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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_