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Diffstat (limited to 'src/Cfe/Language/Construct/Single.agda')
| -rw-r--r-- | src/Cfe/Language/Construct/Single.agda | 61 |
1 files changed, 7 insertions, 54 deletions
diff --git a/src/Cfe/Language/Construct/Single.agda b/src/Cfe/Language/Construct/Single.agda index daa1628..b06be3d 100644 --- a/src/Cfe/Language/Construct/Single.agda +++ b/src/Cfe/Language/Construct/Single.agda @@ -6,70 +6,23 @@ import Relation.Binary.PropositionalEquality as ≡ module Cfe.Language.Construct.Single {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 over renaming (Carrier to C) open import Cfe.Language over hiding (_≈_) open import Data.List -open import Data.List.Relation.Binary.Equality.Setoid over open import Data.Product as Product +open import Data.Unit open import Level -private - ∷-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 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 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 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 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 - ≋-trans-symₗ {_} {_} {_} {_ ∷ _} {_ ∷ _} {_ ∷ _} = uncurry (≡.cong₂ _∷_) - ∘ Product.map ≈-trans-symₗ ≋-trans-symₗ - ∘ ∷-inj - - ≋-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 - → ≋-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) - -｛_｝ : C → Language (c ⊔ ℓ) (c ⊔ ℓ) +｛_｝ : C → Language (c ⊔ ℓ) 0ℓ ｛ c ｝ = record - { Carrier = [ c ] ≋_ - ; _≈_ = λ l≋m l≋n → ∃[ m≋n ] ≋-trans l≋m m≋n ≡.≡ l≋n + { Carrier = λ l → ∃[ a ] (l ≡.≡ [ a ] × a ≈ c) + ; _≈_ = λ _ _ → ⊤ ; isEquivalence = record - { refl = ≋-refl , ≋-trans-reflₗ - ; sym = Product.map ≋-sym ≋-trans-symₗ - ; trans = Product.zip ≋-trans ≋-trans-transₗ + { refl = tt + ; sym = λ _ → tt + ; trans = λ _ _ → tt } } |