310 lines
14 KiB
Plaintext
310 lines
14 KiB
Plaintext
module Analysis.Sign where
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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; _,_)
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open import Data.List using (List; _∷_; []; foldr; cartesianProduct; cartesianProductWith)
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open import Data.List.Membership.Propositional as MemProp using () renaming (_∈_ to _∈ˡ_)
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open import Relation.Binary.PropositionalEquality using (_≡_; refl; 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 Function using (_∘_)
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open import Language
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open import Lattice
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open import Utils using (Pairwise)
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open import Showable using (Showable; show)
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import Lattice.FiniteValueMap
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data Sign : Set where
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+ : Sign
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- : Sign
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0ˢ : Sign
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instance
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showable : Showable Sign
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showable = record
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{ show = (λ
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{ + → "+"
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; - → "-"
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; 0ˢ → "0"
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})
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}
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-- g for siGn; s is used for strings and i is not very descriptive.
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_≟ᵍ_ : IsDecidable (_≡_ {_} {Sign})
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_≟ᵍ_ + + = yes refl
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_≟ᵍ_ + - = no (λ ())
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_≟ᵍ_ + 0ˢ = no (λ ())
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_≟ᵍ_ - + = no (λ ())
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_≟ᵍ_ - - = yes refl
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_≟ᵍ_ - 0ˢ = no (λ ())
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_≟ᵍ_ 0ˢ + = no (λ ())
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_≟ᵍ_ 0ˢ - = no (λ ())
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_≟ᵍ_ 0ˢ 0ˢ = yes refl
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-- embelish 'sign' with a top and bottom element.
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open import Lattice.AboveBelow Sign _≡_ (record { ≈-refl = refl; ≈-sym = sym; ≈-trans = trans }) _≟ᵍ_ as AB
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using ()
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renaming
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( AboveBelow to SignLattice
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; ≈-dec to ≈ᵍ-dec
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; ⊥ to ⊥ᵍ
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; ⊤ to ⊤ᵍ
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; [_] to [_]ᵍ
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; _≈_ to _≈ᵍ_
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; ≈-⊥-⊥ to ≈ᵍ-⊥ᵍ-⊥ᵍ
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; ≈-⊤-⊤ to ≈ᵍ-⊤ᵍ-⊤ᵍ
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; ≈-lift to ≈ᵍ-lift
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; ≈-refl to ≈ᵍ-refl
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)
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-- 'sign' has no underlying lattice structure, so use the 'plain' above-below lattice.
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open AB.Plain 0ˢ using ()
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renaming
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( finiteHeightLattice to finiteHeightLatticeᵍ
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; isLattice to isLatticeᵍ
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; fixedHeight to fixedHeightᵍ
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; _≼_ to _≼ᵍ_
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; _⊔_ to _⊔ᵍ_
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)
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open IsLattice isLatticeᵍ using ()
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renaming
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( ≼-trans to ≼ᵍ-trans
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)
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plus : SignLattice → SignLattice → SignLattice
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plus ⊥ᵍ _ = ⊥ᵍ
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plus _ ⊥ᵍ = ⊥ᵍ
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plus ⊤ᵍ _ = ⊤ᵍ
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plus _ ⊤ᵍ = ⊤ᵍ
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plus [ + ]ᵍ [ + ]ᵍ = [ + ]ᵍ
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plus [ + ]ᵍ [ - ]ᵍ = ⊤ᵍ
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plus [ + ]ᵍ [ 0ˢ ]ᵍ = [ + ]ᵍ
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plus [ - ]ᵍ [ + ]ᵍ = ⊤ᵍ
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plus [ - ]ᵍ [ - ]ᵍ = [ - ]ᵍ
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plus [ - ]ᵍ [ 0ˢ ]ᵍ = [ - ]ᵍ
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plus [ 0ˢ ]ᵍ [ + ]ᵍ = [ + ]ᵍ
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plus [ 0ˢ ]ᵍ [ - ]ᵍ = [ - ]ᵍ
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plus [ 0ˢ ]ᵍ [ 0ˢ ]ᵍ = [ 0ˢ ]ᵍ
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-- this is incredibly tedious: 125 cases per monotonicity proof, and tactics
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-- are hard. postulate for now.
