A≈B = record
{ f = A≈B.f
; cong = A≈B.cong₁
}
where
module A≈B = _≈_ A≈B
≤-trans : ∀ {a aℓ b bℓ c cℓ} → B.Trans (_≤_ {a} {aℓ}) (_≤_ {b} {bℓ} {c} {cℓ}) _≤_
≤-trans A≤B B≤C = record
{ f = B≤C.f ∘ A≤B.f
; cong = B≤C.cong ∘ A≤B.cong
}
where
module A≤B = _≤_ A≤B
module B≤C = _≤_ B≤C
≤-antisym : ∀ {a aℓ b bℓ} → B.Antisym (_≤_ {a} {aℓ} {b} {bℓ}) _≤_ _≈_
≤-antisym A≤B B≤A = record
{ f = A≤B.f
; f⁻¹ = B≤A.f
; cong₁ = A≤B.cong
; cong₂ = B≤A.cong
}
where
module A≤B = _≤_ A≤B
module B≤A = _≤_ B≤A
≤-min : ∀ {b bℓ} → B.Min (_≤_ {b = b} {bℓ}) ∅
≤-min A = record
{ f = λ ()
; cong = λ {_} {_} {}
}
≤-isPartialOrder : ∀ a aℓ → B.IsPartialOrder (_≈_ {a} {aℓ}) _≤_
≤-isPartialOrder a aℓ = record
{ isPreorder = record
{ isEquivalence = ≈-isEquivalence
; reflexive = ≤-reflexive
; trans = ≤-trans
}
; antisym = ≤-antisym
}
poset : ∀ a aℓ → B.Poset (suc (c ⊔ a ⊔ aℓ)) (c ⊔ ℓ ⊔ a ⊔ aℓ) (c ⊔ a ⊔ aℓ)
poset a aℓ = record { isPartialOrder = ≤-isPartialOrder a aℓ }
null : ∀ {a} {aℓ} → Language a aℓ → Set a
null A = [] ∈ A
first : ∀ {a} {aℓ} → Language a aℓ → C → Set (c ⊔ a)
first A x = ∃[ l ] x ∷ l ∈ A
flast : ∀ {a} {aℓ} → Language a aℓ → C → Set (c ⊔ a)
flast A x = ∃[ l ] (l ≡.≢ [] × ∃[ l′ ] l ++ x ∷ l′ ∈ A)
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