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------------------------------------------------------------------------
-- Agda Helium
--
-- Base definitions for the denotational semantics.
------------------------------------------------------------------------
{-# OPTIONS --safe --without-K #-}
module Helium.Semantics.Denotational.Core
{ℓ′}
(State : Set ℓ′)
where
open import Algebra.Core
open import Data.Bool as Bool using (Bool)
open import Data.Fin hiding (lift)
open import Data.Nat using (ℕ; zero; suc)
import Data.Nat.Properties as ℕₚ
open import Data.Product hiding (_<*>_; _,′_)
open import Data.Product.Nary.NonDependent
open import Data.Sum using (_⊎_; inj₁; inj₂; [_,_]′)
open import Data.Unit using (⊤)
open import Level renaming (suc to ℓsuc) hiding (zero)
open import Function using (_∘_; _∘₂_; _|>_)
open import Function.Nary.NonDependent.Base
open import Relation.Nullary.Decidable using (True)
private
variable
ℓ ℓ₁ ℓ₂ : Level
τ τ′ : Set ℓ
mapAll : ∀ {m ls} {Γ : Sets m ls} {l l′}
(f : ∀ {ℓ} → Set ℓ → Set (l ℓ))
(g : ∀ {ℓ} → Set ℓ → Set (l′ ℓ))
(h : ∀ {a} {A : Set a} → f A → g A) →
Product⊤ m (smap l f m Γ) →
Product⊤ m (smap l′ g m Γ)
mapAll {zero} f g h xs = xs
mapAll {suc m} f g h (x , xs) = h x , mapAll f g h xs
update : ∀ {n ls} {Γ : Sets n ls} i → Projₙ Γ i → Product⊤ n Γ → Product⊤ n Γ
update zero y (_ , xs) = y , xs
update (suc i) y (x , xs) = x , update i y xs
PureExpr : ∀ n {ls} → Sets n ls → Set ℓ → Set (ℓ ⊔ ⨆ n ls ⊔ ℓ′)
PureExpr n Γ τ = (σ : State) → (ρ : Product⊤ n Γ) → τ
Expr : ∀ n {ls} → Sets n ls → Set ℓ → Set (ℓ ⊔ ⨆ n ls ⊔ ℓ′)
Expr n Γ τ = (σ : State) → (ρ : Product⊤ n Γ) → State × τ
record Reference n {ls} (Γ : Sets n ls) (τ : Set ℓ) : Set (ℓ ⊔ ⨆ n ls ⊔ ℓ′) where
field
get : PureExpr n Γ τ
set : τ → (σ : State) → (ρ : Product⊤ n Γ) → State × Product⊤ n Γ
Function : ∀ n {ls} → Sets n ls → Set ℓ → Set (ℓ ⊔ ⨆ n ls ⊔ ℓ′)
Function = Expr
Procedure : ∀ n {ls} → Sets n ls → Set (⨆ n ls ⊔ ℓ′)
Procedure n Γ = Function n Γ ⊤
Block : ∀ n {ls} → Sets n ls → Set ℓ → Set (ℓ ⊔ ⨆ n ls ⊔ ℓ′)
Block n Γ τ = (σ : State) → (ρ : Product⊤ n Γ) → State × (Product⊤ n Γ ⊎ τ)
-- Expressions
pure′ : ∀ {n ls} {Γ : Sets n ls} → τ → PureExpr n Γ τ
pure′ v σ ρ = v
pure : ∀ {n ls} {Γ : Sets n ls} → τ → Expr n Γ τ
pure v σ ρ = σ , v
↓_ : ∀ {n ls} {Γ : Sets n ls} → PureExpr n Γ τ → Expr n Γ τ
(↓ e) σ ρ = σ , e σ ρ
_<*>_ : ∀ {n ls} {Γ : Sets n ls} → Expr n Γ (τ → τ′) → Expr n Γ τ → Expr n Γ τ′