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postulate plus-Monoˡ : ∀ (s₂ : SignLattice) → Monotonic _≼ᵍ_ _≼ᵍ_ (λ s₁ → plus s₁ s₂)
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postulate plus-Monoʳ : ∀ (s₁ : SignLattice) → Monotonic _≼ᵍ_ _≼ᵍ_ (plus s₁)
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minus : SignLattice → SignLattice → SignLattice
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minus ⊥ᵍ _ = ⊥ᵍ
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minus _ ⊥ᵍ = ⊥ᵍ
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minus ⊤ᵍ _ = ⊤ᵍ
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minus _ ⊤ᵍ = ⊤ᵍ
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minus [ + ]ᵍ [ + ]ᵍ = ⊤ᵍ
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minus [ + ]ᵍ [ - ]ᵍ = [ + ]ᵍ
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minus [ + ]ᵍ [ 0ˢ ]ᵍ = [ + ]ᵍ
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minus [ - ]ᵍ [ + ]ᵍ = [ - ]ᵍ
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minus [ - ]ᵍ [ - ]ᵍ = ⊤ᵍ
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minus [ - ]ᵍ [ 0ˢ ]ᵍ = [ - ]ᵍ
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minus [ 0ˢ ]ᵍ [ + ]ᵍ = [ - ]ᵍ
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minus [ 0ˢ ]ᵍ [ - ]ᵍ = [ + ]ᵍ
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minus [ 0ˢ ]ᵍ [ 0ˢ ]ᵍ = [ 0ˢ ]ᵍ
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postulate minus-Monoˡ : ∀ (s₂ : SignLattice) → Monotonic _≼ᵍ_ _≼ᵍ_ (λ s₁ → minus s₁ s₂)
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postulate minus-Monoʳ : ∀ (s₁ : SignLattice) → Monotonic _≼ᵍ_ _≼ᵍ_ (minus s₁)
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module WithProg (prog : Program) where
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open Program prog
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-- The variable -> sign map is a finite value-map with keys strings. Use a bundle to avoid explicitly specifying operators.
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module VariableSignsFiniteMap = Lattice.FiniteValueMap.WithKeys _≟ˢ_ isLatticeᵍ vars
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open VariableSignsFiniteMap
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using ()
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renaming
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( FiniteMap to VariableSigns
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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 ≈ᵍ-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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)
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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.WithUniqueKeysAndFixedHeight _≟ˢ_ isLatticeᵍ vars-Unique ≈ᵍ-dec _ fixedHeightᵍ
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using ()
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renaming
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( isFiniteHeightLattice to isFiniteHeightLatticeᵛ
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)
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≈ᵛ-dec = ≈ᵍ-dec⇒≈ᵛ-dec ≈ᵍ-dec
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joinSemilatticeᵛ = IsFiniteHeightLattice.joinSemilattice isFiniteHeightLatticeᵛ
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fixedHeightᵛ = IsFiniteHeightLattice.fixedHeight isFiniteHeightLatticeᵛ
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⊥ᵛ = proj₁ (proj₁ (proj₁ fixedHeightᵛ))
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-- Finally, the map we care about is (state -> (variables -> sign)). 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])
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renaming
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( FiniteMap to StateVariables
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; isLattice to isLatticeᵐ
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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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≈ᵐ-dec = ≈ᵛ-dec⇒≈ᵐ-dec ≈ᵛ-dec
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fixedHeightᵐ = IsFiniteHeightLattice.fixedHeight isFiniteHeightLatticeᵐ
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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 → VariableSigns
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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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renaming
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( f' to joinAll
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; f'-Monotonic to joinAll-Mono
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)
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-- With 'join' in hand, we need to perform abstract evaluation.
