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Diffstat (limited to 'src/Cfe/Language/Construct/Single.agda')
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diff --git a/src/Cfe/Language/Construct/Single.agda b/src/Cfe/Language/Construct/Single.agda new file mode 100644 index 0000000..f54e015 --- /dev/null +++ b/src/Cfe/Language/Construct/Single.agda @@ -0,0 +1,88 @@ +{-# OPTIONS --without-K --safe #-} + +open import Function +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) + where + +open Setoid setoid renaming (Carrier to A) + +open import Cfe.Language setoid +open import Data.List +open import Data.List.Relation.Binary.Equality.Setoid setoid +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 ≡.refl = ≡.refl , ≡.refl + + ≋-trans-injₗ : {x l₁ l₂ : List A} → {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 {_} {_} {_} {[]} [] = [] , ≡.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 {_} {_} {[]} = ≡.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 l₁≋l₂ l₂≋l₃ ≡.≡ l₁≋l₃ + → ≋-trans l₁≋l₃ (≋-sym l₂≋l₃) ≡.≡ l₁≋l₂ + ≋-trans-symₗ {_} {_} {_} {[]} {[]} {[]} _ = ≡.refl + ≋-trans-symₗ {_} {_} {_} {_ ∷ _} {_ ∷ _} {_ ∷ _} = uncurry (≡.cong₂ _∷_) + ∘ Product.map ≈-trans-symₗ ≋-trans-symₗ + ∘ ∷-inj + + ≋-trans-transₗ : {l₁ l₂ l₃ l₄ : List A} + → {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 + → ≋-trans l₁≋l₂ (≋-trans l₂≋l₃ l₃≋l₄) ≡.≡ l₁≋l₄ + ≋-trans-transₗ {l₁≋l₂ = []} {[]} {[]} {[]} {[]} _ _ = ≡.refl + ≋-trans-transₗ {l₁≋l₂ = _ ∷ _} {_ ∷ _} {_ ∷ _} {_ ∷ _} {_ ∷ _} = uncurry (≡.cong₂ _∷_) + ∘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 + } + } |