_<*>_ f e σ ρ = f σ ρ |> λ (σ , f) → map₂ f (e σ ρ)
!_ : ∀ {n ls} {Γ : Sets n ls} → Reference n Γ τ → PureExpr n Γ τ
! r = Reference.get r
↓!_ : ∀ {n ls} {Γ : Sets n ls} → Reference n Γ τ → Expr n Γ τ
↓! r = ↓ ! r
call : ∀ {m n ls₁ ls₂} {Γ : Sets m ls₁} {Δ : Sets n ls₂} → Function m Γ τ → Expr n Δ (Product m Γ) → Expr n Δ τ
call f e σ ρ = e σ ρ |> map₂ (toProduct⊤ _) |> uncurry f
declare : ∀ {n ls} {Γ : Sets n ls} → Expr n Γ τ → Expr (suc n) (τ , Γ) τ′ → Expr n Γ τ′
declare e s σ ρ = e σ ρ |> map₂ (_, ρ) |> uncurry s
-- References
var : ∀ {n ls} {Γ : Sets n ls} i → Reference n Γ (Projₙ Γ i)
var i = record
{ get = λ σ ρ → projₙ _ i (toProduct _ ρ)
; set = λ v → curry (map₂ (update i v))
}
!#_ : ∀ {n ls} {Γ : Sets n ls} m {m<n : True (suc m ℕₚ.≤? n)} → PureExpr n Γ (Projₙ Γ (#_ m {n} {m<n}))
(!# m) {m<n} = ! var (#_ m {m<n = m<n})
_,′_ : ∀ {n ls} {Γ : Sets n ls} → Reference n Γ τ → Reference n Γ τ′ → Reference n Γ (τ × τ′)
&x ,′ &y = record
{ get = λ σ ρ → Reference.get &x σ ρ , Reference.get &y σ ρ
; set = λ (x , y) σ ρ → uncurry (Reference.set &y y) (Reference.set &x x σ ρ)
}
-- Blocks
infixr 1 _∙_
infixl 2 if_then_else_
infix 4 _≔_ _⟵_
skip : ∀ {n ls} {Γ : Sets n ls} → Block n Γ τ
skip σ ρ = σ , inj₁ ρ
invoke : ∀ {m n ls₁ ls₂} {Γ : Sets m ls₁} {Δ : Sets n ls₂} → Procedure m Γ → Expr n Δ (Product m Γ) → Block n Δ τ
invoke f e σ ρ = call f e σ ρ |> map₂ (λ _ → inj₁ ρ)
_≔_ : ∀ {n ls} {Γ : Sets n ls} → Reference n Γ τ → Expr n Γ τ → Block n Γ τ′
(&x ≔ e) σ ρ = e σ ρ |> λ (σ , x) → Reference.set &x x σ ρ |> map₂ inj₁
_⟵_ : ∀ {n ls} {Γ : Sets n ls} → Reference n Γ τ → Op₁ τ → Block n Γ τ′
&x ⟵ e = &x ≔ ⦇ e (↓ (! &x)) ⦈
if_then_else_ : ∀ {n ls} {Γ : Sets n ls} → Expr n Γ Bool → Block n Γ τ → Block n Γ τ → Block n Γ τ
(if e then b₁ else b₂) σ ρ = e σ ρ |> λ (σ , b) → Bool.if b then b₁ σ ρ else b₂ σ ρ
for : ∀ {n ls} {Γ : Sets n ls} m → Block (suc n) (Fin m , Γ) τ → Block n Γ τ
for zero b σ ρ = σ , inj₁ ρ
for (suc m) b σ ρ with b σ (zero , ρ)
... | σ′ , inj₂ x = σ′ , inj₂ x
... | σ′ , inj₁ (_ , ρ′) = for m b′ σ′ ρ′
where
b′ : Block (suc _) (Fin m , _) _
b′ σ (i , ρ) with b σ (suc i , ρ)
... | σ′ , inj₂ x = σ′ , inj₂ x
... | σ′ , inj₁ (_ , ρ′) = σ′ , inj₁ (i , ρ′)
_∙_ : ∀ {n ls} {Γ : Sets n ls} → Block n Γ τ → Expr n Γ τ → Expr n Γ τ
(b ∙ e) σ τ = b σ τ |> λ (σ , ρ⊎x) → [ e σ , σ ,_ ]′ ρ⊎x
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