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------------------------------------------------------------------------
-- Agda Helium
--
-- Base definitions for the axiomatic semantics
------------------------------------------------------------------------
{-# OPTIONS --safe --without-K #-}
open import Helium.Data.Pseudocode.Types using (RawPseudocode)
module Helium.Semantics.Axiomatic.Core
{b₁ b₂ i₁ i₂ i₃ r₁ r₂ r₃}
(rawPseudocode : RawPseudocode b₁ b₂ i₁ i₂ i₃ r₁ r₂ r₃)
where
private
open module C = RawPseudocode rawPseudocode
open import Data.Bool as Bool using (Bool)
open import Data.Fin as Fin using (Fin; zero; suc)
open import Data.Fin.Patterns
open import Data.Nat as ℕ using (ℕ; suc)
import Data.Nat.Induction as Natᵢ
import Data.Nat.Properties as ℕₚ
open import Data.Product as × using (_×_; _,_; uncurry)
open import Data.Sum using (_⊎_)
open import Data.Unit using (⊤)
open import Data.Vec as Vec using (Vec; []; _∷_; _++_; lookup)
open import Data.Vec.Relation.Unary.All as All using (All; []; _∷_)
open import Function using (_on_)
open import Helium.Data.Pseudocode.Core
open import Helium.Data.Pseudocode.Properties
import Induction.WellFounded as Wf
open import Level using (_⊔_; Lift; lift)
import Relation.Binary.Construct.On as On
open import Relation.Binary.PropositionalEquality using (_≡_; refl; cong; cong₂)
open import Relation.Nullary using (Dec; does; yes; no)
open import Relation.Nullary.Decidable.Core using (True; toWitness; map′)
open import Relation.Nullary.Product using (_×-dec_)
open import Relation.Unary using (_⊆_)
private
variable
t t′ : Type
m n : ℕ
Γ Δ Σ ts : Vec Type m
⟦_⟧ₜ : Type → Set (b₁ ⊔ i₁ ⊔ r₁)
⟦_⟧ₜ′ : Vec Type n → Set (b₁ ⊔ i₁ ⊔ r₁)
⟦ bool ⟧ₜ = Lift (b₁ ⊔ i₁ ⊔ r₁) Bool
⟦ int ⟧ₜ = Lift (b₁ ⊔ r₁) ℤ
⟦ fin n ⟧ₜ = Lift (b₁ ⊔ i₁ ⊔ r₁) (Fin n)
⟦ real ⟧ₜ = Lift (b₁ ⊔ i₁) ℝ
⟦ bit ⟧ₜ = Lift (i₁ ⊔ r₁) Bit
⟦ bits n ⟧ₜ = Vec ⟦ bit ⟧ₜ n
⟦ tuple n ts ⟧ₜ = ⟦ ts ⟧ₜ′
⟦ array t n ⟧ₜ = Vec ⟦ t ⟧ₜ n
⟦ [] ⟧ₜ′ = Lift (b₁ ⊔ i₁ ⊔ r₁) ⊤
⟦ t ∷ ts ⟧ₜ′ = ⟦ t ⟧ₜ × ⟦ ts ⟧ₜ′
fetch : ∀ i → ⟦ Γ ⟧ₜ′ → ⟦ lookup Γ i ⟧ₜ
fetch {Γ = _ ∷ _} 0F (x , _) = x
fetch {Γ = _ ∷ _} (suc i) (_ , xs) = fetch i xs
Transform : Vec Type m → Type → Set (b₁ ⊔ i₁ ⊔ r₁)
Transform ts t = ⟦ ts ⟧ₜ′ → ⟦ t ⟧ₜ
Predicate : Vec Type m → Set (b₁ ⊔ i₁ ⊔ r₁)
Predicate ts = ⟦ ts ⟧ₜ′ → Bool
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