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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
-
-∙-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
-
-∙-mono : ∀ {a b} → _∙_ Preserves₂ _≤_ {a} ⟶ _≤_ {b} ⟶ _≤_
-∙-mono X≤Y U≤V = record
-  { f = λ
-    { record { l₁∈A = l₁∈A ; l₂∈B = l₂∈B ; eq = eq } → record
-      { l₁∈A = X≤Y.f l₁∈A
-      ; l₂∈B = U≤V.f l₂∈B
-      ; eq = eq
-      }
-    }
-  }
-  where
-  module X≤Y = _≤_ X≤Y
-  module U≤V = _≤_ U≤V
-
-∙-unique : ∀ {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 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
-
-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}
-  }