Rewrite Forward analysis to use statement-based evaluators.
To keep old (expression-based) analyses working, switch to using instance search and provide "adapters" that auto-construct statement analyzers from expression analyzers. Signed-off-by: Danila Fedorin <danila.fedorin@gmail.com>
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@ -8,208 +8,28 @@ module Analysis.Forward
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(≈ˡ-dec : IsDecidable _≈ˡ_) where
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open import Data.Empty using (⊥-elim)
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open import Data.String using (String) renaming (_≟_ to _≟ˢ_)
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open import Data.Nat using (suc)
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open import Data.Product using (_×_; proj₁; proj₂; _,_)
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open import Data.Sum using (inj₁; inj₂)
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open import Data.List using (List; _∷_; []; foldr; foldl; cartesianProduct; cartesianProductWith)
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open import Data.List.Membership.Propositional as MemProp using () renaming (_∈_ to _∈ˡ_)
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open import Data.String using (String)
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open import Data.Product using (_,_)
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open import Data.List using (_∷_; []; foldr; foldl)
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open import Data.List.Relation.Unary.Any as Any using ()
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open import Relation.Binary.PropositionalEquality using (_≡_; refl; cong; sym; trans; subst)
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open import Relation.Nullary using (¬_; Dec; yes; no)
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open import Data.Unit using (⊤)
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open import Relation.Binary.PropositionalEquality using (_≡_; refl; cong; sym; subst)
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open import Relation.Nullary using (yes; no)
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open import Function using (_∘_; flip)
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import Chain
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open import Utils using (Pairwise; _⇒_; _∨_)
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import Lattice.FiniteValueMap
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open IsFiniteHeightLattice isFiniteHeightLatticeˡ
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using ()
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renaming
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( isLattice to isLatticeˡ
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; fixedHeight to fixedHeightˡ
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; _≼_ to _≼ˡ_
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; ≈-sym to ≈ˡ-sym
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)
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using () renaming (isLattice to isLatticeˡ)
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module WithProg (prog : Program) where
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open import Analysis.Forward.Lattices isFiniteHeightLatticeˡ ≈ˡ-dec prog
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open import Analysis.Forward.Evaluation isFiniteHeightLatticeˡ ≈ˡ-dec prog
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open Program prog
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-- The variable -> abstract value (e.g. sign) map is a finite value-map
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-- with keys strings. Use a bundle to avoid explicitly specifying operators.
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-- It's helpful to export these via 'public' since consumers tend to
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-- use various variable lattice operations.
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module VariableValuesFiniteMap = Lattice.FiniteValueMap.WithKeys _≟ˢ_ isLatticeˡ vars
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open VariableValuesFiniteMap
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using ()
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renaming
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( FiniteMap to VariableValues
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; isLattice to isLatticeᵛ
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; _≈_ to _≈ᵛ_
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; _⊔_ to _⊔ᵛ_
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; _≼_ to _≼ᵛ_
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; ≈₂-dec⇒≈-dec to ≈ˡ-dec⇒≈ᵛ-dec
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; _∈_ to _∈ᵛ_
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; _∈k_ to _∈kᵛ_
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; _updating_via_ to _updatingᵛ_via_
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; locate to locateᵛ
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; m₁≼m₂⇒m₁[k]≼m₂[k] to m₁≼m₂⇒m₁[k]ᵛ≼m₂[k]ᵛ
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; ∈k-dec to ∈k-decᵛ
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; all-equal-keys to all-equal-keysᵛ
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)
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public
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open IsLattice isLatticeᵛ
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using ()
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renaming
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( ⊔-Monotonicˡ to ⊔ᵛ-Monotonicˡ
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; ⊔-Monotonicʳ to ⊔ᵛ-Monotonicʳ
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; ⊔-idemp to ⊔ᵛ-idemp
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)
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open Lattice.FiniteValueMap.IterProdIsomorphism _≟ˢ_ isLatticeˡ
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using ()
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renaming
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( Provenance-union to Provenance-unionᵐ
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)
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open Lattice.FiniteValueMap.IterProdIsomorphism.WithUniqueKeysAndFixedHeight _≟ˢ_ isLatticeˡ vars-Unique ≈ˡ-dec _ fixedHeightˡ
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using ()
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renaming
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( isFiniteHeightLattice to isFiniteHeightLatticeᵛ
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; ⊥-contains-bottoms to ⊥ᵛ-contains-bottoms
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)
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private
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≈ᵛ-dec = ≈ˡ-dec⇒≈ᵛ-dec ≈ˡ-dec
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joinSemilatticeᵛ = IsFiniteHeightLattice.joinSemilattice isFiniteHeightLatticeᵛ
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fixedHeightᵛ = IsFiniteHeightLattice.fixedHeight isFiniteHeightLatticeᵛ
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⊥ᵛ = Chain.Height.⊥ fixedHeightᵛ
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-- Finally, the map we care about is (state -> (variables -> value)). Bring that in.
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module StateVariablesFiniteMap = Lattice.FiniteValueMap.WithKeys _≟_ isLatticeᵛ states
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open StateVariablesFiniteMap
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using (_[_]; []-∈; m₁≼m₂⇒m₁[ks]≼m₂[ks]; m₁≈m₂⇒k∈m₁⇒k∈km₂⇒v₁≈v₂)
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renaming
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( FiniteMap to StateVariables
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; isLattice to isLatticeᵐ
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; _≈_ to _≈ᵐ_
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; _∈_ to _∈ᵐ_
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; _∈k_ to _∈kᵐ_
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; locate to locateᵐ
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; _≼_ to _≼ᵐ_
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; ≈₂-dec⇒≈-dec to ≈ᵛ-dec⇒≈ᵐ-dec
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; m₁≼m₂⇒m₁[k]≼m₂[k] to m₁≼m₂⇒m₁[k]ᵐ≼m₂[k]ᵐ
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)
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open Lattice.FiniteValueMap.IterProdIsomorphism.WithUniqueKeysAndFixedHeight _≟_ isLatticeᵛ states-Unique ≈ᵛ-dec _ fixedHeightᵛ
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using ()
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renaming
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( isFiniteHeightLattice to isFiniteHeightLatticeᵐ
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)
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open IsFiniteHeightLattice isFiniteHeightLatticeᵐ
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using ()
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renaming
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( ≈-sym to ≈ᵐ-sym
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)
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private
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≈ᵐ-dec = ≈ᵛ-dec⇒≈ᵐ-dec ≈ᵛ-dec
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fixedHeightᵐ = IsFiniteHeightLattice.fixedHeight isFiniteHeightLatticeᵐ
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-- We now have our (state -> (variables -> value)) map.
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-- Define a couple of helpers to retrieve values from it. Specifically,
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-- since the State type is as specific as possible, it's always possible to
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-- retrieve the variable values at each state.
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states-in-Map : ∀ (s : State) (sv : StateVariables) → s ∈kᵐ sv
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states-in-Map s sv@(m , ksv≡states) rewrite ksv≡states = states-complete s
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variablesAt : State → StateVariables → VariableValues
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variablesAt s sv = proj₁ (locateᵐ {s} {sv} (states-in-Map s sv))
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variablesAt-∈ : ∀ (s : State) (sv : StateVariables) → (s , variablesAt s sv) ∈ᵐ sv
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variablesAt-∈ s sv = proj₂ (locateᵐ {s} {sv} (states-in-Map s sv))
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variablesAt-≈ : ∀ s sv₁ sv₂ → sv₁ ≈ᵐ sv₂ → variablesAt s sv₁ ≈ᵛ variablesAt s sv₂
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variablesAt-≈ s sv₁ sv₂ sv₁≈sv₂ =
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m₁≈m₂⇒k∈m₁⇒k∈km₂⇒v₁≈v₂ sv₁ sv₂ sv₁≈sv₂
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(states-in-Map s sv₁) (states-in-Map s sv₂)
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-- build up the 'join' function, which follows from Exercise 4.26's
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--
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-- L₁ → (A → L₂)
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--
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-- Construction, with L₁ = (A → L₂), and f = id
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joinForKey : State → StateVariables → VariableValues
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joinForKey k states = foldr _⊔ᵛ_ ⊥ᵛ (states [ incoming k ])
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-- The per-key join is made up of map key accesses (which are monotonic)
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-- and folds using the join operation (also monotonic)
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joinForKey-Mono : ∀ (k : State) → Monotonic _≼ᵐ_ _≼ᵛ_ (joinForKey k)
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joinForKey-Mono k {fm₁} {fm₂} fm₁≼fm₂ =
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foldr-Mono joinSemilatticeᵛ joinSemilatticeᵛ (fm₁ [ incoming k ]) (fm₂ [ incoming k ]) _⊔ᵛ_ ⊥ᵛ ⊥ᵛ
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(m₁≼m₂⇒m₁[ks]≼m₂[ks] fm₁ fm₂ (incoming k) fm₁≼fm₂)
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(⊔ᵛ-idemp ⊥ᵛ) ⊔ᵛ-Monotonicʳ ⊔ᵛ-Monotonicˡ
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-- The name f' comes from the formulation of Exercise 4.26.
