From 01ec93c5a03f6c4c660aa593b4c00afccc48907a Mon Sep 17 00:00:00 2001
From: Chloe Brown <chloe.brown.00@outlook.com>
Date: Mon, 8 Feb 2021 17:40:04 +0000
Subject: Make languages records with more properties.

---
 src/Cfe/Language/Construct/Union.agda | 102 ++++++++++++++++++++++++++++++++++
 1 file changed, 102 insertions(+)
 create mode 100644 src/Cfe/Language/Construct/Union.agda

(limited to 'src/Cfe/Language/Construct/Union.agda')

diff --git a/src/Cfe/Language/Construct/Union.agda b/src/Cfe/Language/Construct/Union.agda
new file mode 100644
index 0000000..44d4c3f
--- /dev/null
+++ b/src/Cfe/Language/Construct/Union.agda
@@ -0,0 +1,102 @@
+{-# OPTIONS --without-K --safe #-}
+
+open import Relation.Binary
+import Cfe.Language
+
+module Cfe.Language.Construct.Union
+  {c ℓ a aℓ b bℓ} (setoid : Setoid c ℓ)
+  (A : Cfe.Language.Language setoid a aℓ)
+  (B : Cfe.Language.Language setoid b bℓ)
+  where
+
+open import Data.Empty
+open import Data.List
+open import Data.List.Relation.Binary.Equality.Setoid setoid
+open import Data.Product as Product
+open import Data.Sum as Sum
+open import Function
+open import Level
+open import Cfe.Language setoid
+open Language
+
+open Setoid setoid renaming (Carrier to C)
+
+infix 4 _≈ᵁ_
+infix 4 _∪_
+
+Union : List C → Set (a ⊔ b)
+Union l = 𝕃 A l ⊎ 𝕃 B l
+
+_≈ᵁ_ : {l : List C} → Rel (Union l) (aℓ ⊔ bℓ)
+(inj₁ x) ≈ᵁ (inj₁ y) = Lift bℓ (_≈ᴸ_ A x y)
+(inj₁ _) ≈ᵁ (inj₂ _) = Lift (aℓ ⊔ bℓ) ⊥
+(inj₂ _) ≈ᵁ (inj₁ _) = Lift (aℓ ⊔ bℓ) ⊥
+(inj₂ x) ≈ᵁ (inj₂ y) = Lift aℓ (_≈ᴸ_ B x y)
+
+⤖ᵁ : {l₁ l₂ : List C} → l₁ ≋ l₂ → Union l₁ → Union l₂
+⤖ᵁ l₁≋l₂ = Sum.map (⤖ A l₁≋l₂) (⤖ B l₁≋l₂)
+
+_∪_ : Language (a ⊔ b) (aℓ ⊔ bℓ)
+_∪_ = record
+  { 𝕃 = Union
+  ; _≈ᴸ_ = _≈ᵁ_
+  ; ⤖ = ⤖ᵁ
+  ; isLanguage = record
+    { ≈ᴸ-isEquivalence = record
+      { refl = λ {x} → case x return (λ x → _≈ᵁ_ x x) of λ
+        { (inj₁ x) → lift (≈ᴸ-refl A)
+        ; (inj₂ y) → lift (≈ᴸ-refl B)
+        }
+      ; sym = λ {x} {y} x≈ᵁy →
+        case (∃[ x ](∃[ y ](x ≈ᵁ y)) ∋ x , y , x≈ᵁy)
+        return (λ (x , y , _) → y ≈ᵁ x) of λ
+          { (inj₁ x , inj₁ y , lift x≈ᵁy) → lift (≈ᴸ-sym A x≈ᵁy)
+          ; (inj₂ y₁ , inj₂ y , lift x≈ᵁy) → lift (≈ᴸ-sym B x≈ᵁy)
