Tighten exported definitions in Forward.agda
Signed-off-by: Danila Fedorin <danila.fedorin@gmail.com>
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@ -78,10 +78,11 @@ module WithProg (prog : Program) where
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; ⊥-contains-bottoms to ⊥ᵛ-contains-bottoms
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)
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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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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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@ -109,45 +110,46 @@ module WithProg (prog : Program) where
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( ≈-sym to ≈ᵐ-sym
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)
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≈ᵐ-dec = ≈ᵛ-dec⇒≈ᵐ-dec ≈ᵛ-dec
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fixedHeightᵐ = IsFiniteHeightLattice.fixedHeight isFiniteHeightLatticeᵐ
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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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-- 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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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 : 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 : 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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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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-- 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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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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-- 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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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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@ -158,11 +160,12 @@ module WithProg (prog : Program) where
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; f'-k∈ks-≡ to joinAll-k∈ks-≡
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)
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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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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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@ -170,7 +173,7 @@ module WithProg (prog : Program) where
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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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module _ {{evaluator : Evaluator}} where
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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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@ -253,7 +256,10 @@ 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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module _ {{latticeInterpretationˡ : LatticeInterpretation isLatticeˡ}} where
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open WithEvaluator
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open WithEvaluator using (result; analyze; result≈analyze-result) public
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private module WithInterpretation {{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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@ -294,6 +300,8 @@ module WithProg (prog : Program) where
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⟦⟧ᵛ-⊔ᵛ-∨ {vs₁ = vs'} {vs₂ = foldr _⊔ᵛ_ ⊥ᵛ vss'} ρ
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(inj₂ (⟦⟧ᵛ-foldr ⟦vs⟧ρ vs∈vss'))
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open WithInterpretation
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module _ {{evaluator : Evaluator}} {{interpretation : LatticeInterpretation isLatticeˡ}} where
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open Evaluator evaluator
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open LatticeInterpretation interpretation
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@ -312,7 +320,7 @@ module WithProg (prog : Program) where
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field
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valid : IsValid
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module _ {{validInterpretation : ValidInterpretation}} where
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module WithValidInterpretation {{validInterpretation : ValidInterpretation}} where
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open ValidInterpretation validInterpretation
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updateVariablesFromStmt-matches : ∀ {bs vs ρ₁ ρ₂} → ρ₁ , bs ⇒ᵇ ρ₂ → ⟦ vs ⟧ᵛ ρ₁ → ⟦ updateVariablesFromStmt bs vs ⟧ᵛ ρ₂
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@ -391,3 +399,5 @@ module WithProg (prog : Program) where
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analyze-correct : ∀ {ρ : Env} → [] , rootStmt ⇒ˢ ρ → ⟦ variablesAt finalState result ⟧ᵛ ρ
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analyze-correct {ρ} ∅,s⇒ρ = walkTrace {initialState} {finalState} {[]} {ρ} ⟦joinAll-initialState⟧ᵛ∅ (trace ∅,s⇒ρ)
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open WithValidInterpretation using (analyze-correct) public
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