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open StateVariablesFiniteMap.GeneralizedUpdate states isLatticeᵐ (λ x → x) (λ a₁≼a₂ → a₁≼a₂) joinForKey joinForKey-Mono states
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using ()
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renaming
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( f' to joinAll
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; f'-Monotonic to joinAll-Mono
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; f'-k∈ks-≡ to joinAll-k∈ks-≡
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)
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private
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variablesAt-joinAll : ∀ (s : State) (sv : StateVariables) →
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variablesAt s (joinAll sv) ≡ joinForKey s sv
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variablesAt-joinAll s sv
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with (vs , s,vs∈usv) ← locateᵐ {s} {joinAll sv} (states-in-Map s (joinAll sv)) =
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joinAll-k∈ks-≡ {l = sv} (states-complete s) s,vs∈usv
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record Evaluator : Set where
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field
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eval : Expr → VariableValues → L
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eval-Mono : ∀ (e : Expr) → Monotonic _≼ᵛ_ _≼ˡ_ (eval e)
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-- With 'join' in hand, we need to perform abstract evaluation.
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private module WithEvaluator {{evaluator : Evaluator}} where
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open Evaluator evaluator
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-- For a particular evaluation function, we need to perform an evaluation
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-- for an assignment, and update the corresponding key. Use Exercise 4.26's
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-- generalized update to set the single key's value.
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module _ (k : String) (e : Expr) where
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open VariableValuesFiniteMap.GeneralizedUpdate vars isLatticeᵛ (λ x → x) (λ a₁≼a₂ → a₁≼a₂) (λ _ → eval e) (λ _ {vs₁} {vs₂} vs₁≼vs₂ → eval-Mono e {vs₁} {vs₂} vs₁≼vs₂) (k ∷ [])
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using ()
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renaming
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( f' to updateVariablesFromExpression
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; f'-Monotonic to updateVariablesFromExpression-Mono
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; f'-k∈ks-≡ to updateVariablesFromExpression-k∈ks-≡
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; f'-k∉ks-backward to updateVariablesFromExpression-k∉ks-backward
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)
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public
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-- The per-state update function makes use of the single-key setter,
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-- updateVariablesFromExpression, for the case where the statement
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-- is an assignment.
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--
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-- This per-state function adjusts the variables in that state,
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-- also monotonically; we derive the for-each-state update from
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-- the Exercise 4.26 again.
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updateVariablesFromStmt : BasicStmt → VariableValues → VariableValues
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updateVariablesFromStmt (k ← e) vs = updateVariablesFromExpression k e vs
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updateVariablesFromStmt noop vs = vs
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updateVariablesFromStmt-Monoʳ : ∀ (bs : BasicStmt) → Monotonic _≼ᵛ_ _≼ᵛ_ (updateVariablesFromStmt bs)
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updateVariablesFromStmt-Monoʳ (k ← e) {vs₁} {vs₂} vs₁≼vs₂ = updateVariablesFromExpression-Mono k e {vs₁} {vs₂} vs₁≼vs₂
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updateVariablesFromStmt-Monoʳ noop vs₁≼vs₂ = vs₁≼vs₂
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private module WithStmtEvaluator {{evaluator : StmtEvaluator}} where
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open StmtEvaluator evaluator
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updateVariablesForState : State → StateVariables → VariableValues
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updateVariablesForState s sv =
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foldl (flip updateVariablesFromStmt) (variablesAt s sv) (code s)
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foldl (flip (eval s)) (variablesAt s sv) (code s)
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updateVariablesForState-Monoʳ : ∀ (s : State) → Monotonic _≼ᵐ_ _≼ᵛ_ (updateVariablesForState s)
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updateVariablesForState-Monoʳ s {sv₁} {sv₂} sv₁≼sv₂ =
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@ -220,7 +40,7 @@ module WithProg (prog : Program) where
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vs₁≼vs₂ = m₁≼m₂⇒m₁[k]ᵐ≼m₂[k]ᵐ sv₁ sv₂ sv₁≼sv₂ s,vs₁∈sv₁ s,vs₂∈sv₂
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in
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foldl-Mono' (IsLattice.joinSemilattice isLatticeᵛ) bss
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(flip updateVariablesFromStmt) updateVariablesFromStmt-Monoʳ
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(flip (eval s)) (eval-Monoʳ s)
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vs₁≼vs₂
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open StateVariablesFiniteMap.GeneralizedUpdate states isLatticeᵐ (λ x → x) (λ a₁≼a₂ → a₁≼a₂) updateVariablesForState updateVariablesForState-Monoʳ states
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@ -256,146 +76,68 @@ module WithProg (prog : Program) where
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with (vs , s,vs∈usv) ← locateᵐ {s} {updateAll sv} (states-in-Map s (updateAll sv)) =
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updateAll-k∈ks-≡ {l = sv} (states-complete s) s,vs∈usv
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open WithEvaluator
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open WithEvaluator using (result; analyze; result≈analyze-result) public
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module WithValidInterpretation {{latticeInterpretationˡ : LatticeInterpretation isLatticeˡ}}
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{{validEvaluator : ValidStmtEvaluator evaluator latticeInterpretationˡ}} where
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open ValidStmtEvaluator validEvaluator
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private module _ {{latticeInterpretationˡ : LatticeInterpretation isLatticeˡ}} where
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open LatticeInterpretation latticeInterpretationˡ
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using ()
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renaming
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( ⟦_⟧ to ⟦_⟧ˡ
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; ⟦⟧-respects-≈ to ⟦⟧ˡ-respects-≈ˡ
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; ⟦⟧-⊔-∨ to ⟦⟧ˡ-⊔ˡ-∨
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)
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public
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eval-fold-valid : ∀ {s bss vs ρ₁ ρ₂} → ρ₁ , bss ⇒ᵇˢ ρ₂ → ⟦ vs ⟧ᵛ ρ₁ → ⟦ foldl (flip (eval s)) vs bss ⟧ᵛ ρ₂
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eval-fold-valid {_} [] ⟦vs⟧ρ = ⟦vs⟧ρ
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eval-fold-valid {s} {bs ∷ bss'} {vs} {ρ₁} {ρ₂} (ρ₁,bs⇒ρ ∷ ρ,bss'⇒ρ₂) ⟦vs⟧ρ₁ =
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eval-fold-valid
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{bss = bss'} {eval s bs vs} ρ,bss'⇒ρ₂
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(valid ρ₁,bs⇒ρ ⟦vs⟧ρ₁)
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⟦_⟧ᵛ : VariableValues → Env → Set
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⟦_⟧ᵛ vs ρ = ∀ {k l} → (k , l) ∈ᵛ vs → ∀ {v} → (k , v) Language.∈ ρ → ⟦ l ⟧ˡ v
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updateVariablesForState-matches : ∀ {s sv ρ₁ ρ₂} → ρ₁ , (code s) ⇒ᵇˢ ρ₂ → ⟦ variablesAt s sv ⟧ᵛ ρ₁ → ⟦ updateVariablesForState s sv ⟧ᵛ ρ₂
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updateVariablesForState-matches = eval-fold-valid
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⟦⊥ᵛ⟧ᵛ∅ : ⟦ ⊥ᵛ ⟧ᵛ []
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⟦⊥ᵛ⟧ᵛ∅ _ ()
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updateAll-matches : ∀ {s sv ρ₁ ρ₂} → ρ₁ , (code s) ⇒ᵇˢ ρ₂ → ⟦ variablesAt s sv ⟧ᵛ ρ₁ → ⟦ variablesAt s (updateAll sv) ⟧ᵛ ρ₂
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updateAll-matches {s} {sv} ρ₁,bss⇒ρ₂ ⟦vs⟧ρ₁
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rewrite variablesAt-updateAll s sv =
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updateVariablesForState-matches {s} {sv} ρ₁,bss⇒ρ₂ ⟦vs⟧ρ₁
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⟦⟧ᵛ-respects-≈ᵛ : ∀ {vs₁ vs₂ : VariableValues} → vs₁ ≈ᵛ vs₂ → ⟦ vs₁ ⟧ᵛ ⇒ ⟦ vs₂ ⟧ᵛ
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⟦⟧ᵛ-respects-≈ᵛ {m₁ , _} {m₂ , _}
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(m₁⊆m₂ , m₂⊆m₁) ρ ⟦vs₁⟧ρ {k} {l} k,l∈m₂ {v} k,v∈ρ =
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let
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(l' , (l≈l' , k,l'∈m₁)) = m₂⊆m₁ _ _ k,l∈m₂
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⟦l'⟧v = ⟦vs₁⟧ρ k,l'∈m₁ k,v∈ρ
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in
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⟦⟧ˡ-respects-≈ˡ (≈ˡ-sym l≈l') v ⟦l'⟧v
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stepTrace : ∀ {s₁ ρ₁ ρ₂} → ⟦ joinForKey s₁ result ⟧ᵛ ρ₁ → ρ₁ , (code s₁) ⇒ᵇˢ ρ₂ → ⟦ variablesAt s₁ result ⟧ᵛ ρ₂
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stepTrace {s₁} {ρ₁} {ρ₂} ⟦joinForKey-s₁⟧ρ₁ ρ₁,bss⇒ρ₂ =
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let
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-- I'd use rewrite, but Agda gets a memory overflow (?!).