+          }
+      ; trans = λ {i} {j} {k} i≈ᵁj j≈ᵁk →
+        case ∃[ i ](∃[ j ](∃[ k ](i ≈ᵁ j × j ≈ᵁ k))) ∋ i , j , k , i≈ᵁj , j≈ᵁk
+        return (λ (i , _ , k , _) → i ≈ᵁ k) of λ
+          { (inj₁ _ , inj₁ _ , inj₁ _ , lift x≈ᵁy , lift y≈ᵁz) →
+            lift (≈ᴸ-trans A x≈ᵁy y≈ᵁz)
+          ; (inj₂ _ , inj₂ _ , inj₂ _ , lift x≈ᵁy , lift y≈ᵁz) →
+            lift (≈ᴸ-trans B x≈ᵁy y≈ᵁz)
+          }
+      }
+    ; ⤖-cong = λ {_} {_} {l₁≋l₂} {x} {y} x≈ᵁy →
+      case ∃[ x ](∃[ y ](x ≈ᵁ y)) ∋ x , y , x≈ᵁy
+      return (λ (x , y , _) → (_≈ᵁ_ on ⤖ᵁ l₁≋l₂) x y) of λ
+        { (inj₁ x , inj₁ y , lift x≈ᵁy) → lift (⤖-cong A x≈ᵁy)
+        ; (inj₂ x , inj₂ y , lift x≈ᵁy) → lift (⤖-cong B x≈ᵁy)
+        }
+    ; ⤖-bijective = λ {_} {_} {l₁≋l₂} →
+      ( λ {x} {y} x≈ᵁy →
+        case ∃[ x ](∃[ y ]((_≈ᵁ_ on ⤖ᵁ l₁≋l₂) x y)) ∋ x , y , x≈ᵁy
+        return (λ (x , y , _) → x ≈ᵁ y) of λ
+          { (inj₁ x , inj₁ y , lift x≈ᵁy) → lift (⤖-injective A x≈ᵁy)
+          ; (inj₂ x , inj₂ y , lift x≈ᵁy) → lift (⤖-injective B x≈ᵁy)
+          })
+    , ( λ
+        { (inj₁ x) → Product.map inj₁ lift (⤖-surjective A x)
+        ; (inj₂ x) → Product.map inj₂ lift (⤖-surjective B x)
+        })
+    ; ⤖-refl = λ {_} {x} → case x return (λ x → ⤖ᵁ ≋-refl x ≈ᵁ x) of λ
+      { (inj₁ x) → lift (⤖-refl A)
+      ; (inj₂ y) → lift (⤖-refl B)
+      }
+    ; ⤖-sym = λ {_} {_} {x} {y} {l₁≋l₂} x≈ᵁy →
+      case ∃[ x ](∃[ y ](⤖ᵁ l₁≋l₂ x ≈ᵁ y)) ∋ x , y , x≈ᵁy
+      return (λ (x , y , _) → ⤖ᵁ (≋-sym l₁≋l₂) y ≈ᵁ x) of λ
+        { (inj₁ x , inj₁ y , lift x≈ᵁy) → lift (⤖-sym A x≈ᵁy)
+        ; (inj₂ x , inj₂ y , lift x≈ᵁy) → lift (⤖-sym B x≈ᵁy)
+        }
+    ; ⤖-trans = λ {_} {_} {_} {x} {y} {z} {l₁≋l₂} {l₂≋l₃} x≈ᵁy y≈ᵁz →
+      case (∃[ x ](∃[ y ](∃[ z ]((⤖ᵁ l₁≋l₂ x ≈ᵁ y) × (⤖ᵁ l₂≋l₃ y ≈ᵁ z))))) ∋
+           x , y , z , x≈ᵁy , y≈ᵁz
+      return (λ (x , _ , z , _ , _) → ⤖ᵁ (≋-trans l₁≋l₂ l₂≋l₃) x ≈ᵁ z) of λ
+        { (inj₁ x , inj₁ y , inj₁ z , lift x≈ᵁy , lift y≈ᵁz) →
+          lift (⤖-trans A x≈ᵁy y≈ᵁz)
+        ; (inj₂ x , inj₂ y , inj₂ z , lift x≈ᵁy , lift y≈ᵁz) →
+          lift (⤖-trans B x≈ᵁy y≈ᵁz)
+        }
+    }
+  }
-- 
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