agda-spa/Analysis/Sign.agda

module Analysis.Sign where

open import Data.String using (String) renaming (_≟_ to _≟ˢ_)
open import Data.Product using (_×_; proj₁; _,_)
open import Data.List using (List; _∷_; []; foldr; cartesianProduct; cartesianProductWith)
open import Relation.Binary.PropositionalEquality using (_≡_; refl; sym; trans)
open import Relation.Nullary using (¬_; Dec; yes; no)
open import Data.Unit using (⊤)

open import Language
open import Lattice
open import Utils using (Pairwise)
import Lattice.Bundles.FiniteValueMap

private module FixedHeightFiniteMap = Lattice.Bundles.FiniteValueMap.FromFiniteHeightLattice

data Sign : Set where
    + : Sign
    - : Sign
    0ˢ : Sign

-- g for siGn; s is used for strings and i is not very descriptive.
_≟ᵍ_ : IsDecidable (_≡_ {_} {Sign})
_≟ᵍ_ + + = yes refl
_≟ᵍ_ + - = no (λ ())
_≟ᵍ_ + 0ˢ = no (λ ())
_≟ᵍ_ - + = no (λ ())
_≟ᵍ_ - - = yes refl
_≟ᵍ_ - 0ˢ = no (λ ())
_≟ᵍ_ 0ˢ + = no (λ ())
_≟ᵍ_ 0ˢ - = no (λ ())
_≟ᵍ_ 0ˢ 0ˢ = yes refl

-- embelish 'sign' with a top and bottom element.
open import Lattice.AboveBelow Sign _≡_ (record { ≈-refl = refl; ≈-sym = sym; ≈-trans = trans }) _≟ᵍ_ as AB
    using ()
    renaming
        ( AboveBelow to SignLattice
        ; ≈-dec to ≈ᵍ-dec
        ; ⊥ to ⊥ᵍ
        ; ⊤ to ⊤ᵍ
        ; [_] to [_]ᵍ
        ; ≈-⊥-⊥ to ≈ᵍ-⊥ᵍ-⊥ᵍ
        ; ≈-⊤-⊤ to ≈ᵍ-⊤ᵍ-⊤ᵍ
        ; ≈-lift to ≈ᵍ-lift
        )
-- 'sign' has no underlying lattice structure, so use the 'plain' above-below lattice.
open AB.Plain using () renaming (finiteHeightLattice to finiteHeightLatticeᵍ-if-inhabited)

finiteHeightLatticeᵍ = finiteHeightLatticeᵍ-if-inhabited 0ˢ

open FiniteHeightLattice finiteHeightLatticeᵍ
    using ()
    renaming
        ( _≼_ to _≼ᵍ_
        ; _≈_ to _≈ᵍ_
        ; _⊔_ to _⊔ᵍ_
        ; ≈-refl to ≈ᵍ-refl
        )

plus : SignLattice → SignLattice → SignLattice
plus ⊥ᵍ _ = ⊥ᵍ
plus _ ⊥ᵍ = ⊥ᵍ
plus ⊤ᵍ _ = ⊤ᵍ
plus _ ⊤ᵍ = ⊤ᵍ
plus [ + ]ᵍ [ + ]ᵍ = [ + ]ᵍ
plus [ + ]ᵍ [ - ]ᵍ = ⊤ᵍ
plus [ + ]ᵍ [ 0ˢ ]ᵍ = [ + ]ᵍ
plus [ - ]ᵍ [ + ]ᵍ = ⊤ᵍ
plus [ - ]ᵍ [ - ]ᵍ = [ - ]ᵍ
plus [ - ]ᵍ [ 0ˢ ]ᵍ = [ - ]ᵍ
plus [ 0ˢ ]ᵍ [ + ]ᵍ = [ + ]ᵍ
plus [ 0ˢ ]ᵍ [ - ]ᵍ = [ - ]ᵍ
plus [ 0ˢ ]ᵍ [ 0ˢ ]ᵍ = [ 0ˢ ]ᵍ

-- this is incredibly tedious: 125 cases per monotonicity proof, and tactics
-- are hard. postulate for now.
postulate plus-Monoˡ : ∀ (s₂ : SignLattice) → Monotonic _≼ᵍ_ _≼ᵍ_ (λ s₁ → plus s₁ s₂)
postulate plus-Monoʳ : ∀ (s₁ : SignLattice) → Monotonic _≼ᵍ_ _≼ᵍ_ (plus s₁)

module _ (prog : Program) where
    open Program prog

    -- The variable -> sign map is a finite value-map with keys strings. Use a bundle to avoid explicitly specifying operators.
    open FixedHeightFiniteMap String SignLattice _≟ˢ_ finiteHeightLatticeᵍ vars-Unique ≈ᵍ-dec
        using ()
        renaming
            ( finiteHeightLattice to finiteHeightLatticeᵛ
            ; FiniteMap to VariableSigns
            ; _≈_ to _≈ᵛ_
            ; _⊔_ to _⊔ᵛ_
            ; ≈-dec to ≈ᵛ-dec
            )
    open FiniteHeightLattice finiteHeightLatticeᵛ
        using ()
        renaming
            ( ⊔-Monotonicˡ to ⊔ᵛ-Monotonicˡ
            ; ⊔-Monotonicʳ to ⊔ᵛ-Monotonicʳ
            ; _≼_ to _≼ᵛ_
            ; joinSemilattice to joinSemilatticeᵛ
            ; ⊔-idemp to ⊔ᵛ-idemp
            )

    ⊥ᵛ = proj₁ (proj₁ (proj₁ (FiniteHeightLattice.fixedHeight finiteHeightLatticeᵛ)))

    -- Finally, the map we care about is (state -> (variables -> sign)). Bring that in.
    module StateVariablesFiniteMap = FixedHeightFiniteMap State VariableSigns _≟_ finiteHeightLatticeᵛ states-Unique ≈ᵛ-dec
    open StateVariablesFiniteMap
        using (_[_]; m₁≼m₂⇒m₁[ks]≼m₂[ks])
        renaming
            ( finiteHeightLattice to finiteHeightLatticeᵐ
            ; FiniteMap to StateVariables
            ; isLattice to isLatticeᵐ
            )
    open FiniteHeightLattice finiteHeightLatticeᵐ
        using ()
        renaming (_≼_ to _≼ᵐ_)

    -- 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 → VariableSigns
    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
        renaming
            ( f' to joinAll
            ; f'-Monotonic to joinAll-Mono
            )