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------------------------------------------------------------------------
-- Agda Helium
--
-- Base definitions for the denotational semantics.
------------------------------------------------------------------------
{-# OPTIONS --safe --without-K #-}
open import Helium.Data.Pseudocode.Algebra using (RawPseudocode)
module Helium.Semantics.Denotational.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
import Data.Bool as Bool
open import Data.Empty using (⊥-elim)
import Data.Fin as Fin
import Data.Integer as 𝕀
open import Data.Nat using (ℕ)
open import Data.Product using (_×_; _,_; proj₁; proj₂; <_,_>; uncurry)
open import Data.Vec as Vec using (Vec; []; _∷_; map; zipWith)
open import Data.Vec.Relation.Unary.All using (All; []; _∷_)
open import Function
open import Helium.Data.Pseudocode.Core
open import Helium.Semantics.Core rawPseudocode
open import Level
open import Relation.Binary.PropositionalEquality using (sym)
open import Relation.Nullary using (does)
private
variable
n : ℕ
t : Type
Σ Γ ts : Vec Type n
module Semantics (2≉0 : 2≉0) where
expr : Expression Σ Γ t → ⟦ Σ ⟧ₜ′ × ⟦ Γ ⟧ₜ′ → ⟦ t ⟧ₜ
exprs : All (Expression Σ Γ) ts → ⟦ Σ ⟧ₜ′ × ⟦ Γ ⟧ₜ′ → ⟦ ts ⟧ₜ′
ref : Reference Σ Γ t → ⟦ Σ ⟧ₜ′ × ⟦ Γ ⟧ₜ′ → ⟦ t ⟧ₜ
locRef : LocalReference Σ Γ t → ⟦ Σ ⟧ₜ′ × ⟦ Γ ⟧ₜ′ → ⟦ t ⟧ₜ
assign : Reference Σ Γ t → ⟦ t ⟧ₜ → ⟦ Σ ⟧ₜ′ × ⟦ Γ ⟧ₜ′ → ⟦ Σ ⟧ₜ′ × ⟦ Γ ⟧ₜ′ → ⟦ Σ ⟧ₜ′ × ⟦ Γ ⟧ₜ′
locAssign : LocalReference Σ Γ t → ⟦ t ⟧ₜ → ⟦ Σ ⟧ₜ′ × ⟦ Γ ⟧ₜ′ → ⟦ Σ ⟧ₜ′ × ⟦ Γ ⟧ₜ′ → ⟦ Γ ⟧ₜ′
stmt : Statement Σ Γ → ⟦ Σ ⟧ₜ′ × ⟦ Γ ⟧ₜ′ → ⟦ Σ ⟧ₜ′ × ⟦ Γ ⟧ₜ′
locStmt : LocalStatement Σ Γ → ⟦ Σ ⟧ₜ′ × ⟦ Γ ⟧ₜ′ → ⟦ Γ ⟧ₜ′
fun : Function Σ Γ t → ⟦ Σ ⟧ₜ′ × ⟦ Γ ⟧ₜ′ → ⟦ t ⟧ₜ
proc : Procedure Σ Γ → ⟦ Σ ⟧ₜ′ × ⟦ Γ ⟧ₜ′ → ⟦ Σ ⟧ₜ′
expr (lit {t = t} x) = const (Κ[ t ] x)
expr {Σ = Σ} (state i) = fetch i Σ ∘ proj₁
expr {Γ = Γ} (var i) = fetch i Γ ∘ proj₂
expr (e ≟ e₁) = lift ∘ does ∘ uncurry ≈-dec ∘ < expr e , expr e₁ >
expr (e <? e₁) = lift ∘ does ∘ uncurry <-dec ∘ < expr e , expr e₁ >
expr (inv e) = lift ∘ Bool.not ∘ lower ∘ expr e
expr (e && e₁) = lift ∘ uncurry (Bool._∧_ on lower) ∘ < expr e , expr e₁ >
expr (e || e₁) = lift ∘ uncurry (Bool._∨_ on lower) ∘ < expr e , expr e₁ >
expr (not e) = map (lift ∘ 𝔹.¬_ ∘ lower) ∘ expr e
expr (e and e₁) = uncurry (zipWith (lift ∘₂ 𝔹._∧_ on lower)) ∘ < expr e , expr e₁ >
expr (e or e₁) = uncurry (zipWith (lift ∘₂ 𝔹._∨_ on lower)) ∘ < expr e , expr e₁ >
expr [ e ] = (_∷ []) ∘ expr e
expr (unbox e) = Vec.head ∘ expr e
expr (merge e e₁ e₂) = uncurry (uncurry mergeVec) ∘ < < expr e , expr e₁ > , lower ∘ expr e₂ >
expr (slice e e₁) = uncurry sliceVec ∘ < expr e , lower ∘ expr e₁ >
expr (cut e e₁) = uncurry cutVec ∘ < expr e , lower ∘ expr e₁ >