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vars-in-Map : ∀ (k : String) (vs : VariableSigns) →
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k ∈ˡ vars → k ∈kᵛ vs
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vars-in-Map k vs@(m , kvs≡vars) k∈vars rewrite kvs≡vars = k∈vars
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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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eval : ∀ (e : Expr) → (∀ k → k ∈ᵉ e → k ∈ˡ vars) → VariableSigns → SignLattice
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eval (e₁ + e₂) k∈e⇒k∈vars vs =
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plus (eval e₁ (λ k k∈e₁ → k∈e⇒k∈vars k (in⁺₁ k∈e₁)) vs)
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(eval e₂ (λ k k∈e₂ → k∈e⇒k∈vars k (in⁺₂ k∈e₂)) vs)
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eval (e₁ - e₂) k∈e⇒k∈vars vs =
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minus (eval e₁ (λ k k∈e₁ → k∈e⇒k∈vars k (in⁻₁ k∈e₁)) vs)
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(eval e₂ (λ k k∈e₂ → k∈e⇒k∈vars k (in⁻₂ k∈e₂)) vs)
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eval (` k) k∈e⇒k∈vars vs = proj₁ (locateᵛ {k} {vs} (vars-in-Map k vs (k∈e⇒k∈vars k here)))
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eval (# 0) _ _ = [ 0ˢ ]ᵍ
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eval (# (suc n')) _ _ = [ + ]ᵍ
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eval-Mono : ∀ (e : Expr) (k∈e⇒k∈vars : ∀ k → k ∈ᵉ e → k ∈ˡ vars) → Monotonic _≼ᵛ_ _≼ᵍ_ (eval e k∈e⇒k∈vars)
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eval-Mono (e₁ + e₂) k∈e⇒k∈vars {vs₁} {vs₂} vs₁≼vs₂ =
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let
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-- TODO: can this be done with less boilerplate?
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k∈e₁⇒k∈vars = λ k k∈e₁ → k∈e⇒k∈vars k (in⁺₁ k∈e₁)
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k∈e₂⇒k∈vars = λ k k∈e₂ → k∈e⇒k∈vars k (in⁺₂ k∈e₂)
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g₁vs₁ = eval e₁ k∈e₁⇒k∈vars vs₁
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g₂vs₁ = eval e₂ k∈e₂⇒k∈vars vs₁
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g₁vs₂ = eval e₁ k∈e₁⇒k∈vars vs₂
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g₂vs₂ = eval e₂ k∈e₂⇒k∈vars vs₂
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in
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≼ᵍ-trans
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{plus g₁vs₁ g₂vs₁} {plus g₁vs₂ g₂vs₁} {plus g₁vs₂ g₂vs₂}
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(plus-Monoˡ g₂vs₁ {g₁vs₁} {g₁vs₂} (eval-Mono e₁ k∈e₁⇒k∈vars {vs₁} {vs₂} vs₁≼vs₂))
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(plus-Monoʳ g₁vs₂ {g₂vs₁} {g₂vs₂} (eval-Mono e₂ k∈e₂⇒k∈vars {vs₁} {vs₂} vs₁≼vs₂))
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eval-Mono (e₁ - e₂) k∈e⇒k∈vars {vs₁} {vs₂} vs₁≼vs₂ =
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let
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-- TODO: here too -- can this be done with less boilerplate?
