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{-# OPTIONS --without-K --safe #-}
open import Relation.Binary
module Cfe.Language.Construct.Concatenate
{c ℓ} (over : Setoid c ℓ)
where
open import Algebra
open import Cfe.Language over as 𝕃
open import Data.Empty
open import Data.List hiding (null)
open import Data.List.Relation.Binary.Equality.Setoid over
open import Data.List.Properties
open import Data.Product as Product
open import Data.Unit using (⊤)
open import Function
open import Level
import Relation.Binary.PropositionalEquality as ≡
open import Relation.Nullary
open import Relation.Unary hiding (_∈_)
import Relation.Binary.Indexed.Heterogeneous as I
open Setoid over using () renaming (Carrier to C; _≈_ to _∼_; refl to ∼-refl; sym to ∼-sym; trans to ∼-trans)
module Compare where
data Compare : List C → List C → List C → List C → Set (c ⊔ ℓ) where
back : ∀ {xs zs} → (xs≋zs : xs ≋ zs) → Compare [] xs [] zs
left : ∀ {w ws xs z zs} → Compare ws xs [] zs → (w∼z : w ∼ z) → Compare (w ∷ ws) xs [] (z ∷ zs)
right : ∀ {x xs y ys zs} → Compare [] xs ys zs → (x∼y : x ∼ y) → Compare [] (x ∷ xs) (y ∷ ys) zs
front : ∀ {w ws xs y ys zs} → Compare ws xs ys zs → (w∼y : w ∼ y) → Compare (w ∷ ws) xs (y ∷ ys) zs
isLeft : ∀ {ws xs ys zs} → Compare ws xs ys zs → Set
isLeft (back xs≋zs) = ⊥
isLeft (left cmp w∼z) = ⊤
isLeft (right cmp x∼y) = ⊥
isLeft (front cmp w∼y) = isLeft cmp
isRight : ∀ {ws xs ys zs} → Compare ws xs ys zs → Set
isRight (back xs≋zs) = ⊥
isRight (left cmp w∼z) = ⊥
isRight (right cmp x∼y) = ⊤
isRight (front cmp w∼y) = isRight cmp
isEqual : ∀ {ws xs ys zs} → Compare ws xs ys zs → Set
isEqual (back xs≋zs) = ⊤
isEqual (left cmp w∼z) = ⊥
isEqual (right cmp x∼y) = ⊥
isEqual (front cmp w∼y) = isEqual cmp
<?> : ∀ {ws xs ys zs} → (cmp : Compare ws xs ys zs) → Tri (isLeft cmp) (isEqual cmp) (isRight cmp)
<?> (back xs≋zs) = tri≈ id _ id
<?> (left cmp w∼z) = tri< _ id id
<?> (right cmp x∼y) = tri> id id _
<?> (front cmp w∼y) = <?> cmp
compare : ∀ ws xs ys zs → ws ++ xs ≋ ys ++ zs → Compare ws xs ys zs
compare [] xs [] zs eq = back eq
compare [] (x ∷ xs) (y ∷ ys) zs (x∼y ∷ eq) = right (compare [] xs ys zs eq) x∼y
compare (w ∷ ws) xs [] (z ∷ zs) (w∼z ∷ eq) = left (compare ws xs [] zs eq) w∼z
compare (w ∷ ws) xs (y ∷ ys) zs (w∼y ∷ eq) = front (compare ws xs ys zs eq) w∼y
left-split : ∀ {ws xs ys zs} → (cmp : Compare ws xs ys zs) → isLeft cmp → ∃[ w ] ∃[ ws′ ] ws ≋ ys ++ w ∷ ws′ × w ∷ ws′ ++ xs ≋ zs
