diff options
Diffstat (limited to 'src/Cfe/Language/Construct')
| -rw-r--r-- | src/Cfe/Language/Construct/Concatenate.agda | 51 | ||||
| -rw-r--r-- | src/Cfe/Language/Construct/Single.agda | 71 | ||||
| -rw-r--r-- | src/Cfe/Language/Construct/Union.agda | 99 |
3 files changed, 73 insertions, 148 deletions
diff --git a/src/Cfe/Language/Construct/Concatenate.agda b/src/Cfe/Language/Construct/Concatenate.agda index b75f822..29e635d 100644 --- a/src/Cfe/Language/Construct/Concatenate.agda +++ b/src/Cfe/Language/Construct/Concatenate.agda @@ -4,56 +4,37 @@ open import Relation.Binary import Cfe.Language module Cfe.Language.Construct.Concatenate - {c ℓ a aℓ b bℓ} (setoid : Setoid c ℓ) - (A : Cfe.Language.Language setoid a aℓ) - (B : Cfe.Language.Language setoid b bℓ) + {c ℓ a aℓ b bℓ} (over : Setoid c ℓ) + (A : Cfe.Language.Language over a aℓ) + (B : Cfe.Language.Language over b bℓ) where open import Data.Empty open import Data.List -open import Data.List.Relation.Binary.Equality.Setoid setoid +open import Data.List.Relation.Binary.Equality.Setoid over open import Data.Product as Product open import Function open import Level -open import Cfe.Language setoid -open Language +open import Cfe.Language over -open Setoid setoid renaming (Carrier to C) +open Setoid over renaming (Carrier to C) infix 4 _≈ᶜ_ infix 4 _∙_ Concat : List C → Set (c ⊔ ℓ ⊔ a ⊔ b) -Concat l = ∃[ l₁ ](l₁ ∈ A × ∃[ l₂ ](l₂ ∈ B × (l₁ ++ l₂ ≋ l))) +Concat l = ∃[ l₁ ] l₁ ∈ A × ∃[ l₂ ] l₂ ∈ B × l₁ ++ l₂ ≋ l -_≈ᶜ_ : {l : List C} → Rel (Concat l) (c ⊔ ℓ ⊔ aℓ ⊔ bℓ) -(l₁ , l₁∈A , l₂ , l₂∈B , l₁++l₂) ≈ᶜ (l₁′ , l₁′∈A , l₂′ , l₂′∈B , l₁′++l₂′) = - ∃[ l₁≋l₁′ ](_≈ᴸ_ A (⤖ A l₁≋l₁′ l₁∈A) l₁′∈A) - × ∃[ l₂≋l₂′ ](_≈ᴸ_ B (⤖ B l₂≋l₂′ l₂∈B) l₂′∈B) +_≈ᶜ_ : {l₁ l₂ : List C} → REL (Concat l₁) (Concat l₂) (aℓ ⊔ bℓ) +(_ , l₁∈A , _ , l₂∈B , _) ≈ᶜ (_ , l₁′∈A , _ , l₂′∈B , _) = (≈ᴸ A l₁∈A l₁′∈A) × (≈ᴸ B l₂∈B l₂′∈B) -⤖ᶜ : {l₁ l₂ : List C} → l₁ ≋ l₂ → Concat l₁ → Concat l₂ -⤖ᶜ l₁≋l₂ = map₂ (map₂ (map₂ (map₂ (flip ≋-trans l₁≋l₂)))) - -_∙_ : Language (c ⊔ ℓ ⊔ a ⊔ b) (c ⊔ ℓ ⊔ aℓ ⊔ bℓ) +_∙_ : Language (c ⊔ ℓ ⊔ a ⊔ b) (aℓ ⊔ bℓ) _∙_ = record - { 𝕃 = Concat - ; _≈ᴸ_ = _≈ᶜ_ - ; ⤖ = ⤖ᶜ - ; isLanguage = record - { ≈ᴸ-isEquivalence = record - { refl = (≋-refl , ⤖-refl A) , (≋-refl , ⤖-refl B) - ; sym = Product.map (Product.map ≋-sym (⤖-sym A)) - (Product.map ≋-sym (⤖-sym B)) - ; trans = Product.zip (Product.zip ≋-trans (⤖-trans A)) - (Product.zip ≋-trans (⤖-trans B)) - } - ; ⤖-cong = id - ; ⤖-bijective = λ {_} {_} {l₁≋l₂} → id , λ l₂∈𝕃 → - ⤖ᶜ (≋-sym l₁≋l₂) l₂∈𝕃 , (≋-refl , ⤖-refl A) , (≋-refl , ⤖-refl B) - ; ⤖-refl = (≋-refl , ⤖-refl A) , (≋-refl , ⤖-refl B) - ; ⤖-sym = Product.map (Product.map ≋-sym (⤖-sym A)) - (Product.map ≋-sym (⤖-sym B)) - ; ⤖-trans = Product.zip (Product.zip ≋-trans (⤖-trans A)) - (Product.zip ≋-trans (⤖-trans B)) + { Carrier = Concat + ; _≈_ = _≈ᶜ_ + ; isEquivalence = record + { refl = ≈ᴸ-refl A , ≈ᴸ-refl B + ; sym = Product.map (≈ᴸ-sym A) (≈ᴸ-sym B) + ; trans = Product.zip (≈ᴸ-trans A) (≈ᴸ-trans B) } } diff --git a/src/Cfe/Language/Construct/Single.agda b/src/Cfe/Language/Construct/Single.agda index f54e015..daa1628 100644 --- a/src/Cfe/Language/Construct/Single.agda +++ b/src/Cfe/Language/Construct/Single.agda @@ -5,56 +5,47 @@ open import Relation.Binary import Relation.Binary.PropositionalEquality as ≡ module Cfe.Language.Construct.Single - {a ℓ} (setoid : Setoid a ℓ) - (≈-trans-bijₗ : ∀ {a b c : Setoid.Carrier setoid} - → {b≈c : Setoid._≈_ setoid b c} - → Bijective ≡._≡_ ≡._≡_ (flip (Setoid.trans setoid {a}) b≈c)) - (≈-trans-reflₗ : ∀ {a b : Setoid.Carrier setoid} {a≈b : Setoid._≈_ setoid a b} - → Setoid.trans setoid a≈b (Setoid.refl setoid) ≡.≡ a≈b) - (≈-trans-symₗ : ∀ {a b c : Setoid.Carrier setoid} - → {a≈b : Setoid._≈_ setoid a b} - → {a≈c : Setoid._≈_ setoid a c} - → {b≈c : Setoid._≈_ setoid b c} - → Setoid.trans setoid a≈b b≈c ≡.≡ a≈c - → Setoid.trans setoid a≈c (Setoid.sym setoid b≈c) ≡.≡ a≈b) - (≈-trans-transₗ : ∀ {a b c d : Setoid.Carrier setoid} - → {a≈b : Setoid._≈_ setoid a b} - → {a≈c : Setoid._≈_ setoid a c} - → {a≈d : Setoid._≈_ setoid a d} - → {b≈c : Setoid._≈_ setoid b c} - → {c≈d : Setoid._≈_ setoid c d} - → Setoid.trans setoid a≈b b≈c ≡.≡ a≈c - → Setoid.trans setoid a≈c c≈d ≡.≡ a≈d - → Setoid.trans setoid a≈b (Setoid.trans setoid b≈c c≈d) ≡.