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⟦joinAll-result⟧ρ₁ =
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subst (λ vs → ⟦ vs ⟧ᵛ ρ₁)
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(sym (variablesAt-joinAll s₁ result))
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⟦joinForKey-s₁⟧ρ₁
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⟦analyze-result⟧ρ₂ =
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updateAll-matches {sv = joinAll result}
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ρ₁,bss⇒ρ₂ ⟦joinAll-result⟧ρ₁
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analyze-result≈result =
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≈ᵐ-sym {result} {updateAll (joinAll result)}
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result≈analyze-result
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analyze-s₁≈s₁ =
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variablesAt-≈ s₁ (updateAll (joinAll result))
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result (analyze-result≈result)
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in
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⟦⟧ᵛ-respects-≈ᵛ {variablesAt s₁ (updateAll (joinAll result))} {variablesAt s₁ result} (analyze-s₁≈s₁) ρ₂ ⟦analyze-result⟧ρ₂
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⟦⟧ᵛ-⊔ᵛ-∨ : ∀ {vs₁ vs₂ : VariableValues} → (⟦ vs₁ ⟧ᵛ ∨ ⟦ vs₂ ⟧ᵛ) ⇒ ⟦ vs₁ ⊔ᵛ vs₂ ⟧ᵛ
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⟦⟧ᵛ-⊔ᵛ-∨ {vs₁} {vs₂} ρ ⟦vs₁⟧ρ∨⟦vs₂⟧ρ {k} {l} k,l∈vs₁₂ {v} k,v∈ρ
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with ((l₁ , l₂) , (refl , (k,l₁∈vs₁ , k,l₂∈vs₂)))
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← Provenance-unionᵐ vs₁ vs₂ k,l∈vs₁₂
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with ⟦vs₁⟧ρ∨⟦vs₂⟧ρ
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... | inj₁ ⟦vs₁⟧ρ = ⟦⟧ˡ-⊔ˡ-∨ {l₁} {l₂} v (inj₁ (⟦vs₁⟧ρ k,l₁∈vs₁ k,v∈ρ))
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... | inj₂ ⟦vs₂⟧ρ = ⟦⟧ˡ-⊔ˡ-∨ {l₁} {l₂} v (inj₂ (⟦vs₂⟧ρ k,l₂∈vs₂ k,v∈ρ))
|
||||
|
||||
⟦⟧ᵛ-foldr : ∀ {vs : VariableValues} {vss : List VariableValues} {ρ : Env} →
|
||||
⟦ vs ⟧ᵛ ρ → vs ∈ˡ vss → ⟦ foldr _⊔ᵛ_ ⊥ᵛ vss ⟧ᵛ ρ
|
||||
⟦⟧ᵛ-foldr {vs} {vs ∷ vss'} {ρ = ρ} ⟦vs⟧ρ (Any.here refl) =
|
||||
⟦⟧ᵛ-⊔ᵛ-∨ {vs₁ = vs} {vs₂ = foldr _⊔ᵛ_ ⊥ᵛ vss'} ρ (inj₁ ⟦vs⟧ρ)
|
||||
⟦⟧ᵛ-foldr {vs} {vs' ∷ vss'} {ρ = ρ} ⟦vs⟧ρ (Any.there vs∈vss') =
|
||||
⟦⟧ᵛ-⊔ᵛ-∨ {vs₁ = vs'} {vs₂ = foldr _⊔ᵛ_ ⊥ᵛ vss'} ρ
|
||||
(inj₂ (⟦⟧ᵛ-foldr ⟦vs⟧ρ vs∈vss'))
|
||||
|
||||
module _ {{evaluator : Evaluator}} {{interpretation : LatticeInterpretation isLatticeˡ}} where
|
||||
open Evaluator evaluator
|
||||
open LatticeInterpretation interpretation
|
||||
|
||||
IsValid : Set
|
||||
IsValid = ∀ {vs ρ e v} → ρ , e ⇒ᵉ v → ⟦ vs ⟧ᵛ ρ → ⟦ eval e vs ⟧ˡ v
|
||||
|
||||
record ValidInterpretation : Set₁ where
|
||||
field
|
||||
{{evaluator}} : Evaluator
|
||||
{{interpretation}} : LatticeInterpretation isLatticeˡ
|
||||
|
||||
open Evaluator evaluator public
|
||||
open LatticeInterpretation interpretation public
|
||||
|
||||
field
|
||||
valid : IsValid
|
||||
|
||||
module WithValidInterpretation {{validInterpretation : ValidInterpretation}} where
|
||||
open ValidInterpretation validInterpretation
|
||||
|
||||
updateVariablesFromStmt-matches : ∀ {bs vs ρ₁ ρ₂} → ρ₁ , bs ⇒ᵇ ρ₂ → ⟦ vs ⟧ᵛ ρ₁ → ⟦ updateVariablesFromStmt bs vs ⟧ᵛ ρ₂
|
||||
updateVariablesFromStmt-matches {_} {vs} {ρ₁} {ρ₁} (⇒ᵇ-noop ρ₁) ⟦vs⟧ρ₁ = ⟦vs⟧ρ₁
|
||||
updateVariablesFromStmt-matches {_} {vs} {ρ₁} {_} (⇒ᵇ-← ρ₁ k e v ρ,e⇒v) ⟦vs⟧ρ₁ {k'} {l} k',l∈vs' {v'} k',v'∈ρ₂
|
||||
with k ≟ˢ k' | k',v'∈ρ₂
|
||||
... | yes refl | here _ v _
|
||||
rewrite updateVariablesFromExpression-k∈ks-≡ k e {l = vs} (Any.here refl) k',l∈vs' =
|
||||
valid ρ,e⇒v ⟦vs⟧ρ₁
|
||||
... | yes k≡k' | there _ _ _ _ _ k'≢k _ = ⊥-elim (k'≢k (sym k≡k'))
|
||||
... | no k≢k' | here _ _ _ = ⊥-elim (k≢k' refl)
|
||||
... | no k≢k' | there _ _ _ _ _ _ k',v'∈ρ₁ =
|
||||
walkTrace : ∀ {s₁ s₂ ρ₁ ρ₂} → ⟦ joinForKey s₁ result ⟧ᵛ ρ₁ → Trace {graph} s₁ s₂ ρ₁ ρ₂ → ⟦ variablesAt s₂ result ⟧ᵛ ρ₂
|
||||
walkTrace {s₁} {s₁} {ρ₁} {ρ₂} ⟦joinForKey-s₁⟧ρ₁ (Trace-single ρ₁,bss⇒ρ₂) =
|
||||
stepTrace {s₁} {ρ₁} {ρ₂} ⟦joinForKey-s₁⟧ρ₁ ρ₁,bss⇒ρ₂
|
||||
walkTrace {s₁} {s₂} {ρ₁} {ρ₂} ⟦joinForKey-s₁⟧ρ₁ (Trace-edge {ρ₂ = ρ} {idx₂ = s} ρ₁,bss⇒ρ s₁→s₂ tr) =
|
||||
let
|
||||
k'∉[k] = (λ { (Any.here refl) → k≢k' refl })
|
||||
k',l∈vs = updateVariablesFromExpression-k∉ks-backward k e {l = vs} k'∉[k] k',l∈vs'
|
||||
⟦result-s₁⟧ρ =
|
||||
stepTrace {s₁} {ρ₁} {ρ} ⟦joinForKey-s₁⟧ρ₁ ρ₁,bss⇒ρ
|
||||
s₁∈incomingStates =
|
||||
[]-∈ result (edge⇒incoming s₁→s₂)
|
||||
(variablesAt-∈ s₁ result)
|
||||
⟦joinForKey-s⟧ρ =
|
||||
⟦⟧ᵛ-foldr ⟦result-s₁⟧ρ s₁∈incomingStates
|
||||
in
|
||||
⟦vs⟧ρ₁ k',l∈vs k',v'∈ρ₁
|
||||
walkTrace ⟦joinForKey-s⟧ρ tr
|
||||
|
||||
updateVariablesFromStmt-fold-matches : ∀ {bss vs ρ₁ ρ₂} → ρ₁ , bss ⇒ᵇˢ ρ₂ → ⟦ vs ⟧ᵛ ρ₁ → ⟦ foldl (flip updateVariablesFromStmt) vs bss ⟧ᵛ ρ₂
|
||||
updateVariablesFromStmt-fold-matches [] ⟦vs⟧ρ = ⟦vs⟧ρ
|
||||