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k∈e₁⇒k∈vars = λ k k∈e₁ → k∈e⇒k∈vars k (in⁻₁ k∈e₁)
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k∈e₂⇒k∈vars = λ k k∈e₂ → k∈e⇒k∈vars k (in⁻₂ k∈e₂)
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g₁vs₁ = eval e₁ k∈e₁⇒k∈vars vs₁
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g₂vs₁ = eval e₂ k∈e₂⇒k∈vars vs₁
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g₁vs₂ = eval e₁ k∈e₁⇒k∈vars vs₂
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g₂vs₂ = eval e₂ k∈e₂⇒k∈vars vs₂
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in
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≼ᵍ-trans
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{minus g₁vs₁ g₂vs₁} {minus g₁vs₂ g₂vs₁} {minus g₁vs₂ g₂vs₂}
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(minus-Monoˡ g₂vs₁ {g₁vs₁} {g₁vs₂} (eval-Mono e₁ k∈e₁⇒k∈vars {vs₁} {vs₂} vs₁≼vs₂))
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(minus-Monoʳ g₁vs₂ {g₂vs₁} {g₂vs₂} (eval-Mono e₂ k∈e₂⇒k∈vars {vs₁} {vs₂} vs₁≼vs₂))
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eval-Mono (` k) k∈e⇒k∈vars {vs₁} {vs₂} vs₁≼vs₂ =
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let
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(v₁ , k,v₁∈vs₁) = locateᵛ {k} {vs₁} (vars-in-Map k vs₁ (k∈e⇒k∈vars k here))
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(v₂ , k,v₂∈vs₂) = locateᵛ {k} {vs₂} (vars-in-Map k vs₂ (k∈e⇒k∈vars k here))
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in
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m₁≼m₂⇒m₁[k]ᵛ≼m₂[k]ᵛ vs₁ vs₂ vs₁≼vs₂ k,v₁∈vs₁ k,v₂∈vs₂
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eval-Mono (# 0) _ _ = ≈ᵍ-refl
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eval-Mono (# (suc n')) _ _ = ≈ᵍ-refl
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private module _ (k : String) (e : Expr) (k∈e⇒k∈vars : ∀ k → k ∈ᵉ e → k ∈ˡ vars) where
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open VariableSignsFiniteMap.GeneralizedUpdate vars isLatticeᵛ (λ x → x) (λ a₁≼a₂ → a₁≼a₂) (λ _ → eval e k∈e⇒k∈vars) (λ _ {vs₁} {vs₂} vs₁≼vs₂ → eval-Mono e k∈e⇒k∈vars {vs₁} {vs₂} vs₁≼vs₂) (k ∷ [])
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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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)
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public
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updateVariablesForState : State → StateVariables → VariableSigns
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updateVariablesForState s sv
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-- More weirdness here. Apparently, capturing the with-equality proof
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-- using 'in p' makes code that reasons about this function (below)
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-- throw ill-typed with-abstraction errors. Instead, make use of the
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-- fact that later with-clauses are generalized over earlier ones to
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-- construct a specialization of vars-complete for (code s).
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with code s | (λ k → vars-complete {k} s)
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... | k ← e | k∈codes⇒k∈vars =
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let
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(vs , s,vs∈sv) = locateᵐ {s} {sv} (states-in-Map s sv)
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in
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updateVariablesFromExpression k e (λ k k∈e → k∈codes⇒k∈vars k (in←₂ k∈e)) vs
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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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with code s | (λ k → vars-complete {k} s)
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... | k ← e | k∈codes⇒k∈vars =
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let
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(vs₁ , s,vs₁∈sv₁) = locateᵐ {s} {sv₁} (states-in-Map s sv₁)
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(vs₂ , s,vs₂∈sv₂) = locateᵐ {s} {sv₂} (states-in-Map s sv₂)
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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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updateVariablesFromExpression-Mono k e (λ k k∈e → k∈codes⇒k∈vars k (in←₂ k∈e)) {vs₁} {vs₂} 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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renaming
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( f' to updateAll
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; f'-Monotonic to updateAll-Mono
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)
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analyze : StateVariables → StateVariables
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analyze = updateAll ∘ joinAll
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analyze-Mono : Monotonic _≼ᵐ_ _≼ᵐ_ analyze
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analyze-Mono {sv₁} {sv₂} sv₁≼sv₂ = updateAll-Mono {joinAll sv₁} {joinAll sv₂} (joinAll-Mono {sv₁} {sv₂} sv₁≼sv₂)
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open import Fixedpoint ≈ᵐ-dec isFiniteHeightLatticeᵐ analyze (λ {m₁} {m₂} m₁≼m₂ → analyze-Mono {m₁} {m₂} m₁≼m₂)
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using ()
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renaming (aᶠ to signs)
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-- Debugging code: print the resulting map.
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output = show signs
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