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+------------------------------------------------------------------------
+-- Agda Helium
+--
+-- Basic properties of the pseudocode data types
+------------------------------------------------------------------------
+
+{-# OPTIONS --without-K --safe #-}
+
+module Helium.Data.Pseudocode.Properties where
+
+import Data.Nat as ℕ
+open import Data.Product using (_,_; uncurry)
+open import Data.Vec using ([]; _∷_)
+open import Function using (_∋_)
+open import Helium.Data.Pseudocode.Core
+import Relation.Binary.Consequences as Consequences
+open import Relation.Binary.PropositionalEquality using (_≡_; refl; cong; cong₂)
+open import Relation.Nullary using (Dec; yes; no)
+open import Relation.Nullary.Decidable.Core using (map′)
+open import Relation.Nullary.Product using (_×-dec_)
+
+infixl 4 _≟ᵗ_ _≟ˢ_
+
+_≟ᵗ_ : ∀ (t t′ : Type) → Dec (t ≡ t′)
+bool ≟ᵗ bool = yes refl
+bool ≟ᵗ int = no (λ ())
+bool ≟ᵗ fin n = no (λ ())
+bool ≟ᵗ real = no (λ ())
+bool ≟ᵗ bit = no (λ ())
+bool ≟ᵗ bits n = no (λ ())
+bool ≟ᵗ tuple n x = no (λ ())
+bool ≟ᵗ array t′ n = no (λ ())
+int ≟ᵗ bool = no (λ ())
+int ≟ᵗ int = yes refl
+int ≟ᵗ fin n = no (λ ())
+int ≟ᵗ real = no (λ ())
+int ≟ᵗ bit = no (λ ())
+int ≟ᵗ bits n = no (λ ())
+int ≟ᵗ tuple n x = no (λ ())
+int ≟ᵗ array t′ n = no (λ ())
+fin n ≟ᵗ bool = no (λ ())
+fin n ≟ᵗ int = no (λ ())
+fin m ≟ᵗ fin n = map′ (cong fin) (λ { refl → refl }) (m ℕ.≟ n)
+fin n ≟ᵗ real = no (λ ())
+fin n ≟ᵗ bit = no (λ ())
+fin n ≟ᵗ bits n₁ = no (λ ())
+fin n ≟ᵗ tuple n₁ x = no (λ ())
+fin n ≟ᵗ array t′ n₁ = no (λ ())
+real ≟ᵗ bool = no (λ ())
+real ≟ᵗ int = no (λ ())
+real ≟ᵗ fin n = no (λ ())
+real ≟ᵗ real = yes refl
+real ≟ᵗ bit = no (λ ())
+real ≟ᵗ bits n = no (λ ())
+real ≟ᵗ tuple n x = no (λ ())
+real ≟ᵗ array t′ n = no (λ ())
+bit ≟ᵗ bool = no (λ ())
+bit ≟ᵗ int = no (λ ())
+bit ≟ᵗ fin n = no (λ ())
+bit ≟ᵗ real = no (λ ())
+bit ≟ᵗ bit = yes refl
+bit ≟ᵗ bits n = no (λ ())
+bit ≟ᵗ tuple n x = no (λ ())
+bit ≟ᵗ array t n = no (λ ())
+bits n ≟ᵗ bool = no (λ ())
+bits n ≟ᵗ int = no (λ ())
+bits n ≟ᵗ fin n₁ = no (λ ())
+bits n ≟ᵗ real = no (λ ())
+bits m ≟ᵗ bit = no (λ ())
+bits m ≟ᵗ bits n = map′ (cong bits) (λ { refl → refl }) (m ℕ.≟ n)
+bits n ≟ᵗ tuple n₁ x = no (λ ())
+bits n ≟ᵗ array t′ n₁ = no (λ ())
+tuple n x ≟ᵗ bool = no (λ ())
+tuple n x ≟ᵗ int = no (λ ())
+tuple n x ≟ᵗ fin n₁ = no (λ ())
+tuple n x ≟ᵗ real = no (λ ())
+tuple n x ≟ᵗ bit = no (λ ())
+tuple n x ≟ᵗ bits n₁ = no (λ ())
+tuple _ [] ≟ᵗ tuple _ [] = yes refl
+tuple _ [] ≟ᵗ tuple _ (y ∷ ys) = no (λ ())
+tuple _ (x ∷ xs) ≟ᵗ tuple _ [] = no (λ ())
+tuple _ (x ∷ xs) ≟ᵗ tuple _ (y ∷ ys) = map′ (λ { (refl , refl) → refl }) (λ { refl → refl , refl }) (x ≟ᵗ y ×-dec tuple _ xs ≟ᵗ tuple _ ys)
+tuple n x ≟ᵗ array t′ n₁ = no (λ ())
+array t n ≟ᵗ bool = no (λ ())
+array t n ≟ᵗ int = no (λ ())
+array t n ≟ᵗ fin n₁ = no (λ ())
+array t n ≟ᵗ real = no (λ ())
+array t n ≟ᵗ bit = no (λ ())
+array t n ≟ᵗ bits n₁ = no (λ ())
+array t n ≟ᵗ tuple n₁ x = no (λ ())
+array t m ≟ᵗ array t′ n = map′ (uncurry (cong₂ array)) (λ { refl → refl , refl }) (t ≟ᵗ t′ ×-dec m ℕ.≟ n)
+
+_≟ˢ_ : ∀ (t t′ : Sliced) → Dec (t ≡ t′)
+bits ≟ˢ bits = yes refl
+bits ≟ˢ array x = no (λ ())
+array x ≟ˢ bits = no (λ ())
+array x ≟ˢ array y = map′ (cong array) (λ { refl → refl }) (x ≟ᵗ y)
+
+bits-injective : ∀ {m n} → (Type ∋ bits m) ≡ bits n → m ≡ n
+bits-injective refl = refl
+
+array-injective₁ : ∀ {t t′ m n} → (Type ∋ array t m) ≡ array t′ n → t ≡ t′
+array-injective₁ refl = refl
+
+array-injective₂ : ∀ {t t′ m n} → (Type ∋ array t m) ≡ array t′ n → m ≡ n
+array-injective₂ refl = refl
+
+typeEqRecomp : ∀ {t t′} → .(eq : t ≡ t′) → t ≡ t′
+typeEqRecomp = Consequences.dec⇒recomputable _≟ᵗ_