updateVariablesFromStmt-fold-matches {bs ∷ bss'} {vs} {ρ₁} {ρ₂} (ρ₁,bs⇒ρ ∷ ρ,bss'⇒ρ₂) ⟦vs⟧ρ₁ =
|
||||
updateVariablesFromStmt-fold-matches
|
||||
{bss'} {updateVariablesFromStmt bs vs} ρ,bss'⇒ρ₂
|
||||
(updateVariablesFromStmt-matches ρ₁,bs⇒ρ ⟦vs⟧ρ₁)
|
||||
joinForKey-initialState-⊥ᵛ : joinForKey initialState result ≡ ⊥ᵛ
|
||||
joinForKey-initialState-⊥ᵛ = cong (λ ins → foldr _⊔ᵛ_ ⊥ᵛ (result [ ins ])) initialState-pred-∅
|
||||
|
||||
updateVariablesForState-matches : ∀ {s sv ρ₁ ρ₂} → ρ₁ , (code s) ⇒ᵇˢ ρ₂ → ⟦ variablesAt s sv ⟧ᵛ ρ₁ → ⟦ updateVariablesForState s sv ⟧ᵛ ρ₂
|
||||
updateVariablesForState-matches =
|
||||
updateVariablesFromStmt-fold-matches
|
||||
⟦joinAll-initialState⟧ᵛ∅ : ⟦ joinForKey initialState result ⟧ᵛ []
|
||||
⟦joinAll-initialState⟧ᵛ∅ = subst (λ vs → ⟦ vs ⟧ᵛ []) (sym joinForKey-initialState-⊥ᵛ) ⟦⊥ᵛ⟧ᵛ∅
|
||||
|
||||
updateAll-matches : ∀ {s sv ρ₁ ρ₂} → ρ₁ , (code s) ⇒ᵇˢ ρ₂ → ⟦ variablesAt s sv ⟧ᵛ ρ₁ → ⟦ variablesAt s (updateAll sv) ⟧ᵛ ρ₂
|
||||
updateAll-matches {s} {sv} ρ₁,bss⇒ρ₂ ⟦vs⟧ρ₁
|
||||
rewrite variablesAt-updateAll s sv =
|
||||
updateVariablesForState-matches {s} {sv} ρ₁,bss⇒ρ₂ ⟦vs⟧ρ₁
|
||||
analyze-correct : ∀ {ρ : Env} → [] , rootStmt ⇒ˢ ρ → ⟦ variablesAt finalState result ⟧ᵛ ρ
|
||||
analyze-correct {ρ} ∅,s⇒ρ = walkTrace {initialState} {finalState} {[]} {ρ} ⟦joinAll-initialState⟧ᵛ∅ (trace ∅,s⇒ρ)
|
||||
|
||||
|
||||
stepTrace : ∀ {s₁ ρ₁ ρ₂} → ⟦ joinForKey s₁ result ⟧ᵛ ρ₁ → ρ₁ , (code s₁) ⇒ᵇˢ ρ₂ → ⟦ variablesAt s₁ result ⟧ᵛ ρ₂
|
||||
stepTrace {s₁} {ρ₁} {ρ₂} ⟦joinForKey-s₁⟧ρ₁ ρ₁,bss⇒ρ₂ =
|
||||
let
|
||||
-- I'd use rewrite, but Agda gets a memory overflow (?!).
|
||||
⟦joinAll-result⟧ρ₁ =
|
||||
subst (λ vs → ⟦ vs ⟧ᵛ ρ₁)
|
||||
(sym (variablesAt-joinAll s₁ result))
|
||||
⟦joinForKey-s₁⟧ρ₁
|
||||
⟦analyze-result⟧ρ₂ =
|
||||
updateAll-matches {sv = joinAll result}
|
||||
ρ₁,bss⇒ρ₂ ⟦joinAll-result⟧ρ₁
|
||||
analyze-result≈result =
|
||||
≈ᵐ-sym {result} {updateAll (joinAll result)}
|
||||
result≈analyze-result
|
||||
analyze-s₁≈s₁ =
|
||||
variablesAt-≈ s₁ (updateAll (joinAll result))
|
||||
result (analyze-result≈result)
|
||||
in
|
||||
⟦⟧ᵛ-respects-≈ᵛ {variablesAt s₁ (updateAll (joinAll result))} {variablesAt s₁ result} (analyze-s₁≈s₁) ρ₂ ⟦analyze-result⟧ρ₂
|
||||
|
||||
walkTrace : ∀ {s₁ s₂ ρ₁ ρ₂} → ⟦ joinForKey s₁ result ⟧ᵛ ρ₁ → Trace {graph} s₁ s₂ ρ₁ ρ₂ → ⟦ variablesAt s₂ result ⟧ᵛ ρ₂
|
||||
walkTrace {s₁} {s₁} {ρ₁} {ρ₂} ⟦joinForKey-s₁⟧ρ₁ (Trace-single ρ₁,bss⇒ρ₂) =
|
||||
stepTrace {s₁} {ρ₁} {ρ₂} ⟦joinForKey-s₁⟧ρ₁ ρ₁,bss⇒ρ₂
|
||||
walkTrace {s₁} {s₂} {ρ₁} {ρ₂} ⟦joinForKey-s₁⟧ρ₁ (Trace-edge {ρ₂ = ρ} {idx₂ = s} ρ₁,bss⇒ρ s₁→s₂ tr) =
|
||||
let
|
||||
⟦result-s₁⟧ρ =
|
||||
stepTrace {s₁} {ρ₁} {ρ} ⟦joinForKey-s₁⟧ρ₁ ρ₁,bss⇒ρ
|
||||
s₁∈incomingStates =
|
||||
[]-∈ result (edge⇒incoming s₁→s₂)
|
||||
(variablesAt-∈ s₁ result)
|
||||
⟦joinForKey-s⟧ρ =
|
||||
⟦⟧ᵛ-foldr ⟦result-s₁⟧ρ s₁∈incomingStates
|
||||
in
|
||||
walkTrace ⟦joinForKey-s⟧ρ tr
|
||||
|
||||
joinForKey-initialState-⊥ᵛ : joinForKey initialState result ≡ ⊥ᵛ
|
||||
joinForKey-initialState-⊥ᵛ = cong (λ ins → foldr _⊔ᵛ_ ⊥ᵛ (result [ ins ])) initialState-pred-∅
|
||||
|
||||
⟦joinAll-initialState⟧ᵛ∅ : ⟦ joinForKey initialState result ⟧ᵛ []
|
||||
⟦joinAll-initialState⟧ᵛ∅ = subst (λ vs → ⟦ vs ⟧ᵛ []) (sym joinForKey-initialState-⊥ᵛ) ⟦⊥ᵛ⟧ᵛ∅
|
||||
|
||||
analyze-correct : ∀ {ρ : Env} → [] , rootStmt ⇒ˢ ρ → ⟦ variablesAt finalState result ⟧ᵛ ρ
|
||||
analyze-correct {ρ} ∅,s⇒ρ = walkTrace {initialState} {finalState} {[]} {ρ} ⟦joinAll-initialState⟧ᵛ∅ (trace ∅,s⇒ρ)
|
||||
|
||||
open WithValidInterpretation using (analyze-correct) public
|
||||
open WithStmtEvaluator using (result; analyze; result≈analyze-result) public
|
||||
open WithStmtEvaluator.WithValidInterpretation using (analyze-correct) public
|
||||
|
||||
100
Analysis/Forward/Adapters.agda
Normal file
100
Analysis/Forward/Adapters.agda
Normal file
@ -0,0 +1,100 @@
|
||||
open import Language hiding (_[_])
|
||||
open import Lattice
|
||||
|
||||
module Analysis.Forward.Adapters
|
||||
{L : Set} {h}
|
||||
{_≈ˡ_ : L → L → Set} {_⊔ˡ_ : L → L → L} {_⊓ˡ_ : L → L → L}
|
||||
(isFiniteHeightLatticeˡ : IsFiniteHeightLattice L h _≈ˡ_ _⊔ˡ_ _⊓ˡ_)
|
||||
(≈ˡ-dec : IsDecidable _≈ˡ_)
|
||||
(prog : Program) where
|
||||
|
||||
open import Analysis.Forward.Lattices isFiniteHeightLatticeˡ ≈ˡ-dec prog
|
||||
open import Analysis.Forward.Evaluation isFiniteHeightLatticeˡ ≈ˡ-dec prog
|
||||
|
||||
open import Data.Empty using (⊥-elim)
|
||||
open import Data.String using (String) renaming (_≟_ to _≟ˢ_)
|
||||
open import Data.Product using (_,_)
|
||||
open import Data.List using (_∷_; []; foldr; foldl)
|
||||
open import Data.List.Relation.Unary.Any as Any using ()
|
||||
open import Relation.Binary.PropositionalEquality using (_≡_; refl; cong; sym; subst)
|
||||
open import Relation.Nullary using (yes; no)
|
||||
open import Function using (_∘_; flip)
|
||||
|
||||
open IsFiniteHeightLattice isFiniteHeightLatticeˡ
|
||||
using ()
|
||||
renaming
|
||||
( isLattice to isLatticeˡ
|
||||
; _≼_ to _≼ˡ_
|
||||
)
|
||||
open Program prog
|
||||
|
||||
-- Now, allow StmtEvaluators to be auto-constructed from ExprEvaluators.