expr (cast eq e) = castVec eq ∘ expr e
expr (- e) = neg ∘ expr e
expr (e + e₁) = uncurry add ∘ < expr e , expr e₁ >
expr (e * e₁) = uncurry mul ∘ < expr e , expr e₁ >
expr (e ^ x) = flip pow x ∘ expr e
expr (e >> n) = lift ∘ flip (shift 2≉0) n ∘ lower ∘ expr e
expr (rnd e) = lift ∘ ⌊_⌋ ∘ lower ∘ expr e
expr (fin {ms = ms} f e) = lift ∘ f ∘ lowerFin ms ∘ expr e
expr (asInt e) = lift ∘ 𝕀⇒ℤ ∘ 𝕀.+_ ∘ Fin.toℕ ∘ lower ∘ expr e
expr nil = const _
expr (cons {ts = ts} e e₁) = uncurry (cons′ ts) ∘ < expr e , expr e₁ >
expr (head {ts = ts} e) = head′ ts ∘ expr e
expr (tail {ts = ts} e) = tail′ ts ∘ expr e
expr (call f es) = fun f ∘ < proj₁ , exprs es >
expr (if e then e₁ else e₂) = uncurry (uncurry Bool.if_then_else_) ∘ < < lower ∘ expr e , expr e₁ > , expr e₂ >
exprs [] = const _
exprs (e ∷ []) = expr e
exprs (e ∷ e₁ ∷ es) = < expr e , exprs (e₁ ∷ es) >
ref {Σ = Σ} (state i) = fetch i Σ ∘ proj₁
ref {Γ = Γ} (var i) = fetch i Γ ∘ proj₂
ref [ r ] = (_∷ []) ∘ ref r
ref (unbox r) = Vec.head ∘ ref r
ref (merge r r₁ e) = uncurry (uncurry mergeVec) ∘ < < ref r , ref r₁ > , lower ∘ expr e >
ref (slice r e) = uncurry sliceVec ∘ < ref r , lower ∘ expr e >
ref (cut r e) = uncurry cutVec ∘ < ref r , lower ∘ expr e >
ref (cast eq r) = castVec eq ∘ ref r
ref nil = const _
ref (cons {ts = ts} r r₁) = uncurry (cons′ ts) ∘ < ref r , ref r₁ >
ref (head {ts = ts} r) = head′ ts ∘ ref r
ref (tail {ts = ts} r) = tail′ ts ∘ ref r
locRef {Γ = Γ} (var i) = fetch i Γ ∘ proj₂
locRef [ r ] = (_∷ []) ∘ locRef r
locRef (unbox r) = Vec.head ∘ locRef r
locRef (merge r r₁ e) = uncurry (uncurry mergeVec) ∘ < < locRef r , locRef r₁ > , lower ∘ expr e >
locRef (slice r e) = uncurry sliceVec ∘ < locRef r , lower ∘ expr e >
locRef (cut r e) = uncurry cutVec ∘ < locRef r , lower ∘ expr e >
locRef (cast eq r) = castVec eq ∘ locRef r
locRef nil = const _
locRef (cons {ts = ts} r r₁) = uncurry (cons′ ts) ∘ < locRef r , locRef r₁ >
locRef (head {ts = ts} r) = head′ ts ∘ locRef r
locRef (tail {ts = ts} r) = tail′ ts ∘ locRef r
assign {Σ = Σ} (state i) val σ,γ = < updateAt i Σ val ∘ proj₁ , proj₂ >
assign {Γ = Γ} (var i) val σ,γ = < proj₁ , updateAt i Γ val ∘ proj₂ >
assign [ r ] val σ,γ = assign r (Vec.head val) σ,γ
assign (unbox r) val σ,γ = assign r (val ∷ []) σ,γ
assign (merge r r₁ e) val σ,γ = assign r₁ (cutVec val (lower (expr e σ,γ))) σ,γ ∘ assign r (sliceVec val (lower (expr e σ,γ))) σ,γ
assign (slice r e) val σ,γ = assign r (mergeVec val (cutVec (ref r σ,γ) (lower (expr e σ,γ))) (lower (expr e σ,γ))) σ,γ
assign (cut r e) val σ,γ = assign r (mergeVec (sliceVec (ref r σ,γ) (lower (expr e σ,γ))) val (lower (expr e σ,γ))) σ,γ
assign (cast eq r) val σ,γ = assign r (castVec (sym eq) val) σ,γ
assign nil val σ,γ = id