≡ a≈d) + {c ℓ} (over : Setoid c ℓ) + (≈-trans-bijₗ : ∀ {a b c b≈c} + → Bijective ≡._≡_ ≡._≡_ (flip (Setoid.trans over {a} {b} {c}) b≈c)) + (≈-trans-reflₗ : ∀ {a b a≈b} + → Setoid.trans over {a} a≈b (Setoid.refl over {b}) ≡.≡ a≈b) + (≈-trans-symₗ : ∀ {a b c a≈b a≈c b≈c} + → Setoid.trans over {a} {b} {c} a≈b b≈c ≡.≡ a≈c + → Setoid.trans over a≈c (Setoid.sym over b≈c) ≡.≡ a≈b) + (≈-trans-transₗ : ∀ {a b c d a≈b a≈c a≈d b≈c c≈d} + → Setoid.trans over {a} {b} a≈b b≈c ≡.≡ a≈c + → Setoid.trans over {a} {c} {d} a≈c c≈d ≡.≡ a≈d + → Setoid.trans over a≈b (Setoid.trans over b≈c c≈d) ≡.≡ a≈d) where -open Setoid setoid renaming (Carrier to A) +open Setoid over renaming (Carrier to C) -open import Cfe.Language setoid +open import Cfe.Language over hiding (_≈_) open import Data.List -open import Data.List.Relation.Binary.Equality.Setoid setoid +open import Data.List.Relation.Binary.Equality.Setoid over open import Data.Product as Product open import Level private - ∷-inj : {a b : A} {l₁ l₂ : List A} {a≈b a≈b′ : a ≈ b} {l₁≋l₂ l₁≋l₂′ : l₁ ≋ l₂} → ≡._≡_ {A = a ∷ l₁ ≋ b ∷ l₂} (a≈b ∷ l₁≋l₂) (a≈b′ ∷ l₁≋l₂′) → (a≈b ≡.≡ a≈b′) × (l₁≋l₂ ≡.≡ l₁≋l₂′) + ∷-inj : {a b : C} {l₁ l₂ : List C} {a≈b a≈b′ : a ≈ b} {l₁≋l₂ l₁≋l₂′ : l₁ ≋ l₂} → ≡._≡_ {A = a ∷ l₁ ≋ b ∷ l₂} (a≈b ∷ l₁≋l₂) (a≈b′ ∷ l₁≋l₂′) → (a≈b ≡.≡ a≈b′) × (l₁≋l₂ ≡.≡ l₁≋l₂′) ∷-inj ≡.refl = ≡.refl , ≡.refl - ≋-trans-injₗ : {x l₁ l₂ : List A} → {l₁≋l₂ : l₁ ≋ l₂} → Injective ≡._≡_ ≡._≡_ (flip (≋-trans {x}) l₁≋l₂) + ≋-trans-injₗ : {x l₁ l₂ : List C} → {l₁≋l₂ : l₁ ≋ l₂} → Injective ≡._≡_ ≡._≡_ (flip (≋-trans {x}) l₁≋l₂) ≋-trans-injₗ {_} {_} {_} {_} {[]} {[]} _ = ≡.refl ≋-trans-injₗ {_} {_} {_} {_ ∷ _} {_ ∷ _} {_ ∷ _} = uncurry (≡.cong₂ _∷_) ∘ Product.map (proj₁ ≈-trans-bijₗ) ≋-trans-injₗ ∘ ∷-inj - ≋-trans-surₗ : {x l₁ l₂ : List A} → {l₁≋l₂ : l₁ ≋ l₂} → Surjective {A = x ≋ l₁} ≡._≡_ ≡._≡_ (flip (≋-trans {x}) l₁≋l₂) + ≋-trans-surₗ : {x l₁ l₂ : List C} → {l₁≋l₂ : l₁ ≋ l₂} → Surjective {A = x ≋ l₁} ≡._≡_ ≡._≡_ (flip (≋-trans {x}) l₁≋l₂) ≋-trans-surₗ {_} {_} {_} {[]} [] = [] , ≡.refl ≋-trans-surₗ {_} {_} {_} {_ ∷ _} (a≈c ∷ x≋l₂) = Product.zip _∷_ (≡.cong₂ _∷_) (proj₂ ≈-trans-bijₗ a≈c) (≋-trans-surₗ x≋l₂) - ≋-trans-reflₗ : {l₁ l₂ : List A} {l₁≋l₂ : l₁ ≋ l₂} → ≋-trans l₁≋l₂ ≋-refl ≡.