|
||||
module ExprToStmtAdapter {{ exprEvaluator : ExprEvaluator }} where
|
||||
open ExprEvaluator exprEvaluator
|
||||
using ()
|
||||
renaming
|
||||
( eval to evalᵉ
|
||||
; eval-Monoʳ to evalᵉ-Monoʳ
|
||||
)
|
||||
|
||||
-- For a particular evaluation function, we need to perform an evaluation
|
||||
-- for an assignment, and update the corresponding key. Use Exercise 4.26's
|
||||
-- generalized update to set the single key's value.
|
||||
private module _ (k : String) (e : Expr) where
|
||||
open VariableValuesFiniteMap.GeneralizedUpdate vars isLatticeᵛ (λ x → x) (λ a₁≼a₂ → a₁≼a₂) (λ _ → evalᵉ e) (λ _ {vs₁} {vs₂} vs₁≼vs₂ → evalᵉ-Monoʳ e {vs₁} {vs₂} vs₁≼vs₂) (k ∷ [])
|
||||
using ()
|
||||
renaming
|
||||
( f' to updateVariablesFromExpression
|
||||
; f'-Monotonic to updateVariablesFromExpression-Mono
|
||||
; f'-k∈ks-≡ to updateVariablesFromExpression-k∈ks-≡
|
||||
; f'-k∉ks-backward to updateVariablesFromExpression-k∉ks-backward
|
||||
)
|
||||
public
|
||||
|
||||
-- The per-state update function makes use of the single-key setter,
|
||||
-- updateVariablesFromExpression, for the case where the statement
|
||||
-- is an assignment.
|
||||
--
|
||||
-- This per-state function adjusts the variables in that state,
|
||||
-- also monotonically; we derive the for-each-state update from
|
||||
-- the Exercise 4.26 again.
|
||||
|
||||
evalᵇ : State → BasicStmt → VariableValues → VariableValues
|
||||
evalᵇ _ (k ← e) vs = updateVariablesFromExpression k e vs
|
||||
evalᵇ _ noop vs = vs
|
||||
|
||||
evalᵇ-Monoʳ : ∀ (s : State) (bs : BasicStmt) → Monotonic _≼ᵛ_ _≼ᵛ_ (evalᵇ s bs)
|
||||
evalᵇ-Monoʳ _ (k ← e) {vs₁} {vs₂} vs₁≼vs₂ = updateVariablesFromExpression-Mono k e {vs₁} {vs₂} vs₁≼vs₂
|
||||
evalᵇ-Monoʳ _ noop vs₁≼vs₂ = vs₁≼vs₂
|
||||
|
||||
instance
|
||||
stmtEvaluator : StmtEvaluator
|
||||
stmtEvaluator = record { eval = evalᵇ ; eval-Monoʳ = evalᵇ-Monoʳ }
|
||||
|
||||
-- Moreover, correct StmtEvaluators can be constructed from correct
|
||||
-- ExprEvaluators.
|
||||
module _ {{latticeInterpretationˡ : LatticeInterpretation isLatticeˡ}}
|
||||
{{isValid : ValidExprEvaluator exprEvaluator latticeInterpretationˡ}} where
|
||||
open ValidExprEvaluator isValid using () renaming (valid to validᵉ)
|
||||
|
||||
evalᵇ-valid : ∀ {s vs ρ₁ ρ₂ bs} → ρ₁ , bs ⇒ᵇ ρ₂ → ⟦ vs ⟧ᵛ ρ₁ → ⟦ evalᵇ s bs vs ⟧ᵛ ρ₂
|
||||
evalᵇ-valid {_} {vs} {ρ₁} {ρ₁} {_} (⇒ᵇ-noop ρ₁) ⟦vs⟧ρ₁ = ⟦vs⟧ρ₁
|
||||
evalᵇ-valid {_} {vs} {ρ₁} {_} {_} (⇒ᵇ-← ρ₁ k e v ρ,e⇒v) ⟦vs⟧ρ₁ {k'} {l} k',l∈vs' {v'} k',v'∈ρ₂
|
||||
with k ≟ˢ k' | k',v'∈ρ₂
|
||||
... | yes refl | here _ v _
|
||||
rewrite updateVariablesFromExpression-k∈ks-≡ k e {l = vs} (Any.here refl) k',l∈vs' =
|
||||
validᵉ ρ,e⇒v ⟦vs⟧ρ₁
|
||||
... | yes k≡k' | there _ _ _ _ _ k'≢k _ = ⊥-elim (k'≢k (sym k≡k'))
|
||||
... | no k≢k' | here _ _ _ = ⊥-elim (k≢k' refl)
|
||||
... | no k≢k' | there _ _ _ _ _ _ k',v'∈ρ₁ =
|
||||
let
|
||||
k'∉[k] = (λ { (Any.here refl) → k≢k' refl })
|
||||
k',l∈vs = updateVariablesFromExpression-k∉ks-backward k e {l = vs} k'∉[k] k',l∈vs'
|
||||
in
|
||||
⟦vs⟧ρ₁ k',l∈vs k',v'∈ρ₁
|
||||
|
||||
instance
|
||||
validStmtEvaluator : ValidStmtEvaluator stmtEvaluator latticeInterpretationˡ
|
||||
validStmtEvaluator = record
|
||||
{ valid = λ {a} {b} {c} {d} → evalᵇ-valid {a} {b} {c} {d}
|
||||
}
|
||||
66
Analysis/Forward/Evaluation.agda
Normal file
66
Analysis/Forward/Evaluation.agda
Normal file
@ -0,0 +1,66 @@
|
||||
open import Language hiding (_[_])
|
||||
open import Lattice
|
||||
|
||||
module Analysis.Forward.Evaluation
|
||||
{L : Set} {h}
|
||||
{_≈ˡ_ : L → L → Set} {_⊔ˡ_ : L → L → L} {_⊓ˡ_ : L → L → L}
|
||||
(isFiniteHeightLatticeˡ : IsFiniteHeightLattice L h _≈ˡ_ _⊔ˡ_ _⊓ˡ_)
|
||||
(≈ˡ-dec : IsDecidable _≈ˡ_)
|
||||
(prog : Program) where
|
||||
|
||||
open import Analysis.Forward.Lattices isFiniteHeightLatticeˡ ≈ˡ-dec prog
|
||||
open import Data.Product using (_,_)
|
||||
|
||||
open IsFiniteHeightLattice isFiniteHeightLatticeˡ
|
||||
using ()
|
||||
renaming
|
||||
( isLattice to isLatticeˡ
|
||||
; _≼_ to _≼ˡ_
|
||||
)
|
||||
open Program prog
|
||||
|
||||
-- The "full" version of the analysis ought to define a function
|
||||
-- that analyzes each basic statement. For some analyses, the state ID
|
||||
-- is used as part of the lattice, so include it here.
|
||||
record StmtEvaluator : Set where
|
||||
field
|
||||
eval : State → BasicStmt → VariableValues → VariableValues
|
||||
eval-Monoʳ : ∀ (s : State) (bs : BasicStmt) → Monotonic _≼ᵛ_ _≼ᵛ_ (eval s bs)
|
||||
|
||||
-- For some "simpler" analyes, all we need to do is analyze the expressions.
|
||||
-- For that purpose, provide a simpler evaluator type.
|
||||
record ExprEvaluator : Set where
|
||||
field
|
||||
eval : Expr → VariableValues → L
|
||||
eval-Monoʳ : ∀ (e : Expr) → Monotonic _≼ᵛ_ _≼ˡ_ (eval e)
|
||||
|
||||
-- Evaluators have a notion of being "valid", in which the (symbolic)
|
||||
-- manipulations on lattice elements they perform match up with
|
||||
-- the semantics. Define what it means to be valid for statement and
|
||||
-- expression-based evaluators. Define "IsValidExprEvaluator"
|
||||
-- and "IsValidStmtEvaluator" standalone so that users can use them
|
||||
-- in their type expressions.