assign (cons {ts = ts} r r₁) val σ,γ = assign r₁ (tail′ ts val) σ,γ ∘ assign r (head′ ts val) σ,γ
assign (head {ts = ts} r) val σ,γ = assign r (cons′ ts val (ref (tail r) σ,γ)) σ,γ
assign (tail {ts = ts} r) val σ,γ = assign r (cons′ ts (ref (head r) σ,γ) val) σ,γ
locAssign {Γ = Γ} (var i) val σ,γ = updateAt i Γ val ∘ proj₂
locAssign [ r ] val σ,γ = locAssign r (Vec.head val) σ,γ
locAssign (unbox r) val σ,γ = locAssign r (val ∷ []) σ,γ
locAssign (merge r r₁ e) val σ,γ = locAssign r₁ (cutVec val (lower (expr e σ,γ))) σ,γ ∘ < proj₁ , locAssign r (sliceVec val (lower (expr e σ,γ))) σ,γ >
locAssign (slice r e) val σ,γ = locAssign r (mergeVec val (cutVec (locRef r σ,γ) (lower (expr e σ,γ))) (lower (expr e σ,γ))) σ,γ
locAssign (cut r e) val σ,γ = locAssign r (mergeVec (sliceVec (locRef r σ,γ) (lower (expr e σ,γ))) val (lower (expr e σ,γ))) σ,γ
locAssign (cast eq r) val σ,γ = locAssign r (castVec (sym eq) val) σ,γ
locAssign nil val σ,γ = proj₂
locAssign (cons {ts = ts} r r₁) val σ,γ = locAssign r₁ (tail′ ts val) σ,γ ∘ < proj₁ , locAssign r (head′ ts val) σ,γ >
locAssign (head {ts = ts} r) val σ,γ = locAssign r (cons′ ts val (locRef (tail r) σ,γ)) σ,γ
locAssign (tail {ts = ts} r) val σ,γ = locAssign r (cons′ ts (locRef (head r) σ,γ) val) σ,γ
stmt (s ∙ s₁) = stmt s₁ ∘ stmt s
stmt skip = id
stmt (ref ≔ val) = uncurry (uncurry (assign ref)) ∘ < < expr val , id > , id >
stmt {Γ = Γ} (declare e s) = < proj₁ , tail′ Γ ∘ proj₂ > ∘ stmt s ∘ < proj₁ , uncurry (cons′ Γ) ∘ < expr e , proj₂ > >
stmt (invoke p es) = < proc p ∘ < proj₁ , exprs es > , proj₂ >
stmt (if e then s) = uncurry (uncurry Bool.if_then_else_) ∘ < < lower ∘ expr e , stmt s > , id >
stmt (if e then s else s₁) = uncurry (uncurry Bool.if_then_else_) ∘ < < lower ∘ expr e , stmt s > , stmt s₁ >
stmt {Γ = Γ} (for m s) = Vec.foldl _ (flip λ i → (< proj₁ , tail′ Γ ∘ proj₂ > ∘ stmt s ∘ < proj₁ , cons′ Γ (lift i) ∘ proj₂ >) ∘_) id (Vec.allFin m)
locStmt (s ∙ s₁) = locStmt s₁ ∘ < proj₁ , locStmt s >
locStmt skip = proj₂
locStmt (ref ≔ val) = uncurry (uncurry (locAssign ref)) ∘ < < expr val , id > , id >
locStmt {Γ = Γ} (declare e s) = tail′ Γ ∘ locStmt s ∘ < proj₁ , uncurry (cons′ Γ) ∘ < expr e , proj₂ > >
locStmt (if e then s) = uncurry (uncurry Bool.if_then_else_) ∘ < < lower ∘ expr e , locStmt s > , proj₂ >
locStmt (if e then s else s₁) = uncurry (uncurry Bool.if_then_else_) ∘ < < lower ∘ expr e , locStmt s > , locStmt s₁ >
locStmt {Γ = Γ} (for m s) = proj₂ ∘ Vec.foldl _ (flip λ i → (< proj₁ , tail′ Γ ∘ locStmt s > ∘ < proj₁ , cons′ Γ (lift i) ∘ proj₂ >) ∘_) id (Vec.allFin m)
fun {Γ = Γ} (declare e f) = fun f ∘ < proj₁ , uncurry (cons′ Γ) ∘ < expr e , proj₂ > >
fun (s ∙return e) = expr e ∘ < proj₁ , locStmt s >
proc (s ∙end) = proj₁ ∘ stmt s
|
