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{-# OPTIONS --safe --without-K #-}
open import Helium.Data.Pseudocode
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.Maybe using (Maybe; just; nothing; map; _>>=_)
open import Data.Nat using (ℕ; zero; suc)
import Data.Nat.Properties as ℕₚ
open import Data.Product using (_×_; _,_; map₂; uncurry)
open import Data.Product.Nary.NonDependent
open import Data.Unit using (⊤)
open import Level renaming (suc to ℓsuc) hiding (lift)
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
Expr : ∀ n {ls} → Sets n ls → Set ℓ → Set (ℓ ⊔ ⨆ n ls ⊔ ℓ′)
Expr _ Γ τ = (σ : State) → (ρ : Product⊤ _ Γ) → Maybe (State × τ)
record Reference n {ls} (Γ : Sets n ls) (τ : Set ℓ) : Set (ℓ ⊔ ⨆ n ls ⊔ ℓ′) where
field
get : Expr n Γ τ
set : (σ : State) → (ρ : Product⊤ _ Γ) → τ → Maybe (State × Product⊤ _ Γ)
Statement : ∀ n {ls} → Sets n ls → Set ℓ → Set (ℓ ⊔ ⨆ n ls ⊔ ℓ′)
Statement n Γ τ = (cont : Expr n Γ τ) → Expr n Γ τ
ForStatement : ∀ n {ls} → Sets n ls → Set ℓ → ℕ → Set (ℓ ⊔ ⨆ n ls ⊔ ℓ′)
ForStatement n Γ τ m = (cont break : Expr n Γ τ) → Expr (suc n) (Fin m , Γ) τ
Function : ∀ n {ls} → Sets n ls → Set ℓ → Set (ℓ ⊔ ⨆ n ls ⊔ ℓ′)
Function = Statement
Procedure : ∀ n {ls} → Sets n ls → Set (⨆ n ls ⊔ ℓ′)
Procedure n Γ = Function n Γ ⊤
-- Expressions
unknown : ∀ {n ls} {Γ : Sets n ls} → Expr n Γ τ
unknown σ ρ = nothing
pure : ∀ {n ls} {Γ : Sets n ls} → τ → Expr n Γ τ
pure v σ ρ = just (σ , v)
apply : ∀ {n ls} {Γ : Sets n ls} → (τ → τ′) → Expr n Γ τ → Expr n Γ τ′
apply f e σ ρ = map (map₂ f) (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) → apply f e σ ρ
!_ : ∀ {n ls} {Γ : Sets n ls} → Reference n Γ τ → Expr n Γ τ
! r = Reference.get 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 >>= λ (σ , v) → f unknown σ (toProduct⊤ _ v)
-- References
var : ∀ {n ls} {Γ : Sets n ls} i → Reference n Γ (Projₙ Γ i)
var i = record
{ get = λ σ ρ → just (σ , projₙ _ i (toProduct _ ρ))
; set = λ σ ρ v → just (σ , update i v ρ)
}
!#_ : ∀ {n ls} {Γ : Sets n ls} m {m<n : True (suc m ℕₚ.≤? n)} → Expr n Γ (Projₙ Γ (#_ m {n} {m<n}))
(!# m) {m<n} = ! (var (#_ m {m<n = m<n}))
wknRef : ∀ {m ls} {Γ : Sets m ls} → Reference m Γ τ → Reference (suc m) (τ′ , Γ) τ
wknRef &x = record
{ get = λ σ (_ , ρ) → Reference.get &x σ ρ
; set = λ σ (v , ρ) x → Reference.set &x σ ρ x >>= λ (σ , ρ) → just (σ , (v , ρ))
}
-- Statements
infixr 9 _∙_
skip : ∀ {n ls} {Γ : Sets n ls} → Statement n Γ τ
skip cont = cont
return : ∀ {n ls} {Γ : Sets n ls} → Expr n Γ τ → Statement n Γ τ
return e _ = e
_≔_ : ∀ {n ls} {Γ : Sets n ls} → Reference n Γ τ → Expr n Γ τ → Statement n Γ τ′
(ref ≔ e) cont σ ρ = e σ ρ >>= λ (σ , v) → Reference.set ref σ ρ v >>= λ (σ , v) → cont σ v
label : ∀ {n ls} {Γ : Sets n ls} → smap _ (Reference n Γ) n Γ ⇉ Statement n Γ τ → Statement n Γ τ
label {n = n} s = uncurry⊤ₙ n s vars
where
vars : ∀ {n ls} {Γ : Sets n ls} → Product⊤ n (smap _ (Reference n Γ) n Γ)
vars {zero} = _
vars {suc n} = var (# 0) , mapAll _ _ wknRef vars
declare : ∀ {n ls} {Γ : Sets n ls} → Expr n Γ τ → Statement (suc n) (τ , Γ) τ′ → Statement n Γ τ′
declare e s cont σ ρ = e σ ρ >>= λ (σ , v) → s (λ σ (_ , ρ) → cont σ ρ) σ (v , ρ)
if_then_else_ : ∀ {n ls} {Γ : Sets n ls} → Expr n Γ Bool → Statement n Γ τ → Statement n Γ τ → Statement n Γ τ
(if e then b₁ else b₂) cont σ ρ = e σ ρ >>= λ (σ , b) → Bool.if b then b₁ cont σ ρ else b₂ cont σ ρ
for : ∀ {n ls} {Γ : Sets n ls} m → ForStatement n Γ τ m → Statement n Γ τ
for zero s cont σ ρ = cont σ ρ
for (suc m) s cont σ ρ = s (for m (λ cont break σ (i , ρ) → s cont break σ (suc i , ρ)) cont) cont σ (# 0 , ρ)
_∙_ : ∀ {n ls} {Γ : Sets n ls} → Op₂ (Statement n Γ τ)
(s ∙ t) cont = s (t cont)
-- For statements
infixr 9 _∙′_
lift : ∀ {m n ls} {Γ : Sets n ls} → Statement (suc n) (Fin m , Γ) τ → ForStatement n Γ τ m
lift s cont _ = s (λ σ (_ , ρ) → cont σ ρ)
continue : ∀ {m n ls} {Γ : Sets n ls} → ForStatement n Γ τ m
continue cont break σ (_ , ρ) = cont σ ρ
break : ∀ {m n ls} {Γ : Sets n ls} → ForStatement n Γ τ m
break cont break σ (_ , ρ) = break σ ρ
_∙′_ : ∀ {m n ls} {Γ : Sets n ls} → Op₂ (ForStatement n Γ τ m)
(s ∙′ t) cont break σ (i , ρ) = s (λ σ ρ → t cont break σ (i , ρ)) break σ (i , ρ)
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