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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 using (Bool; T)
open import Data.Fin as Fin using (zero; suc)
import Data.Fin.Properties as Finₚ
-- open import Data.Nat as ℕ using (zero; suc)
import Data.Nat as ℕ
import Data.Nat.Properties as ℕₚ
open import Data.Product using (∃; _×_; _,_; <_,_>)
open import Data.Sum using (_⊎_)
open import Data.Unit using (⊤)
open import Data.Vec using (Vec; _∷_; lookup)
open import Data.Vec.Relation.Unary.All as All using (All; []; _∷_)
open import Function using (_$_)
open import Helium.Data.Pseudocode.Core
open import Helium.Semantics.Core rawPseudocode
open import Level using (_⊔_; Lift)
open import Relation.Binary.PropositionalEquality using (refl)
open import Relation.Nullary using (yes; no)
open import Relation.Unary using (Decidable; _⊆_)
module _ {o} (Σ : Vec Type o) {n} (Γ : Vec Type n) where
data Term {m} (Δ : Vec Type m) : Type → Set (b₁ ⊔ i₁ ⊔ r₁) where
state : ∀ i → Term Δ (lookup Σ i)
var : ∀ i → Term Δ (lookup Γ i)
meta : ∀ i → Term Δ (lookup Δ i)
funct : ∀ {m ts t} → (⟦ tuple m ts ⟧ₜˡ → ⟦ t ⟧ₜˡ) → All (Term Δ) ts → Term Δ t
infixl 7 _⇒_
infixl 6 _∧_
infixl 5 _∨_
data Assertion {m} (Δ : Vec Type m) : Set (b₁ ⊔ i₁ ⊔ r₁) where
_∧_ : Assertion Δ → Assertion Δ → Assertion Δ
_∨_ : Assertion Δ → Assertion Δ → Assertion Δ
_⇒_ : Assertion Δ → Assertion Δ → Assertion Δ
all : ∀ {t} → Assertion (t ∷ Δ) → Assertion Δ
some : ∀ {t} → Assertion (t ∷ Δ) → Assertion Δ
pred : ∀ {m ts} → (⟦ tuple m ts ⟧ₜˡ → Bool) → All (Term Δ) ts → Assertion Δ
module _ {o} {Σ : Vec Type o} {n} {Γ : Vec Type n} {m} {Δ : Vec Type m} where
⟦_⟧ : ∀ {t} → Term Σ Γ Δ t → ⟦ Σ ⟧ₜˡ′ → ⟦ Γ ⟧ₜˡ′ → ⟦ Δ ⟧ₜˡ′ → ⟦ t ⟧ₜˡ
⟦_⟧′ : ∀ {n ts} → All (Term Σ Γ Δ) ts → ⟦ Σ ⟧ₜˡ′ → ⟦ Γ ⟧ₜˡ′ → ⟦ Δ ⟧ₜˡ′ → ⟦ tuple n ts ⟧ₜˡ
⟦ state i ⟧ σ γ δ = fetchˡ Σ σ i
⟦ var i ⟧ σ γ δ = fetchˡ Γ γ i
⟦ meta i ⟧ σ γ δ = fetchˡ Δ δ i
⟦ funct f ts ⟧ σ γ δ = f (⟦ ts ⟧′ σ γ δ)
⟦ [] ⟧′ σ γ δ = _
⟦ (t ∷ []) ⟧′ σ γ δ = ⟦ t ⟧ σ γ δ
⟦ (t ∷ t′ ∷ ts) ⟧′ σ γ δ = ⟦ t ⟧ σ γ δ , ⟦ t′ ∷ ts ⟧′ σ γ δ
termSubstState : ∀ {t} → Term Σ Γ Δ t → ∀ j → Term Σ Γ Δ (lookup Σ j) → Term Σ Γ Δ t
termSubstState (state i) j t′ with j Fin.≟ i
... | yes refl = t′
... | no _ = state i
termSubstState (var i) j t′ = var i
termSubstState (meta i) j t′ = meta i
termSubstState (funct f ts) j t′ = funct f (helper ts)
where
helper : ∀ {n ts} → All (Term Σ Γ Δ) {n} ts → All (Term Σ Γ Δ) ts
helper [] = []
helper (t ∷ ts) = termSubstState t j t′ ∷ helper ts
termSubstVar : ∀ {t} → Term Σ Γ Δ t → ∀ j → Term Σ Γ Δ (lookup Γ j) → Term Σ Γ Δ t
termSubstVar (state i) j t′ = state i
termSubstVar (var i) j t′ with j Fin.≟ i
... | yes refl = t′
... | no _ = var i
termSubstVar (meta i) j t′ = meta i
termSubstVar (funct f ts) j t′ = funct f (helper ts)
where
helper : ∀ {n ts} → All (Term Σ Γ Δ) {n} ts → All (Term Σ Γ Δ) ts
helper [] = []
helper (t ∷ ts) = termSubstVar t j t′ ∷ helper ts
termElimVar : ∀ {t t′} → Term Σ (t′ ∷ Γ) Δ t → Term Σ Γ Δ t′ → Term Σ Γ Δ t
termElimVar (state i) t′ = state i
termElimVar (var zero) t′ = t′
termElimVar (var (suc i)) t′ = var i
termElimVar (meta i) t′ = meta i
termElimVar (funct f ts) t′ = funct f (helper ts)
where
helper : ∀ {n ts} → All (Term _ _ _) {n} ts → All (Term _ _ _) ts
