From c32c75ab3d5628163a4ece83e38d85152bf9e189 Mon Sep 17 00:00:00 2001 From: Greg Brown Date: Tue, 8 Mar 2022 17:14:41 +0000 Subject: Add semantics of Hoare logic assertions. --- src/Helium/Semantics/Axiomatic/Assertion.agda | 53 +++++++++++++++++---------- 1 file changed, 34 insertions(+), 19 deletions(-) diff --git a/src/Helium/Semantics/Axiomatic/Assertion.agda b/src/Helium/Semantics/Axiomatic/Assertion.agda index 70feb2b..8d66886 100644 --- a/src/Helium/Semantics/Axiomatic/Assertion.agda +++ b/src/Helium/Semantics/Axiomatic/Assertion.agda @@ -16,17 +16,21 @@ module Helium.Semantics.Axiomatic.Assertion open RawPseudocode rawPseudocode open import Data.Bool as Bool using (Bool) +open import Data.Empty.Polymorphic using (⊥) open import Data.Fin as Fin using (suc) open import Data.Fin.Patterns open import Data.Nat using (ℕ; suc) -open import Data.Product using (proj₁; proj₂) +import Data.Nat.Properties as ℕₚ +open import Data.Product using (∃; _×_; _,_; proj₁; proj₂) +open import Data.Sum using (_⊎_) +open import Data.Unit.Polymorphic using (⊤) open import Data.Vec as Vec using (Vec; []; _∷_; _++_) 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.Axiomatic.Core rawPseudocode open import Helium.Semantics.Axiomatic.Term rawPseudocode as Term using (Term) -open import Level using (_⊔_; lift) -open import Relation.Nullary using (does) +open import Level using (_⊔_; Lift; lift; lower) renaming (suc to ℓsuc) private variable @@ -34,19 +38,17 @@ private m n o : ℕ Γ Δ Σ ts : Vec Type m -infixl 7 _[_]↦_ - open Term.Term -data Assertion (Σ : Vec Type o) (Γ : Vec Type n) (Δ : Vec Type m) : Set (b₁ ⊔ i₁ ⊔ r₁) where - _[_]↦_ : ∀ {m t} → Term Σ Γ Δ (asType t m) → Term Σ Γ Δ (fin m) → Term Σ Γ Δ (elemType t) → Assertion Σ Γ Δ +data Assertion (Σ : Vec Type o) (Γ : Vec Type n) (Δ : Vec Type m) : Set (ℓsuc (b₁ ⊔ i₁ ⊔ r₁)) + +data Assertion Σ Γ Δ where all : Assertion Σ Γ (t ∷ Δ) → Assertion Σ Γ Δ some : Assertion Σ Γ (t ∷ Δ) → Assertion Σ Γ Δ pred : Term Σ Γ Δ bool → Assertion Σ Γ Δ - comb : ∀ {n} → (Vec Bool n → Bool) → Vec (Assertion Σ Γ Δ) n → Assertion Σ Γ Δ + comb : ∀ {n} → (Vec (Set (b₁ ⊔ i₁ ⊔ r₁)) n → Set (b₁ ⊔ i₁ ⊔ r₁)) → Vec (Assertion Σ Γ Δ) n → Assertion Σ Γ Δ elimVar : Assertion Σ (t ∷ Γ) Δ → Term Σ Γ Δ t → Assertion Σ Γ Δ -elimVar (ref [ i ]↦ val) t = Term.elimVar ref t [ Term.elimVar i t ]↦ Term.elimVar val t elimVar (all P) t = all (elimVar P (Term.wknMeta t)) elimVar (some P) t = some (elimVar P (Term.wknMeta t)) elimVar (pred p) t = pred (Term.elimVar p t) @@ -57,7 +59,6 @@ elimVar (comb f Ps) t = comb f (helper Ps t) helper (P ∷ Ps) t = elimVar P t ∷ helper Ps t wknVar : Assertion Σ Γ Δ → Assertion Σ (t ∷ Γ) Δ -wknVar (ref [ i ]↦ val) = Term.wknVar ref [ Term.wknVar i ]↦ Term.wknVar val wknVar (all P) = all (wknVar P) wknVar (some P) = some (wknVar P) wknVar (pred p) = pred (Term.wknVar p) @@ -68,7 +69,6 @@ wknVar (comb f Ps) = comb f (helper Ps) helper (P ∷ Ps) = wknVar P ∷ helper Ps wknVars : (ts : Vec Type n) → Assertion Σ Γ Δ → Assertion Σ (ts ++ Γ) Δ -wknVars τs (ref [ i ]↦ val) = Term.wknVars τs ref [ Term.wknVars τs i ]↦ Term.wknVars τs val wknVars τs (all P) = all (wknVars τs P) wknVars τs (some P) = some (wknVars τs