|
||||
|
||||
module _ {{evaluator : ExprEvaluator}} {{interpretation : LatticeInterpretation isLatticeˡ}} where
|
||||
open ExprEvaluator evaluator
|
||||
open LatticeInterpretation interpretation
|
||||
|
||||
IsValidExprEvaluator : Set
|
||||
IsValidExprEvaluator = ∀ {vs ρ e v} → ρ , e ⇒ᵉ v → ⟦ vs ⟧ᵛ ρ → ⟦ eval e vs ⟧ˡ v
|
||||
|
||||
record ValidExprEvaluator (evaluator : ExprEvaluator)
|
||||
(interpretation : LatticeInterpretation isLatticeˡ) : Set₁ where
|
||||
field
|
||||
valid : IsValidExprEvaluator {{evaluator}} {{interpretation}}
|
||||
|
||||
module _ {{evaluator : StmtEvaluator}} {{interpretation : LatticeInterpretation isLatticeˡ}} where
|
||||
open StmtEvaluator evaluator
|
||||
open LatticeInterpretation interpretation
|
||||
|
||||
IsValidStmtEvaluator : Set
|
||||
IsValidStmtEvaluator = ∀ {s vs ρ₁ ρ₂ bs} → ρ₁ , bs ⇒ᵇ ρ₂ → ⟦ vs ⟧ᵛ ρ₁ → ⟦ eval s bs vs ⟧ᵛ ρ₂
|
||||
|
||||
record ValidStmtEvaluator (evaluator : StmtEvaluator)
|
||||
(interpretation : LatticeInterpretation isLatticeˡ) : Set₁ where
|
||||
field
|
||||
valid : IsValidStmtEvaluator {{evaluator}} {{interpretation}}
|
||||
211
Analysis/Forward/Lattices.agda
Normal file
211
Analysis/Forward/Lattices.agda
Normal file
@ -0,0 +1,211 @@
|
||||
open import Language hiding (_[_])
|
||||
open import Lattice
|
||||
|
||||
module Analysis.Forward.Lattices
|
||||
{L : Set} {h}
|
||||
{_≈ˡ_ : L → L → Set} {_⊔ˡ_ : L → L → L} {_⊓ˡ_ : L → L → L}
|
||||
(isFiniteHeightLatticeˡ : IsFiniteHeightLattice L h _≈ˡ_ _⊔ˡ_ _⊓ˡ_)
|
||||
(≈ˡ-dec : IsDecidable _≈ˡ_)
|
||||
(prog : Program) where
|
||||
|
||||
open import Data.String using () renaming (_≟_ to _≟ˢ_)
|
||||
open import Data.Product using (proj₁; proj₂; _,_)
|
||||
open import Data.Sum using (inj₁; inj₂)
|
||||
open import Data.List using (List; _∷_; []; foldr)
|
||||
open import Data.List.Membership.Propositional using () renaming (_∈_ to _∈ˡ_)
|
||||
open import Data.List.Relation.Unary.Any as Any using ()
|
||||
open import Relation.Binary.PropositionalEquality using (_≡_; refl)
|
||||
open import Utils using (Pairwise; _⇒_; _∨_)
|
||||
|
||||
open IsFiniteHeightLattice isFiniteHeightLatticeˡ
|
||||
using ()
|
||||
renaming
|
||||
( isLattice to isLatticeˡ
|
||||
; fixedHeight to fixedHeightˡ
|
||||
; ≈-sym to ≈ˡ-sym
|
||||
)
|
||||
open Program prog
|
||||
|
||||
import Lattice.FiniteValueMap
|
||||
import Chain
|
||||
|
||||
-- The variable -> abstract value (e.g. sign) map is a finite value-map
|
||||
-- with keys strings. Use a bundle to avoid explicitly specifying operators.
|
||||
-- It's helpful to export these via 'public' since consumers tend to
|
||||
-- use various variable lattice operations.
|
||||
module VariableValuesFiniteMap = Lattice.FiniteValueMap.WithKeys _≟ˢ_ isLatticeˡ vars
|
||||
open VariableValuesFiniteMap
|
||||
using ()
|
||||
renaming
|
||||
( FiniteMap to VariableValues
|
||||
; isLattice to isLatticeᵛ
|
||||
; _≈_ to _≈ᵛ_
|
||||
; _⊔_ to _⊔ᵛ_
|
||||
; _≼_ to _≼ᵛ_
|
||||
; ≈₂-dec⇒≈-dec to ≈ˡ-dec⇒≈ᵛ-dec
|
||||
; _∈_ to _∈ᵛ_
|
||||
; _∈k_ to _∈kᵛ_
|
||||
; _updating_via_ to _updatingᵛ_via_
|
||||
; locate to locateᵛ
|
||||
; m₁≼m₂⇒m₁[k]≼m₂[k] to m₁≼m₂⇒m₁[k]ᵛ≼m₂[k]ᵛ
|
||||
; ∈k-dec to ∈k-decᵛ
|
||||
; all-equal-keys to all-equal-keysᵛ
|
||||
)
|
||||
public
|
||||
open IsLattice isLatticeᵛ
|
||||
using ()
|
||||
renaming
|
||||
( ⊔-Monotonicˡ to ⊔ᵛ-Monotonicˡ
|
||||
; ⊔-Monotonicʳ to ⊔ᵛ-Monotonicʳ
|
||||
; ⊔-idemp to ⊔ᵛ-idemp
|
||||
)
|
||||
public
|
||||
open Lattice.FiniteValueMap.IterProdIsomorphism _≟ˢ_ isLatticeˡ
|
||||
using ()
|
||||
renaming
|
||||
( Provenance-union to Provenance-unionᵐ
|
||||
)
|
||||
public
|
||||
open Lattice.FiniteValueMap.IterProdIsomorphism.WithUniqueKeysAndFixedHeight _≟ˢ_ isLatticeˡ vars-Unique ≈ˡ-dec _ fixedHeightˡ
|
||||
using ()
|
||||
renaming
|
||||
( isFiniteHeightLattice to isFiniteHeightLatticeᵛ
|
||||
; ⊥-contains-bottoms to ⊥ᵛ-contains-bottoms
|
||||
)
|
||||
public
|
||||
|
||||
≈ᵛ-dec = ≈ˡ-dec⇒≈ᵛ-dec ≈ˡ-dec
|
||||
joinSemilatticeᵛ = IsFiniteHeightLattice.joinSemilattice isFiniteHeightLatticeᵛ
|
||||
fixedHeightᵛ = IsFiniteHeightLattice.fixedHeight isFiniteHeightLatticeᵛ
|
||||
⊥ᵛ = Chain.Height.⊥ fixedHeightᵛ
|
||||
|
||||
-- Finally, the map we care about is (state -> (variables -> value)). Bring that in.
|
||||
module StateVariablesFiniteMap = Lattice.FiniteValueMap.WithKeys _≟_ isLatticeᵛ states
|
||||
open StateVariablesFiniteMap
|
||||
using (_[_]; []-∈; m₁≼m₂⇒m₁[ks]≼m₂[ks]; m₁≈m₂⇒k∈m₁⇒k∈km₂⇒v₁≈v₂)
|
||||
renaming
|
||||
( FiniteMap to StateVariables
|
||||
; isLattice to isLatticeᵐ
|
||||
; _≈_ to _≈ᵐ_
|
||||
; _∈_ to _∈ᵐ_
|
||||
; _∈k_ to _∈kᵐ_
|
||||
; locate to locateᵐ
|
||||
; _≼_ to _≼ᵐ_
|
||||
; ≈₂-dec⇒≈-dec to ≈ᵛ-dec⇒≈ᵐ-dec
|
||||
; m₁≼m₂⇒m₁[k]≼m₂[k] to m₁≼m₂⇒m₁[k]ᵐ≼m₂[k]ᵐ
|
||||
)
|
||||
public
|
||||
open Lattice.FiniteValueMap.IterProdIsomorphism.WithUniqueKeysAndFixedHeight _≟_ isLatticeᵛ states-Unique ≈ᵛ-dec _ fixedHeightᵛ
|
||||
using ()
|
||||
renaming
|
||||
( isFiniteHeightLattice to isFiniteHeightLatticeᵐ
|
||||
)
|
||||
public
|
||||
open IsFiniteHeightLattice isFiniteHeightLatticeᵐ
|
||||
using ()
|
||||
renaming
|
||||
( ≈-sym to ≈ᵐ-sym
|
||||
)
|
||||
public
|
||||
|
||||
≈ᵐ-dec = ≈ᵛ-dec⇒≈ᵐ-dec ≈ᵛ-dec
|
||||
fixedHeightᵐ = IsFiniteHeightLattice.fixedHeight isFiniteHeightLatticeᵐ
|
||||
|
||||
-- We now have our (state -> (variables -> value)) map.
