1 files changed, 85 insertions, 2 deletions

diff --git a/src/Cfe/Language/Construct/Concatenate.agda b/src/Cfe/Language/Construct/Concatenate.agda
index 428e8a4..8dff2ff 100644
--- a/src/Cfe/Language/Construct/Concatenate.agda
+++ b/src/Cfe/Language/Construct/Concatenate.agda
@@ -9,16 +9,19 @@ module Cfe.Language.Construct.Concatenate
 open import Algebra
 open import Cfe.Language over as 𝕃
 open import Data.Empty
-open import Data.List
+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
 open 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)
+open Setoid over using () renaming (Carrier to C; _≈_ to _∼_; refl to ∼-refl; sym to ∼-sym; trans to ∼-trans)
 
 module _
   {a b}
@@ -85,3 +88,83 @@ isMonoid {a} = record
   where
   module X≤Y = _≤_ X≤Y
   module U≤V = _≤_ U≤V
+
+private
+  data Compare : List C → List C → List C → List C → Set (c ⊔ ℓ) where
+    -- left : ∀ {ws₁ w ws₂ xs ys z zs₁ zs₂} → (ws₁≋ys : ws₁ ≋ ys) → (w∼z : w ∼ z) → (ws₂≋zs₁ : ws₂ ≋ zs₁) → (xs≋zs₂ : xs ≋ zs₂) → Compare (ws₁ ++ w ∷ ws₂) xs ys (z ∷ zs₁ ++ zs₂)
+    -- right : ∀ {ws x xs₁ xs₂ ys₁ y ys₂ zs} → (ws≋ys₁ : ws ≋ ys₁) → (x∼y : x ∼ y) → (xs₁≋ys₂ : xs₁ ≋ ys₂) → (xs₂≋zs : xs₂ ≋ zs) → Compare ws (x ∷ xs₁ ++ xs₂) (ys₁ ++ y ∷ ys₂) zs
+    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
+  eq-split (back xs≋zs) e = []
+  eq-split (front cmp w∼y) e = w∼y ∷ eq-split cmp e
+
+∙-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) → proj₁ l∈A∙B ≋ proj₁ l∈A∙B′
+∙-unique-prefix _ _ _ ¬n₁ ([] , l₁∈A , _) _ = ⊥-elim (¬n₁ l₁∈A)
+∙-unique-prefix _ _ _ ¬n₁ (_ ∷ _ , _) ([] , l₁′∈A , _) = ⊥-elim (¬n₁ l₁′∈A)
+∙-unique-prefix A B ∄[l₁∩f₂] _ (c ∷ l₁ , l₁∈A , l₂ , l₂∈B , eq₁) (c′ ∷ l₁′ , l₁′∈A , l₂′ , l₂′∈B , eq₂) with compare (c ∷ l₁) l₂ (c′ ∷ l₁′) l₂′ (≋-trans eq₁ (≋-sym eq₂))
+... | cmp with <?> cmp
+... | tri< l _ _ = ⊥-elim (∄[l₁∩f₂] w ((-, (λ ()) , l₁′∈A , -, A.∈-resp-≋ eq₃ l₁∈A) , (-, B.∈-resp-≋ (≋-sym eq₄) l₂′∈B)))
+  where
+  module A = Language A
+  module B = Language B
+  lsplit = left-split cmp l
+  w = proj₁ lsplit
+  eq₃ = (proj₁ ∘ proj₂ ∘ proj₂) lsplit
+  eq₄ = (proj₂ ∘ proj₂ ∘ proj₂) lsplit
+... | tri≈ _ e _ = eq-split cmp e
+... | tri> _ _ r = ⊥-elim (∄[l₁∩f₂] w ((-, (λ ()) , l₁∈A , -, A.∈-resp-≋ (≋-sym eq₃) l₁′∈A) , (-, (B.∈-resp-≋ eq₄ l₂∈B))))
+  where
+  module A = Language A
+  module B = Language B
+  rsplit = right-split cmp r
+  w = proj₁ rsplit
+  eq₃ = (proj₁ ∘ proj₂ ∘ proj₂) rsplit
+  eq₄ = (proj₂ ∘ proj₂ ∘ proj₂) rsplit