module Main where

open import IO
open import Level using (0ℓ)
open import Data.Nat.Show using (show)
open import Data.List using (List; _∷_; []; foldr)
open import Data.String using (String; _++_) renaming (_≟_ to _≟ˢ_)
open import Data.Unit using (⊤; tt) renaming (_≟_ to _≟ᵘ_)
open import Data.Product using (_,_; _×_; proj₁; proj₂)
open import Data.List.Relation.Unary.All using (_∷_; [])
open import Relation.Binary.PropositionalEquality as Eq using (_≡_; sym; subst; refl; trans)
open import Relation.Nullary using (¬_)

open import Utils using (Unique; push; empty)

xyzw : List String
xyzw = "x" ∷ "y" ∷ "z" ∷ "w" ∷ []

xyzw-Unique : Unique xyzw
xyzw-Unique = push ((λ ()) ∷ (λ ()) ∷ (λ ()) ∷ []) (push ((λ ()) ∷ (λ ()) ∷ []) (push ((λ ()) ∷ []) (push [] empty)))

open import Lattice using (IsFiniteHeightLattice; FiniteHeightLattice; Monotonic)
open import Lattice.AboveBelow ⊤ _≡_ (record { ≈-refl = refl; ≈-sym = sym; ≈-trans = trans }) _≟ᵘ_  as AB using () renaming (≈-dec to ≈ᵘ-dec)
open AB.Plain using () renaming (finiteHeightLattice to finiteHeightLatticeᵘ)
open import Lattice.Bundles.FiniteValueMap String AB.AboveBelow _≟ˢ_ renaming (finiteHeightLattice to finiteHeightLatticeᵐ; FiniteHeightType to FiniteHeightTypeᵐ; ≈-dec to ≈-dec)

fhlᵘ = finiteHeightLatticeᵘ (Data.Unit.tt)

FiniteHeightMap = FiniteHeightTypeᵐ fhlᵘ xyzw-Unique ≈ᵘ-dec

showAboveBelow : AB.AboveBelow → String
showAboveBelow AB.⊤ = "⊤"
showAboveBelow AB.⊥ = "⊥"
showAboveBelow (AB.[_] tt) = "()"

showMap : ∀ {ks : List String} → FiniteHeightMap ks → String
showMap ((kvs , _) , _) = "{" ++ foldr (λ (x , y) rest → x ++ " ↦ " ++ showAboveBelow y ++ ", " ++ rest) "" kvs ++ "}"

fhlⁱᵖ = finiteHeightLatticeᵐ fhlᵘ xyzw-Unique ≈ᵘ-dec

open FiniteHeightLattice fhlⁱᵖ using (_≈_; _⊔_; _⊓_; ⊔-idemp; _≼_; ≈-⊔-cong; ≈-refl; ≈-trans; ≈-sym; ⊔-assoc; ⊔-comm; ⊔-Monotonicˡ)
open import Relation.Binary.Reasoning.Base.Single _≈_ (λ {m} → ≈-refl {m}) (λ {m₁} {m₂} {m₃} → ≈-trans {m₁} {m₂} {m₃}) -- why am I having to eta-expand here?

smallestMap = proj₁ (proj₁ (proj₁ (FiniteHeightLattice.fixedHeight fhlⁱᵖ)))
largestMap = proj₂ (proj₁ (proj₁ (FiniteHeightLattice.fixedHeight fhlⁱᵖ)))

dumb : FiniteHeightMap xyzw
dumb = ((("x" , AB.[_] tt) ∷ ("y" , AB.⊥) ∷ ("z" , AB.⊥) ∷ ("w" , AB.⊥) ∷ [] , xyzw-Unique) , refl)

dumbFunction : FiniteHeightMap xyzw → FiniteHeightMap xyzw
dumbFunction = _⊔_ dumb

dumbFunction-Monotonic : Monotonic _≼_ _≼_ dumbFunction
dumbFunction-Monotonic {m₁} {m₂} m₁≼m₂ = ⊔-Monotonicˡ dumb {m₁} {m₂} m₁≼m₂


open import Fixedpoint {0ℓ} {FiniteHeightMap xyzw} {8} {_≈_} {_⊔_} {_⊓_} (≈-dec fhlᵘ xyzw-Unique ≈ᵘ-dec) (FiniteHeightLattice.isFiniteHeightLattice fhlⁱᵖ) dumbFunction (λ {m₁} {m₂} m₁≼m₂ → dumbFunction-Monotonic {m₁} {m₂} m₁≼m₂)

-- module Fixedpoint {a} {A : Set a}
--                   {h : ℕ}
--                   {_≈_ : A → A → Set a}
--                   {_⊔_ : A → A → A} {_⊓_ : A → A → A}
--                   (≈-dec : IsDecidable _≈_)
--                   (flA : IsFiniteHeightLattice A h _≈_ _⊔_ _⊓_)
--                   (f : A → A)
--                   (Monotonicᶠ : Monotonic (IsFiniteHeightLattice._≼_ flA)
--                                           (IsFiniteHeightLattice._≼_ flA) f) where

main = run {0ℓ} (putStrLn (showMap aᶠ))