|
||||
-- Define a couple of helpers to retrieve values from it. Specifically,
|
||||
-- since the State type is as specific as possible, it's always possible to
|
||||
-- retrieve the variable values at each state.
|
||||
|
||||
states-in-Map : ∀ (s : State) (sv : StateVariables) → s ∈kᵐ sv
|
||||
states-in-Map s sv@(m , ksv≡states) rewrite ksv≡states = states-complete s
|
||||
|
||||
variablesAt : State → StateVariables → VariableValues
|
||||
variablesAt s sv = proj₁ (locateᵐ {s} {sv} (states-in-Map s sv))
|
||||
|
||||
variablesAt-∈ : ∀ (s : State) (sv : StateVariables) → (s , variablesAt s sv) ∈ᵐ sv
|
||||
variablesAt-∈ s sv = proj₂ (locateᵐ {s} {sv} (states-in-Map s sv))
|
||||
|
||||
variablesAt-≈ : ∀ s sv₁ sv₂ → sv₁ ≈ᵐ sv₂ → variablesAt s sv₁ ≈ᵛ variablesAt s sv₂
|
||||
variablesAt-≈ s sv₁ sv₂ sv₁≈sv₂ =
|
||||
m₁≈m₂⇒k∈m₁⇒k∈km₂⇒v₁≈v₂ sv₁ sv₂ sv₁≈sv₂
|
||||
(states-in-Map s sv₁) (states-in-Map s sv₂)
|
||||
|
||||
-- build up the 'join' function, which follows from Exercise 4.26's
|
||||
--
|
||||
-- L₁ → (A → L₂)
|
||||
--
|
||||
-- Construction, with L₁ = (A → L₂), and f = id
|
||||
|
||||
joinForKey : State → StateVariables → VariableValues
|
||||
joinForKey k states = foldr _⊔ᵛ_ ⊥ᵛ (states [ incoming k ])
|
||||
|
||||
-- The per-key join is made up of map key accesses (which are monotonic)
|
||||
-- and folds using the join operation (also monotonic)
|
||||
|
||||
joinForKey-Mono : ∀ (k : State) → Monotonic _≼ᵐ_ _≼ᵛ_ (joinForKey k)
|
||||
joinForKey-Mono k {fm₁} {fm₂} fm₁≼fm₂ =
|
||||
foldr-Mono joinSemilatticeᵛ joinSemilatticeᵛ (fm₁ [ incoming k ]) (fm₂ [ incoming k ]) _⊔ᵛ_ ⊥ᵛ ⊥ᵛ
|
||||
(m₁≼m₂⇒m₁[ks]≼m₂[ks] fm₁ fm₂ (incoming k) fm₁≼fm₂)
|
||||
(⊔ᵛ-idemp ⊥ᵛ) ⊔ᵛ-Monotonicʳ ⊔ᵛ-Monotonicˡ
|
||||
|
||||
-- The name f' comes from the formulation of Exercise 4.26.
|
||||
open StateVariablesFiniteMap.GeneralizedUpdate states isLatticeᵐ (λ x → x) (λ a₁≼a₂ → a₁≼a₂) joinForKey joinForKey-Mono states
|
||||
using ()
|
||||
renaming
|
||||
( f' to joinAll
|
||||
; f'-Monotonic to joinAll-Mono
|
||||
; f'-k∈ks-≡ to joinAll-k∈ks-≡
|
||||
)
|
||||
public
|
||||
|
||||
variablesAt-joinAll : ∀ (s : State) (sv : StateVariables) →
|
||||
variablesAt s (joinAll sv) ≡ joinForKey s sv
|
||||
variablesAt-joinAll s sv
|
||||
with (vs , s,vs∈usv) ← locateᵐ {s} {joinAll sv} (states-in-Map s (joinAll sv)) =
|
||||
joinAll-k∈ks-≡ {l = sv} (states-complete s) s,vs∈usv
|
||||
|
||||
-- Elements of the lattice type L describe individual variables. What
|
||||
-- exactly each lattice element says about the variable is defined
|
||||
-- by a LatticeInterpretation element. We've now constructed the
|
||||
-- (Variable → L) lattice, which describes states, and we need to lift
|
||||
-- the "meaning" of the element lattice to a descriptions of states.
|
||||
module _ {{latticeInterpretationˡ : LatticeInterpretation isLatticeˡ}} where
|
||||
open LatticeInterpretation latticeInterpretationˡ
|
||||
using ()
|
||||
renaming
|
||||
( ⟦_⟧ to ⟦_⟧ˡ
|
||||
; ⟦⟧-respects-≈ to ⟦⟧ˡ-respects-≈ˡ
|
||||
; ⟦⟧-⊔-∨ to ⟦⟧ˡ-⊔ˡ-∨
|
||||
)
|
||||
public
|
||||
|
||||
⟦_⟧ᵛ : VariableValues → Env → Set
|
||||
⟦_⟧ᵛ vs ρ = ∀ {k l} → (k , l) ∈ᵛ vs → ∀ {v} → (k , v) Language.∈ ρ → ⟦ l ⟧ˡ v
|
||||
|
||||
⟦⊥ᵛ⟧ᵛ∅ : ⟦ ⊥ᵛ ⟧ᵛ []
|
||||
⟦⊥ᵛ⟧ᵛ∅ _ ()
|
||||
|
||||
⟦⟧ᵛ-respects-≈ᵛ : ∀ {vs₁ vs₂ : VariableValues} → vs₁ ≈ᵛ vs₂ → ⟦ vs₁ ⟧ᵛ ⇒ ⟦ vs₂ ⟧ᵛ
|
||||
⟦⟧ᵛ-respects-≈ᵛ {m₁ , _} {m₂ , _}
|
||||
(m₁⊆m₂ , m₂⊆m₁) ρ ⟦vs₁⟧ρ {k} {l} k,l∈m₂ {v} k,v∈ρ =
|
||||
let
|
||||
(l' , (l≈l' , k,l'∈m₁)) = m₂⊆m₁ _ _ k,l∈m₂
|
||||
⟦l'⟧v = ⟦vs₁⟧ρ k,l'∈m₁ k,v∈ρ
|
||||
in
|
||||
⟦⟧ˡ-respects-≈ˡ (≈ˡ-sym l≈l') v ⟦l'⟧v
|
||||
|
||||
⟦⟧ᵛ-⊔ᵛ-∨ : ∀ {vs₁ vs₂ : VariableValues} → (⟦ vs₁ ⟧ᵛ ∨ ⟦ vs₂ ⟧ᵛ) ⇒ ⟦ vs₁ ⊔ᵛ vs₂ ⟧ᵛ
|
||||
⟦⟧ᵛ-⊔ᵛ-∨ {vs₁} {vs₂} ρ ⟦vs₁⟧ρ∨⟦vs₂⟧ρ {k} {l} k,l∈vs₁₂ {v} k,v∈ρ
|
||||
with ((l₁ , l₂) , (refl , (k,l₁∈vs₁ , k,l₂∈vs₂)))
|
||||
← Provenance-unionᵐ vs₁ vs₂ k,l∈vs₁₂
|
||||
with ⟦vs₁⟧ρ∨⟦vs₂⟧ρ
|
||||
... | inj₁ ⟦vs₁⟧ρ = ⟦⟧ˡ-⊔ˡ-∨ {l₁} {l₂} v (inj₁ (⟦vs₁⟧ρ k,l₁∈vs₁ k,v∈ρ))
|
||||
... | inj₂ ⟦vs₂⟧ρ = ⟦⟧ˡ-⊔ˡ-∨ {l₁} {l₂} v (inj₂ (⟦vs₂⟧ρ k,l₂∈vs₂ k,v∈ρ))
|
||||
|
||||
⟦⟧ᵛ-foldr : ∀ {vs : VariableValues} {vss : List VariableValues} {ρ : Env} →
|
||||
⟦ vs ⟧ᵛ ρ → vs ∈ˡ vss → ⟦ foldr _⊔ᵛ_ ⊥ᵛ vss ⟧ᵛ ρ
|
||||
⟦⟧ᵛ-foldr {vs} {vs ∷ vss'} {ρ = ρ} ⟦vs⟧ρ (Any.here refl) =
|
||||
⟦⟧ᵛ-⊔ᵛ-∨ {vs₁ = vs} {vs₂ = foldr _⊔ᵛ_ ⊥ᵛ vss'} ρ (inj₁ ⟦vs⟧ρ)
|
||||
⟦⟧ᵛ-foldr {vs} {vs' ∷ vss'} {ρ = ρ} ⟦vs⟧ρ (Any.there vs∈vss') =
|
||||
⟦⟧ᵛ-⊔ᵛ-∨ {vs₁ = vs'} {vs₂ = foldr _⊔ᵛ_ ⊥ᵛ vss'} ρ
|
||||
(inj₂ (⟦⟧ᵛ-foldr ⟦vs⟧ρ vs∈vss'))
|
||||
@ -62,7 +62,7 @@ open import Lattice.AboveBelow Sign _≡_ (record { ≈-refl = refl; ≈-sym = s
|
||||
open AB.Plain 0ˢ using ()
|
||||
renaming
|
||||
( isLattice to isLatticeᵍ
|
||||
; fixedHeight to fixedHeightᵍ
|
||||
; isFiniteHeightLattice to isFiniteHeightLatticeᵍ
|
||||
; _≼_ to _≼ᵍ_
|
||||
; _⊔_ to _⊔ᵍ_
|
||||
; _⊓_ to _⊓ᵍ_
|
||||
@ -171,8 +171,9 @@ instance
|
||||
module WithProg (prog : Program) where
|
||||
open Program prog
|
||||
|
||||
module ForwardWithProg = Analysis.Forward.WithProg (record { isLattice = isLatticeᵍ; fixedHeight = fixedHeightᵍ }) ≈ᵍ-dec prog
|
||||
open ForwardWithProg hiding (analyze-correct)
|
||||
open import Analysis.Forward.Lattices isFiniteHeightLatticeᵍ ≈ᵍ-dec prog
|
||||
open import Analysis.Forward.Evaluation isFiniteHeightLatticeᵍ ≈ᵍ-dec prog
|
||||
open import Analysis.Forward.Adapters isFiniteHeightLatticeᵍ ≈ᵍ-dec prog
|
||||
|
||||
eval : ∀ (e : Expr) → VariableValues → SignLattice
|
||||
eval (e₁ + e₂) vs = plus (eval e₁ vs) (eval e₂ vs)
|
||||
@ -184,8 +185,8 @@ module WithProg (prog : Program) where
|
||||
eval (# 0) _ = [ 0ˢ ]ᵍ
|
||||
eval (# (suc n')) _ = [ + ]ᵍ
|
||||
|
||||
eval-Mono : ∀ (e : Expr) → Monotonic _≼ᵛ_ _≼ᵍ_ (eval e)
|
||||
eval-Mono (e₁ + e₂) {vs₁} {vs₂} vs₁≼vs₂ =
|
||||
eval-Monoʳ : ∀ (e : Expr) → Monotonic _≼ᵛ_ _≼ᵍ_ (eval e)
|
||||
eval-Monoʳ (e₁ + e₂) {vs₁} {vs₂} vs₁≼vs₂ =
|
||||
let
|
||||
-- TODO: can this be done with less boilerplate?
