agda-spa/Analysis/Sign.agda

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module Analysis.Sign where
open import Data.String using (String) renaming (_≟_ to _≟ˢ_)
open import Relation.Binary.PropositionalEquality using (_≡_; refl; sym; trans)
open import Relation.Nullary using (¬_; Dec; yes; no)
open import Language
open import Lattice
import Lattice.Bundles.FiniteValueMap
private module FixedHeightFiniteMap = Lattice.Bundles.FiniteValueMap.FromFiniteHeightLattice
data Sign : Set where
+ : Sign
- : Sign
: 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
module _ (prog : Program) where
open Program prog
-- embelish 'sign' with a top and bottom element.
open import Lattice.AboveBelow Sign _≡_ (record { ≈-refl = refl; ≈-sym = sym; ≈-trans = trans }) _≟ᵍ_ as AB renaming (AboveBelow to SignLattice; ≈-dec to ≈ᵍ-dec)
-- '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
-- 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
renaming
( finiteHeightLattice to finiteHeightLatticeᵛ
; FiniteMap to VariableSigns
; _≈_ to _≈ᵛ_
; ≈-dec to ≈ᵛ-dec
)
-- Finally, the map we care about is (state -> (variables -> sign)). Bring that in.
open FixedHeightFiniteMap State VariableSigns _≟_ finiteHeightLatticeᵛ states-Unique ≈ᵛ-dec
renaming
( finiteHeightLattice to finiteHeightLatticeᵐ
; FiniteMap to StateVariables
)