≡ l₁≋l₂ + ≋-trans-reflₗ : {l₁ l₂ : List C} {l₁≋l₂ : l₁ ≋ l₂} → ≋-trans l₁≋l₂ ≋-refl ≡.≡ l₁≋l₂ ≋-trans-reflₗ {_} {_} {[]} = ≡.refl ≋-trans-reflₗ {_} {_} {a≈b ∷ l₁≋l₂} = ≡.cong₂ _∷_ ≈-trans-reflₗ ≋-trans-reflₗ - ≋-trans-symₗ : {l₁ l₂ l₃ : List A} {l₁≋l₂ : l₁ ≋ l₂} {l₁≋l₃ : l₁ ≋ l₃} {l₂≋l₃ : l₂ ≋ l₃} + ≋-trans-symₗ : {l₁ l₂ l₃ : List C} {l₁≋l₂ : l₁ ≋ l₂} {l₁≋l₃ : l₁ ≋ l₃} {l₂≋l₃ : l₂ ≋ l₃} → ≋-trans l₁≋l₂ l₂≋l₃ ≡.≡ l₁≋l₃ → ≋-trans l₁≋l₃ (≋-sym l₂≋l₃) ≡.≡ l₁≋l₂ ≋-trans-symₗ {_} {_} {_} {[]} {[]} {[]} _ = ≡.refl @@ -62,7 +53,7 @@ private ∘ Product.map ≈-trans-symₗ ≋-trans-symₗ ∘ ∷-inj - ≋-trans-transₗ : {l₁ l₂ l₃ l₄ : List A} + ≋-trans-transₗ : {l₁ l₂ l₃ l₄ : List C} → {l₁≋l₂ : l₁ ≋ l₂} {l₁≋l₃ : l₁ ≋ l₃} {l₁≋l₄ : l₁ ≋ l₄} {l₂≋l₃ : l₂ ≋ l₃} {l₃≋l₄ : l₃ ≋ l₄} → ≋-trans l₁≋l₂ l₂≋l₃ ≡.≡ l₁≋l₃ → ≋-trans l₁≋l₃ l₃≋l₄ ≡.≡ l₁≋l₄ @@ -72,17 +63,13 @@ private ∘₂ uncurry (Product.zip ≈-trans-transₗ ≋-trans-transₗ) ∘₂ curry (Product.map ∷-inj ∷-inj) -｛_｝ : List A → Language (a ⊔ ℓ) (a ⊔ ℓ) -｛ l ｝ = record - { 𝕃 = l ≋_ - ; _≈ᴸ_ = ≡._≡_ - ; ⤖ = flip ≋-trans - ; isLanguage = record - { ≈ᴸ-isEquivalence = ≡.isEquivalence - ; ⤖-cong = λ {_} {_} {l₁≋l₂} → ≡.cong (flip ≋-trans l₁≋l₂) - ; ⤖-bijective = ≋-trans-injₗ , ≋-trans-surₗ - ; ⤖-refl = ≋-trans-reflₗ - ; ⤖-sym = ≋-trans-symₗ - ; ⤖-trans = ≋-trans-transₗ +｛_｝ : C → Language (c ⊔ ℓ) (c ⊔ ℓ) +｛ c ｝ = record + { Carrier = [ c ] ≋_ + ; _≈_ = λ l≋m l≋n → ∃[ m≋n ] ≋-trans l≋m m≋n ≡.≡ l≋n + ; isEquivalence = record + { refl = ≋-refl , ≋-trans-reflₗ + ; sym = Product.map ≋-sym ≋-trans-symₗ + ; trans = Product.zip ≋-trans ≋-trans-transₗ } } diff --git a/src/Cfe/Language/Construct/Union.agda b/src/Cfe/Language/Construct/Union.agda index 44d4c3f..ee8b0f7 100644 --- a/src/Cfe/Language/Construct/Union.agda +++ b/src/Cfe/Language/Construct/Union.agda @@ -4,99 +4,56 @@ open import Relation.Binary import Cfe.Language module Cfe.Language.Construct.Union - {c ℓ a aℓ b bℓ} (setoid : Setoid c ℓ) - (A : Cfe.Language.Language setoid a aℓ) - (B : Cfe.Language.Language setoid b bℓ) + {c ℓ a aℓ b bℓ} (over : Setoid c ℓ) + (A : Cfe.Language.Language over a aℓ) + (B : Cfe.Language.Language over b bℓ) where open import Data.Empty open import Data.List -open import Data.List.Relation.Binary.Equality.Setoid setoid +open import Data.List.Relation.Binary.Equality.Setoid over open import Data.Product as Product open import Data.Sum as Sum open import Function open import Level -open import Cfe.Language setoid -open Language +open import Cfe.Language over hiding (Lift) -open Setoid setoid renaming (Carrier to C) +open Setoid over renaming (Carrier to C) infix 4 _≈ᵁ_ infix 4 _∪_ Union : List C → Set (a ⊔ b) -Union l = 𝕃 A l ⊎ 𝕃 B l +Union l = l ∈ A ⊎ l ∈ B -_≈ᵁ_ : {l : List C} → Rel (Union l) (aℓ ⊔ bℓ) -(inj₁ x) ≈ᵁ (inj₁ y) = Lift bℓ (_≈ᴸ_ A x y) +_≈ᵁ_ : {l₁ l₂ : List C} → REL (Union l₁) (Union l₂) (aℓ ⊔ bℓ) +(inj₁ x) ≈ᵁ (inj₁ y) = Lift bℓ (≈ᴸ A x y) (inj₁ _) ≈ᵁ (inj₂ _) = Lift (aℓ ⊔ bℓ) ⊥ (inj₂ _) ≈ᵁ (inj₁ _) = Lift (aℓ ⊔ bℓ) ⊥ -(inj₂ x) ≈ᵁ (inj₂ y) = Lift aℓ (_≈ᴸ_ B x y) - -⤖ᵁ : {l₁ l₂ : List C} → l₁ ≋ l₂ → Union l₁ → Union l₂ -⤖ᵁ l₁≋l₂ = Sum.map (⤖ A l₁≋l₂) (⤖ B l₁≋l₂) +(inj₂ x) ≈ᵁ (inj₂ y) = Lift aℓ (≈ᴸ B x y) _∪_ : Language (a ⊔ b) (aℓ ⊔ bℓ) _∪_ = record - { 𝕃 = Union - ; _≈ᴸ_ = _≈ᵁ_ - ; ⤖ = ⤖ᵁ - ; isLanguage = record - { ≈ᴸ-isEquivalence = record - { refl = λ {x} → case x return (λ x → _≈ᵁ_ x x) of λ - { (inj₁ x) → lift (≈ᴸ-refl A) - ; (inj₂ y) → lift (≈ᴸ-refl B) - } - ; sym = λ {x} {y} x≈ᵁy → - case (∃[ x ](∃[ y ](x ≈ᵁ y)) ∋ x , y , x≈ᵁy) - return (λ (x , y , _) → y ≈ᵁ x) of λ - { (inj₁ x , inj₁ y , lift x≈ᵁy) → lift (≈ᴸ-sym A x≈ᵁy) - ; (inj₂ y₁ , inj₂ y , lift x≈ᵁy) → lift (≈ᴸ-sym B x≈ᵁy) - } - ; trans = λ {i} {j} {k} i≈ᵁj j≈ᵁk → - case ∃[ i ](∃[ j ](∃[ k ](i ≈ᵁ j × j ≈ᵁ k))) ∋ i , j , k , i≈ᵁj , j≈ᵁk - return (λ (i , _ , k , _) → i ≈ᵁ k) of λ - { (inj₁ _ , inj₁ _ , inj₁ _ , lift x≈ᵁy , lift y≈ᵁz) → - lift (≈ᴸ-trans A x≈ᵁy y≈ᵁz) - ; (inj₂ _ , inj₂ _ , inj₂ _ , lift x≈ᵁy , lift y≈ᵁz) → - lift (≈ᴸ-trans B x≈ᵁy y≈ᵁz) - } - } - ; ⤖-cong = λ {_} {_} {l₁≋l₂} {x} {y} x≈ᵁy → - case ∃[ x ](∃[ y ](x ≈ᵁ y)) ∋ x , y , x≈ᵁy - return (λ (x , y , _) → (_≈ᵁ_ on ⤖ᵁ l₁≋l₂) x y) of λ - { (inj₁ x , inj₁ y , lift x≈ᵁy) → lift (⤖-cong A x≈ᵁy) - ; (inj₂ x , inj₂ y , lift x≈ᵁy) → lift (⤖-cong B x≈ᵁy) - } - ; ⤖-bijective = λ {_} {_} {l₁≋l₂} → - ( λ {x} {y} x≈ᵁy → - case ∃[ x ](∃[ y ]((_≈ᵁ_ on ⤖ᵁ l₁≋l₂) x y)) ∋ x , y , x≈ᵁy - return (λ (x , y , _) → x ≈ᵁ y) of λ - { (inj₁ x , inj₁ y , lift x≈ᵁy) → lift (⤖-injective A x≈ᵁy) - ; (inj₂ x , inj₂ y , lift x≈ᵁy) → lift (⤖-injective B x≈ᵁy) - }) - , ( λ - { (inj₁ x) → Product.map inj₁ lift (⤖-surjective A x) - ; (inj₂ x) → Product.map inj₂ lift (⤖-surjective B x) - }) - ; ⤖-refl = λ {_} {x} → case x return (λ x → ⤖ᵁ ≋-refl x ≈ᵁ x) of λ - { (inj₁ x) → lift (⤖-refl A) - ; (inj₂ y) → lift (⤖-refl B) + { Carrier = Union + ; _≈_ = _≈ᵁ_ + ; isEquivalence = record + { refl = λ {_} {x} → case x return (λ x → x ≈ᵁ x) of λ + { (inj₁ x) → lift (≈ᴸ-refl A) + ; (inj₂ y) → lift (≈ᴸ-refl B) } - ; ⤖-sym = λ {_} {_} {x} {y} {l₁≋l₂} x≈ᵁy → - case ∃[ x ](∃[ y ](⤖ᵁ l₁≋l₂ x ≈ᵁ y)) ∋ x , y , x≈ᵁy - return (λ (x , y , _) → ⤖ᵁ (≋-sym l₁≋l₂) y ≈ᵁ x) of λ - { (inj₁ x , inj₁ y , lift x≈ᵁy) → lift (⤖-sym A x≈ᵁy) - ; (inj₂ x , inj₂ y , lift x≈ᵁy) → lift (⤖-sym B x≈ᵁy) + ; sym = λ {_} {_} {x} {y} x≈ᵁy → + case (∃[ x ] ∃[ y ] x ≈ᵁ y ∋ x , y , x≈ᵁy) + return (λ (x , y , _) → y ≈ᵁ x) of λ + { (inj₁ x , inj₁ y , lift x≈ᵁy) → lift (≈ᴸ-sym A x≈ᵁy) + ; (inj₂ y₁ , inj₂ y , lift x≈ᵁy) → lift (≈ᴸ-sym B x≈ᵁy) } - ; ⤖-trans = λ {_} {_} {_} {x} {y} {z} {l₁≋l₂} {l₂≋l₃} x≈ᵁy y≈ᵁz → - case (∃[ x ](∃[ y ](∃[ z ]((⤖ᵁ l₁≋l₂ x ≈ᵁ y) × (⤖ᵁ l₂≋l₃ y ≈ᵁ z))))) ∋ - x , y , z , x≈ᵁy , y≈ᵁz - return (λ (x , _ , z , _ , _) → ⤖ᵁ (≋-trans l₁≋l₂ l₂≋l₃) x ≈ᵁ z) of λ - { (inj₁ x , inj₁ y , inj₁ z , lift x≈ᵁy , lift y≈ᵁz) → - lift (⤖-trans A x≈ᵁy y≈ᵁz) - ; (inj₂ x , inj₂ y , inj₂ z , lift x≈ᵁy , lift y≈ᵁz) → - lift (⤖-trans B x≈ᵁy y≈ᵁz) + ; trans = λ {_} {_} {_} {x} {y} {z} x≈ᵁy y≈ᵁz → + case ∃[ x ] ∃[ y ] ∃[ z ] x ≈ᵁ y × y ≈ᵁ z ∋ x , y , z , x≈ᵁy , y≈ᵁz + return (λ (x , _ , z , _) → x ≈ᵁ z) of λ + { (inj₁ _ , inj₁ _ , inj₁ _ , lift x≈ᵁy , lift y≈ᵁz) → + lift (≈ᴸ-trans A x≈ᵁy y≈ᵁz) + ; (inj₂ _ , inj₂ _ , inj₂ _ , lift x≈ᵁy , lift y≈ᵁz) → + lift (≈ᴸ-trans B x≈ᵁy y≈ᵁz) } } } |