helper [] = []
helper (t ∷ ts) = termElimVar t t′ ∷ helper ts
termWknVar : ∀ {t t′} → Term Σ Γ Δ t → Term Σ (t′ ∷ Γ) Δ t
termWknVar (state i) = state i
termWknVar (var i) = var (suc i)
termWknVar (meta i) = meta i
termWknVar (funct f ts) = funct f (helper ts)
where
helper : ∀ {n ts} → All (Term _ _ _) {n} ts → All (Term _ _ _) ts
helper [] = []
helper (t ∷ ts) = termWknVar t ∷ helper ts
termWknMeta : ∀ {t t′} → Term Σ Γ Δ t → Term Σ Γ (t′ ∷ Δ) t
termWknMeta (state i) = state i
termWknMeta (var i) = var i
termWknMeta (meta i) = meta (suc i)
termWknMeta (funct f ts) = funct f (helper ts)
where
helper : ∀ {n ts} → All (Term _ _ _) {n} ts → All (Term _ _ _) ts
helper [] = []
helper (t ∷ ts) = termWknMeta t ∷ helper ts
module _ {o} {Σ : Vec Type o} {n} {Γ : Vec Type n} where
infix 4 _∋[_] _⊨_
_∋[_] : ∀ {m Δ} → Assertion Σ Γ {m} Δ → ⟦ Σ ⟧ₜˡ′ × ⟦ Γ ⟧ₜˡ′ × ⟦ Δ ⟧ₜˡ′ → Set (b₁ ⊔ i₁ ⊔ r₁)
P ∧ Q ∋[ s ] = P ∋[ s ] × Q ∋[ s ]
P ∨ Q ∋[ s ] = P ∋[ s ] ⊎ Q ∋[ s ]
P ⇒ Q ∋[ s ] = P ∋[ s ] → Q ∋[ s ]
pred P ts ∋[ σ , γ , δ ] = Lift (b₁ ⊔ i₁ ⊔ r₁) $ T $ P (⟦ ts ⟧′ σ γ δ)
_∋[_] {Δ = Δ} (all P) (σ , γ , δ) = ∀ v → P ∋[ σ , γ , consˡ Δ v δ ]
_∋[_] {Δ = Δ} (some P) (σ , γ , δ) = ∃ λ v → P ∋[ σ , γ , consˡ Δ v δ ]
_⊨_ : ∀ {m Δ} → ⟦ Σ ⟧ₜˡ′ × ⟦ Γ ⟧ₜˡ′ × ⟦ Δ ⟧ₜˡ′ → Assertion Σ Γ {m} Δ → Set (b₁ ⊔ i₁ ⊔ r₁)
s ⊨ P = P ∋[ s ]
asstSubstState : ∀ {m Δ} → Assertion Σ Γ {m} Δ → ∀ j → Term Σ Γ Δ (lookup Σ j) → Assertion Σ Γ Δ
asstSubstState (P ∧ Q) j t = asstSubstState P j t ∧ asstSubstState Q j t
asstSubstState (P ∨ Q) j t = asstSubstState P j t ∨ asstSubstState Q j t
asstSubstState (P ⇒ Q) j t = asstSubstState P j t ⇒ asstSubstState Q j t
asstSubstState (all P) j t = all (asstSubstState P j (termWknMeta t))
asstSubstState (some P) j t = some (asstSubstState P j (termWknMeta t))
asstSubstState (pred p ts) j t = pred p (All.map (λ t′ → termSubstState t′ j t) ts)
asstSubstVar : ∀ {m Δ} → Assertion Σ Γ {m} Δ → ∀ j → Term Σ Γ Δ (lookup Γ j) → Assertion Σ Γ Δ
asstSubstVar (P ∧ Q) j t = asstSubstVar P j t ∧ asstSubstVar Q j t
asstSubstVar (P ∨ Q) j t = asstSubstVar P j t ∨ asstSubstVar Q j t
asstSubstVar (P ⇒ Q) j t = asstSubstVar P j t ⇒ asstSubstVar Q j t
asstSubstVar (all P) j t = all (asstSubstVar P j (termWknMeta t))
asstSubstVar (some P) j t = some (asstSubstVar P j (termWknMeta t))
asstSubstVar (pred p ts) j t = pred p (All.map (λ t′ → termSubstVar t′ j t) ts)
asstElimVar : ∀ {m Δ t′} → Assertion Σ (t′ ∷ Γ) {m} Δ → Term Σ Γ Δ t′ → Assertion Σ Γ Δ
asstElimVar (P ∧ Q) t = asstElimVar P t ∧ asstElimVar Q t
asstElimVar (P ∨ Q) t = asstElimVar P t ∨ asstElimVar Q t
asstElimVar (P ⇒ Q) t = asstElimVar P t ⇒ asstElimVar Q t
asstElimVar (all P) t = all (asstElimVar P (termWknMeta t))
asstElimVar (some P) t = some (asstElimVar P (termWknMeta t))
asstElimVar (pred p ts) t = pred p (All.map (λ t′ → termElimVar t′ t) ts)
module _ {o} {Σ : Vec Type o} where
open Code Σ
data HoareTriple {n Γ m Δ} : Assertion Σ {n} Γ {m} Δ → Statement Γ → Assertion Σ Γ Δ → Set (b₁ ⊔ i₁ ⊔ r₁) where
csqs : ∀ {P₁ P₂ Q₁ Q₂ : Assertion Σ Γ Δ} {s} → (_⊨ P₁) ⊆ (_⊨ P₂) → HoareTriple P₂ s Q₂ → (_⊨ Q₂) ⊆ (_⊨ Q₁) → HoareTriple P₁ s Q₁
_∙_ : ∀ {P Q R s₁ s₂} → HoareTriple P s₁ Q → HoareTriple Q s₂ R → HoareTriple P (s₁ ∙ s₂) R
skip : ∀ {P} → HoareTriple P skip P
|