left-split (left (back xs≋zs) w∼z) _ = -, -, ≋-refl , w∼z ∷ xs≋zs
left-split (left (left cmp w′∼z′) w∼z) _ with left-split (left cmp w′∼z′) _
... | (_ , _ , eq₁ , eq₂) = -, -, ∼-refl ∷ eq₁ , w∼z ∷ eq₂
left-split (front cmp w∼y) l with left-split cmp l
... | (_ , _ , eq₁ , eq₂) = -, -, w∼y ∷ eq₁ , eq₂
right-split : ∀ {ws xs ys zs} → (cmp : Compare ws xs ys zs) → isRight cmp → ∃[ y ] ∃[ ys′ ] ws ++ y ∷ ys′ ≋ ys × xs ≋ y ∷ ys′ ++ zs
right-split (right (back xs≋zs) x∼y) _ = -, -, ≋-refl , x∼y ∷ xs≋zs
right-split (right (right cmp x′∼y′) x∼y) _ with right-split (right cmp x′∼y′) _
... | (_ , _ , eq₁ , eq₂) = -, -, ∼-refl ∷ eq₁ , x∼y ∷ eq₂
right-split (front cmp w∼y) r with right-split cmp r
... | (_ , _ , eq₁ , eq₂) = -, -, w∼y ∷ eq₁ , eq₂
eq-split : ∀ {ws xs ys zs} → (cmp : Compare ws xs ys zs) → isEqual cmp → ws ≋ ys × xs ≋ zs
eq-split (back xs≋zs) e = [] , xs≋zs
eq-split (front cmp w∼y) e = map₁ (w∼y ∷_) (eq-split cmp e)
module _
{a b}
(A : Language a)
(B : Language b)
where
private
module A = Language A
module B = Language B
infix 7 _∙_
record Concat (l : List C) : Set (c ⊔ ℓ ⊔ a ⊔ b) where
field
l₁ : List C
l₂ : List C
l₁∈A : l₁ ∈ A
l₂∈B : l₂ ∈ B
eq : l₁ ++ l₂ ≋ l
_∙_ : Language (c ⊔ ℓ ⊔ a ⊔ b)
_∙_ = record
{ 𝕃 = Concat
; ∈-resp-≋ = λ
{ l≋l′ record { l₁ = _ ; l₂ = _ ; l₁∈A = l₁∈A ; l₂∈B = l₂∈B ; eq = eq } → record
{ l₁∈A = l₁∈A ; l₂∈B = l₂∈B ; eq = ≋-trans eq l≋l′ }
}
}
∙-cong : ∀ {a} → Congruent₂ _≈_ (_∙_ {c ⊔ ℓ ⊔ a})
∙-cong X≈Y U≈V = record
{ f = λ
{ record { l₁∈A = l₁∈X ; l₂∈B = l₂∈Y ; eq = eq } → record
{ l₁∈A = X≈Y.f l₁∈X
; l₂∈B = U≈V.f l₂∈Y
; eq = eq
}
}
; f⁻¹ = λ
{ record { l₁∈A = l₁∈Y ; l₂∈B = l₂∈V ; eq = eq } → record
{ l₁∈A = X≈Y.f⁻¹ l₁∈Y
; l₂∈B = U≈V.f⁻¹ l₂∈V
; eq = eq
}
}
}
where
module X≈Y = _≈_ X≈Y
module U≈V = _≈_ U≈V
∙-assoc : ∀ {a b c} (A : Language a) (B : Language b) (C : Language c) →
(A ∙ B) ∙ C ≈ A ∙ (B ∙ C)
∙-assoc A B C = record
{ f = λ
{ record
{ l₂ = l₃
; l₁∈A = record { l₁ = l₁ ; l₂ = l₂ ; l₁∈A = l₁∈A ; l₂∈B = l₂∈B ; eq = eq₁ }
; l₂∈B = l₃∈C
; eq = eq₂
} → record
{ l₁∈A = l₁∈A
; l₂∈B = record
{ l₁∈A = l₂∈B
; l₂∈B = l₃∈C
; eq = ≋-refl
}
; eq = ≋-trans (≋-sym (≋-reflexive (++-assoc l₁ l₂ l₃))) (≋-trans (++⁺ eq₁ ≋-refl) eq₂)
}
}
; f⁻¹ = λ
{ record
{ l₁ = l₁
; l₁∈A = l₁∈A
; l₂∈B = record { l₁ = l₂ ; l₂ = l₃ ; l₁∈A = l₂∈B ; l₂∈B = l₃∈C ; eq = eq₁ }
; eq = eq₂
} → record
{ l₁∈A = record
{ l₁∈A = l₁∈A
