module Analysis.Sign where

open import Data.Nat using (suc)
open import Data.Product using (proj₁; _,_)
open import Data.Empty using (⊥; ⊥-elim)
open import Data.List.Membership.Propositional as MemProp using () renaming (_∈_ to _∈ˡ_)
open import Relation.Binary.PropositionalEquality using (_≡_; refl; sym; trans; subst)
open import Relation.Nullary using (yes; no)

open import Language
open import Lattice
open import Showable using (Showable; show)
import Analysis.Forward

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

instance
    showable : Showable Sign
    showable = record
        { show = (λ
            { + → "+"
            ; - → "-"
            ; 0ˢ → "0"
            })
        }

-- 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_
        ; ≈-⊥-⊥ to ≈ᵍ-⊥ᵍ-⊥ᵍ
        ; ≈-⊤-⊤ to ≈ᵍ-⊤ᵍ-⊤ᵍ
        ; ≈-lift to ≈ᵍ-lift
        ; ≈-refl to ≈ᵍ-refl
        )
-- 'sign' has no underlying lattice structure, so use the 'plain' above-below lattice.
open AB.Plain 0ˢ using ()
    renaming
        ( isLattice to isLatticeᵍ
        ; fixedHeight to fixedHeightᵍ
        ; _≼_ to _≼ᵍ_
        ; _⊔_ to _⊔ᵍ_
        )

open IsLattice isLatticeᵍ using ()
    renaming
        ( ≼-trans to ≼ᵍ-trans
        )

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)

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

postulate minus-Monoˡ : ∀ (s₂ : SignLattice) → Monotonic _≼ᵍ_ _≼ᵍ_ (λ s₁ → minus s₁ s₂)
postulate minus-Monoʳ : ∀ (s₁ : SignLattice) → Monotonic _≼ᵍ_ _≼ᵍ_ (minus s₁)

module WithProg (prog : Program) where
    open Program prog

    module ForwardWithProg = Analysis.Forward.WithProg (record { isLattice = isLatticeᵍ; fixedHeight = fixedHeightᵍ }) ≈ᵍ-dec prog
    open ForwardWithProg

    eval : ∀ (e : Expr) → VariableValues → SignLattice
    eval (e₁ + e₂) vs = plus (eval e₁ vs) (eval e₂ vs)
    eval (e₁ - e₂) vs = minus (eval e₁ vs) (eval e₂ vs)
    eval (` k) vs
        with ∈k-decᵛ k (proj₁ (proj₁ vs))
    ...   | yes k∈vs = proj₁ (locateᵛ {k} {vs} k∈vs)
    ...   | no _ = ⊤ᵍ
    eval (# 0) _ = [ 0ˢ ]ᵍ
    eval (# (suc n')) _ = [ + ]ᵍ

    eval-Mono : ∀ (e : Expr) → Monotonic _≼ᵛ_ _≼ᵍ_ (eval e)
    eval-Mono (e₁ + e₂) {vs₁} {vs₂} vs₁≼vs₂ =
        let
            -- TODO: can this be done with less boilerplate?
            g₁vs₁ = eval e₁ vs₁
            g₂vs₁ = eval e₂ vs₁
            g₁vs₂ = eval e₁ vs₂
            g₂vs₂ = eval e₂ vs₂
        in
            ≼ᵍ-trans
                {plus g₁vs₁ g₂vs₁} {plus g₁vs₂ g₂vs₁} {plus g₁vs₂ g₂vs₂}
                (plus-Monoˡ g₂vs₁ {g₁vs₁} {g₁vs₂} (eval-Mono e₁ {vs₁} {vs₂} vs₁≼vs₂))
                (plus-Monoʳ g₁vs₂ {g₂vs₁} {g₂vs₂} (eval-Mono e₂ {vs₁} {vs₂} vs₁≼vs₂))
    eval-Mono (e₁ - e₂) {vs₁} {vs₂} vs₁≼vs₂ =
        let
            -- TODO: here too -- can this be done with less boilerplate?
            g₁vs₁ = eval e₁ vs₁
            g₂vs₁ = eval e₂ vs₁
            g₁vs₂ = eval e₁ vs₂
            g₂vs₂ = eval e₂ vs₂
        in
            ≼ᵍ-trans
                {minus g₁vs₁ g₂vs₁} {minus g₁vs₂ g₂vs₁} {minus g₁vs₂ g₂vs₂}
                (minus-Monoˡ g₂vs₁ {g₁vs₁} {g₁vs₂} (eval-Mono e₁ {vs₁} {vs₂} vs₁≼vs₂))
                (minus-Monoʳ g₁vs₂ {g₂vs₁} {g₂vs₂} (eval-Mono e₂ {vs₁} {vs₂} vs₁≼vs₂))
    eval-Mono (` k) {vs₁@((kvs₁ , _) , _)} {vs₂@((kvs₂ , _), _)} vs₁≼vs₂
        with ∈k-decᵛ k kvs₁ | ∈k-decᵛ k kvs₂
    ...   | yes k∈kvs₁ | yes k∈kvs₂ =
            let
                (v₁ , k,v₁∈vs₁) = locateᵛ {k} {vs₁} k∈kvs₁
                (v₂ , k,v₂∈vs₂) = locateᵛ {k} {vs₂} k∈kvs₂
            in
                m₁≼m₂⇒m₁[k]ᵛ≼m₂[k]ᵛ vs₁ vs₂ vs₁≼vs₂ k,v₁∈vs₁ k,v₂∈vs₂
    ...   | yes k∈kvs₁ | no k∉kvs₂ = ⊥-elim (k∉kvs₂ (subst (λ l → k ∈ˡ l) (all-equal-keysᵛ vs₁ vs₂) k∈kvs₁))
    ...   | no k∉kvs₁ | yes k∈kvs₂ = ⊥-elim (k∉kvs₁ (subst (λ l → k ∈ˡ l) (all-equal-keysᵛ vs₂ vs₁) k∈kvs₂))
    ...   | no k∉kvs₁ | no k∉kvs₂ = IsLattice.≈-refl isLatticeᵍ
    eval-Mono (# 0) _ = ≈ᵍ-refl
    eval-Mono (# (suc n')) _ = ≈ᵍ-refl

    open ForwardWithProg.WithEvaluator eval eval-Mono using (result)

    -- For debugging purposes, print out the result.
    output = show result