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

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

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 import Lattice.Bundles.FiniteValueMap String SignLattice _≟ˢ_ renaming (finiteHeightLattice to finiteHeightLatticeᵛ-if-B-finite; FiniteHeightType to FiniteHeightTypeᵛ; _≈_ to _≈ᵛ_; ≈-dec to ≈ᵛ-dec-if-≈ᵍ-dec)


    VariableSigns = FiniteHeightTypeᵛ finiteHeightLatticeᵍ vars-Unique ≈ᵍ-dec
    finiteHeightLatticeᵛ = finiteHeightLatticeᵛ-if-B-finite finiteHeightLatticeᵍ vars-Unique ≈ᵍ-dec
    ≈ᵛ-dec = ≈ᵛ-dec-if-≈ᵍ-dec finiteHeightLatticeᵍ vars-Unique ≈ᵍ-dec

    -- Finally, the map we care about is (state -> (variables -> sign)). Bring that in.
    open import Lattice.Bundles.FiniteValueMap State VariableSigns _≟_ renaming (finiteHeightLattice to finiteHeightLatticeᵐ-if-B-finite; FiniteHeightType to FiniteHeightTypeᵐ)

    StateVariables = FiniteHeightTypeᵐ finiteHeightLatticeᵛ states-Unique ≈ᵛ-dec
    finiteHeightLatticeᵐ = finiteHeightLatticeᵐ-if-B-finite finiteHeightLatticeᵛ states-Unique ≈ᵛ-dec