; l₂∈B = l₂∈B
; eq = ≋-refl
}
; l₂∈B = l₃∈C
; eq = ≋-trans (≋-reflexive (++-assoc l₁ l₂ l₃)) (≋-trans (++⁺ ≋-refl eq₁) eq₂)
}
}
}
∙-identityˡ : ∀ {a} → LeftIdentity _≈_ (𝕃.Lift (ℓ ⊔ a) ｛ε｝) _∙_
∙-identityˡ X = record
{ f = λ
{ record { l₁ = [] ; l₂∈B = l∈X ; eq = eq } → X.∈-resp-≋ eq l∈X
}
; f⁻¹ = λ l∈X → record
{ l₁∈A = lift ≡.refl
; l₂∈B = l∈X
; eq = ≋-refl
}
}
where
module X = Language X
∙-unique-prefix : ∀ {a b} (A : Language a) (B : Language b) → Empty (flast A ∩ first B) → ¬ (null A) → ∀ {l} → (l∈A∙B l∈A∙B′ : l ∈ A ∙ B) → Concat.l₁ l∈A∙B ≋ Concat.l₁ l∈A∙B′ × Concat.l₂ l∈A∙B ≋ Concat.l₂ l∈A∙B′
∙-unique-prefix A B ∄[l₁∩f₂] ¬n₁ l∈A∙B l∈A∙B′ with Compare.compare (Concat.l₁ l∈A∙B) (Concat.l₂ l∈A∙B) (Concat.l₁ l∈A∙B′) (Concat.l₂ l∈A∙B′) (≋-trans (Concat.eq l∈A∙B) (≋-sym (Concat.eq l∈A∙B′)))
... | cmp with Compare.<?> cmp
... | tri< l _ _ = ⊥-elim (∄[l₁∩f₂] w ((-, (λ { ≡.refl → ¬n₁ (Concat.l₁∈A l∈A∙B′)}) , (Concat.l₁∈A l∈A∙B′) , -, A.∈-resp-≋ eq₃ (Concat.l₁∈A l∈A∙B)) , (-, B.∈-resp-≋ (≋-sym eq₄) (Concat.l₂∈B l∈A∙B′))))
where
module A = Language A
module B = Language B
lsplit = Compare.left-split cmp l
w = proj₁ lsplit
eq₃ = (proj₁ ∘ proj₂ ∘ proj₂) lsplit
eq₄ = (proj₂ ∘ proj₂ ∘ proj₂) lsplit
... | tri≈ _ e _ = Compare.eq-split cmp e
... | tri> _ _ r = ⊥-elim (∄[l₁∩f₂] w ((-, (λ { ≡.refl → ¬n₁ (Concat.l₁∈A l∈A∙B)}) , (Concat.l₁∈A l∈A∙B) , -, A.∈-resp-≋ (≋-sym eq₃) (Concat.l₁∈A l∈A∙B′)) , (-, (B.∈-resp-≋ eq₄ (Concat.l₂∈B l∈A∙B)))))
where
module A = Language A
module B = Language B
rsplit = Compare.right-split cmp r
w = proj₁ rsplit
eq₃ = (proj₁ ∘ proj₂ ∘ proj₂) rsplit
eq₄ = (proj₂ ∘ proj₂ ∘ proj₂) rsplit
∙-identityʳ : ∀ {a} → RightIdentity _≈_ (𝕃.Lift (ℓ ⊔ a) ｛ε｝) _∙_
∙-identityʳ X = record
{ f = λ
{ record { l₁ = l₁ ; l₂ = [] ; l₁∈A = l∈X ; eq = eq } → X.∈-resp-≋ (≋-trans (≋-sym (≋-reflexive (++-identityʳ l₁))) eq) l∈X
}
; f⁻¹ = λ {l} l∈X → record
{ l₁∈A = l∈X
; l₂∈B = lift ≡.refl
; eq = ≋-reflexive (++-identityʳ l)
}
}
where
module X = Language X
isMagma : ∀ {a} → IsMagma _≈_ (_∙_ {c ⊔ ℓ ⊔ a})
isMagma {a} = record
{ isEquivalence = ≈-isEquivalence
; ∙-cong = ∙-cong {a}
}
isSemigroup : ∀ {a} → IsSemigroup _≈_ (_∙_ {c ⊔ ℓ ⊔ a})
isSemigroup {a} = record
{ isMagma = isMagma {a}
; assoc = ∙-assoc
}
isMonoid : ∀ {a} → IsMonoid _≈_ _∙_ (𝕃.Lift (ℓ ⊔ a) ｛ε｝)
isMonoid {a} = record
{ isSemigroup = isSemigroup {a}
; identity = ∙-identityˡ {a} , ∙-identityʳ {a}
}
|