P) wknVars τs (pred p) = pred (Term.wknVars τs p) @@ -79,7 +79,6 @@ wknVars τs (comb f Ps) = comb f (helper Ps) helper (P ∷ Ps) = wknVars τs P ∷ helper Ps addVars : Assertion Σ [] Δ → Assertion Σ Γ Δ -addVars (ref [ i ]↦ val) = Term.addVars ref [ Term.addVars i ]↦ Term.addVars val addVars (all P) = all (addVars P) addVars (some P) = some (addVars P) addVars (pred p) = pred (Term.addVars p) @@ -90,7 +89,6 @@ addVars (comb f Ps) = comb f (helper Ps) helper (P ∷ Ps) = addVars P ∷ helper Ps wknMetaAt : ∀ i → Assertion Σ Γ Δ → Assertion Σ Γ (Vec.insert Δ i t) -wknMetaAt i (ref [ j ]↦ val) = Term.wknMetaAt i ref [ Term.wknMetaAt i j ]↦ Term.wknMetaAt i val wknMetaAt i (all P) = all (wknMetaAt (suc i) P) wknMetaAt i (some P) = some (wknMetaAt (suc i) P) wknMetaAt i (pred p) = pred (Term.wknMetaAt i p) @@ -108,7 +106,6 @@ wknMetas (_ ∷ ts) P = wknMetaAt 0F (wknMetas ts P) module _ (2≉0 : Term.2≉0) where -- NOTE: better to induct on e here than in Term? subst : Assertion Σ Γ Δ → {e : Code.Expression Σ Γ t} → Code.CanAssign Σ e → Term Σ Γ Δ t → Assertion Σ Γ Δ - subst (ref [ i ]↦ val) e t = Term.subst 2≉0 ref e t [ Term.subst 2≉0 i e t ]↦ Term.subst 2≉0 val e t subst (all P) e t = all (subst P e (Term.wknMeta t)) subst (some P) e t = some (subst P e (Term.wknMeta t)) subst (pred p) e t = pred (Term.subst 2≉0 p e t) @@ -123,16 +120,16 @@ module Construct where infixl 5 _∨_ true : Assertion Σ Γ Δ - true = comb (λ _ → Bool.true) [] + true = comb (λ _ → ⊤) [] false : Assertion Σ Γ Δ - false = comb (λ _ → Bool.false) [] + false = comb (λ _ → ⊥) [] _∧_ : Assertion Σ Γ Δ → Assertion Σ Γ Δ → Assertion Σ Γ Δ - P ∧ Q = comb (λ { (p ∷ q ∷ []) → p Bool.∧ q }) (P ∷ Q ∷ []) + P ∧ Q = comb (λ { (P ∷ Q ∷ []) → P × Q }) (P ∷ Q ∷ []) _∨_ : Assertion Σ Γ Δ → Assertion Σ Γ Δ → Assertion Σ Γ Δ - P ∨ Q = comb (λ { (p ∷ q ∷ []) → p Bool.∨ q }) (P ∷ Q ∷ []) + P ∨ Q = comb (λ { (P ∷ Q ∷ []) → P ⊎ Q }) (P ∷ Q ∷ []) equal : Term Σ Γ Δ t → Term Σ Γ Δ t → Assertion Σ Γ Δ equal {t = bool} x y = pred Term.[ bool ][ x ≟ y ] @@ -143,6 +140,24 @@ module Construct where equal {t = bits n} x y = pred Term.[ bits n ][ x ≟ y ] equal {t = tuple _ []} x y = true equal {t = tuple _ (t ∷ ts)} x y = equal {t = t} (Term.func₁ proj₁ x) (Term.func₁ proj₁ y) ∧ equal (Term.func₁ proj₂ x) (Term.func₁ proj₂ y) - equal {t = array t n} x y = all (some (Term.wknMeta (Term.wknMeta x) [ meta 1F ]↦ meta 0F ∧ Term.wknMeta (Term.wknMeta y) [ meta 1F ]↦ meta 0F)) + equal {t = array t 0} x y = true + equal {t = array t (suc n)} x y = all (equal {t = t} (index x) (index y)) + where + index = λ v → Term.unbox (array t) $ + Term.func₁ proj₁ $ + Term.cut (array t) + (Term.cast (array t) (ℕₚ.+-comm 1 n) (Term.wknMeta v)) + (meta 0F) open Construct public + +⟦_⟧ : Assertion Σ Γ Δ → ⟦ Σ ⟧ₜ′ → ⟦ Γ ⟧ₜ′ → ⟦ Δ ⟧ₜ′ → Set (b₁ ⊔ i₁ ⊔ r₁) +⟦_⟧′ : Vec (Assertion Σ Γ Δ) n → ⟦ Σ ⟧ₜ′ → ⟦ Γ ⟧ₜ′ → ⟦ Δ ⟧ₜ′ → Vec (Set (b₁ ⊔ i₁ ⊔ r₁)) n + +⟦ all P ⟧ σ γ δ = ∀ x → ⟦ P ⟧ σ γ (x , δ) +⟦ some P ⟧ σ γ δ = ∃ λ x → ⟦ P ⟧ σ γ (x , δ) +⟦ pred p ⟧ σ γ δ = Lift (b₁ ⊔ i₁ ⊔ r₁) (Bool.T (lower (Term.⟦ p ⟧ σ γ δ))) +⟦ comb f Ps ⟧ σ γ δ = f (⟦ Ps ⟧′ σ γ δ) + +⟦ [] ⟧′ σ γ δ = [] +⟦ P ∷ Ps ⟧′ σ γ δ = ⟦ P ⟧ σ γ δ ∷ ⟦ Ps ⟧′ σ γ δ -- cgit v1.2.3