|
||||
g₁vs₁ = eval e₁ vs₁
|
||||
@ -195,9 +196,9 @@ module WithProg (prog : Program) where
|
||||
in
|
||||
≼ᵍ-trans
|
||||
{plus g₁vs₁ g₂vs₁} {plus g₁vs₂ g₂vs₁} {plus g₁vs₂ g₂vs₂}
|
||||
(plus-Monoˡ g₂vs₁ {g₁vs₁} {g₁vs₂} (eval-Mono e₁ {vs₁} {vs₂} vs₁≼vs₂))
|
||||
(plus-Monoʳ g₁vs₂ {g₂vs₁} {g₂vs₂} (eval-Mono e₂ {vs₁} {vs₂} vs₁≼vs₂))
|
||||
eval-Mono (e₁ - e₂) {vs₁} {vs₂} vs₁≼vs₂ =
|
||||
(plus-Monoˡ g₂vs₁ {g₁vs₁} {g₁vs₂} (eval-Monoʳ e₁ {vs₁} {vs₂} vs₁≼vs₂))
|
||||
(plus-Monoʳ g₁vs₂ {g₂vs₁} {g₂vs₂} (eval-Monoʳ e₂ {vs₁} {vs₂} vs₁≼vs₂))
|
||||
eval-Monoʳ (e₁ - e₂) {vs₁} {vs₂} vs₁≼vs₂ =
|
||||
let
|
||||
-- TODO: here too -- can this be done with less boilerplate?
|
||||
g₁vs₁ = eval e₁ vs₁
|
||||
@ -207,9 +208,9 @@ module WithProg (prog : Program) where
|
||||
in
|
||||
≼ᵍ-trans
|
||||
{minus g₁vs₁ g₂vs₁} {minus g₁vs₂ g₂vs₁} {minus g₁vs₂ g₂vs₂}
|
||||
(minus-Monoˡ g₂vs₁ {g₁vs₁} {g₁vs₂} (eval-Mono e₁ {vs₁} {vs₂} vs₁≼vs₂))
|
||||
(minus-Monoʳ g₁vs₂ {g₂vs₁} {g₂vs₂} (eval-Mono e₂ {vs₁} {vs₂} vs₁≼vs₂))
|
||||
eval-Mono (` k) {vs₁@((kvs₁ , _) , _)} {vs₂@((kvs₂ , _), _)} vs₁≼vs₂
|
||||
(minus-Monoˡ g₂vs₁ {g₁vs₁} {g₁vs₂} (eval-Monoʳ e₁ {vs₁} {vs₂} vs₁≼vs₂))
|
||||
(minus-Monoʳ g₁vs₂ {g₂vs₁} {g₂vs₂} (eval-Monoʳ e₂ {vs₁} {vs₂} vs₁≼vs₂))
|
||||
eval-Monoʳ (` k) {vs₁@((kvs₁ , _) , _)} {vs₂@((kvs₂ , _), _)} vs₁≼vs₂
|
||||
with ∈k-decᵛ k kvs₁ | ∈k-decᵛ k kvs₂
|
||||
... | yes k∈kvs₁ | yes k∈kvs₂ =
|
||||
let
|
||||
@ -220,15 +221,15 @@ module WithProg (prog : Program) where
|
||||
... | yes k∈kvs₁ | no k∉kvs₂ = ⊥-elim (k∉kvs₂ (subst (λ l → k ∈ˡ l) (all-equal-keysᵛ vs₁ vs₂) k∈kvs₁))
|
||||
... | no k∉kvs₁ | yes k∈kvs₂ = ⊥-elim (k∉kvs₁ (subst (λ l → k ∈ˡ l) (all-equal-keysᵛ vs₂ vs₁) k∈kvs₂))
|
||||
... | no k∉kvs₁ | no k∉kvs₂ = IsLattice.≈-refl isLatticeᵍ
|
||||
eval-Mono (# 0) _ = ≈ᵍ-refl
|
||||
eval-Mono (# (suc n')) _ = ≈ᵍ-refl
|
||||
eval-Monoʳ (# 0) _ = ≈ᵍ-refl
|
||||
eval-Monoʳ (# (suc n')) _ = ≈ᵍ-refl
|
||||
|
||||
instance
|
||||
SignEval : Evaluator
|
||||
SignEval = record { eval = eval; eval-Mono = eval-Mono }
|
||||
SignEval : ExprEvaluator
|
||||
SignEval = record { eval = eval; eval-Monoʳ = eval-Monoʳ }
|
||||
|
||||
-- For debugging purposes, print out the result.
|
||||
output = show result
|
||||
output = show (Analysis.Forward.WithProg.result isFiniteHeightLatticeᵍ ≈ᵍ-dec prog)
|
||||
|
||||
-- This should have fewer cases -- the same number as the actual 'plus' above.
|
||||
-- But agda only simplifies on first argument, apparently, so we are stuck
|
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@ -280,7 +281,7 @@ module WithProg (prog : Program) where
|
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minus-valid {[ 0ˢ ]ᵍ} {[ 0ˢ ]ᵍ} refl refl = refl
|
||||
minus-valid {[ 0ˢ ]ᵍ} {⊤ᵍ} _ _ = tt
|
||||
|
||||
eval-valid : IsValid
|
||||
eval-valid : IsValidExprEvaluator
|
||||
eval-valid (⇒ᵉ-+ ρ e₁ e₂ z₁ z₂ ρ,e₁⇒z₁ ρ,e₂⇒z₂) ⟦vs⟧ρ =
|
||||
plus-valid (eval-valid ρ,e₁⇒z₁ ⟦vs⟧ρ) (eval-valid ρ,e₂⇒z₂ ⟦vs⟧ρ)
|
||||
eval-valid (⇒ᵉ-- ρ e₁ e₂ z₁ z₂ ρ,e₁⇒z₁ ρ,e₂⇒z₂) ⟦vs⟧ρ =
|
||||
@ -292,4 +293,4 @@ module WithProg (prog : Program) where
|
||||
eval-valid (⇒ᵉ-ℕ ρ 0) _ = refl
|
||||
eval-valid (⇒ᵉ-ℕ ρ (suc n')) _ = (n' , refl)
|
||||
|
||||
analyze-correct = ForwardWithProg.analyze-correct
|
||||
analyze-correct = Analysis.Forward.WithProg.analyze-correct
|
||||
|
||||
Loading…
Reference in New Issue
Block a user