module Lattice where

open import Equivalence
import Chain

import Data.Nat.Properties as NatProps
open import Relation.Binary.PropositionalEquality as Eq using (_≡_; sym; subst)
open Eq.≡-Reasoning using (begin_; _≡⟨⟩_; step-≡; _∎)
open import Relation.Binary.Definitions
open import Relation.Binary.Core using (_Preserves_⟶_ )
open import Relation.Nullary using (Dec; ¬_; yes; no)
open import Data.Nat as Nat using (ℕ; _≤_; _+_; suc)
open import Data.Product using (_×_; Σ; _,_; proj₁; proj₂)
open import Data.Sum using (_⊎_; inj₁; inj₂)
open import Agda.Primitive using (lsuc; Level) renaming (_⊔_ to _⊔ℓ_)
open import Function.Definitions using (Injective)
open import Data.Empty using (⊥)

absurd : ∀ {a} {A : Set a} →  ⊥ → A
absurd ()

IsDecidable : ∀ {a} {A : Set a} (R : A → A → Set a) → Set a
IsDecidable {a} {A} R = ∀ (a₁ a₂ : A) → Dec (R a₁ a₂)

record IsSemilattice {a} (A : Set a)
    (_≈_ : A → A → Set a)
    (_⊔_ : A → A → A) : Set a where

    _≼_ : A → A → Set a
    a ≼ b = Σ A (λ c → (a ⊔ c) ≈ b)

    _≺_ : A → A → Set a
    a ≺ b = (a ≼ b) × (¬ a ≈ b)

    field
        ≈-equiv : IsEquivalence A _≈_
        ≈-⊔-cong : ∀ {a₁ a₂ a₃ a₄} → a₁ ≈ a₂ → a₃ ≈ a₄ → (a₁ ⊔ a₃) ≈ (a₂ ⊔ a₄)

        ⊔-assoc : (x y z : A) → ((x ⊔ y) ⊔ z) ≈ (x ⊔ (y ⊔ z))
        ⊔-comm : (x y : A) → (x ⊔ y) ≈ (y ⊔ x)
        ⊔-idemp : (x : A) → (x ⊔ x) ≈ x

    open IsEquivalence ≈-equiv public

    ≼-refl : ∀ (a : A) → a ≼ a
    ≼-refl a = (a , ⊔-idemp a)

    ≼-cong : ∀ {a₁ a₂ a₃ a₄ : A} → a₁ ≈ a₂ → a₃ ≈ a₄ → a₁ ≼ a₃ → a₂ ≼ a₄
    ≼-cong a₁≈a₂ a₃≈a₄ (c₁ , a₁⊔c₁≈a₃) = (c₁ , ≈-trans (≈-⊔-cong (≈-sym a₁≈a₂) ≈-refl) (≈-trans a₁⊔c₁≈a₃ a₃≈a₄))

    ≺-cong : ∀ {a₁ a₂ a₃ a₄ : A} → a₁ ≈ a₂ → a₃ ≈ a₄ → a₁ ≺ a₃ → a₂ ≺ a₄
    ≺-cong a₁≈a₂ a₃≈a₄ (a₁≼a₃ , a₁̷≈a₃) =
        ( ≼-cong a₁≈a₂ a₃≈a₄ a₁≼a₃
        , λ a₂≈a₄ → a₁̷≈a₃ (≈-trans a₁≈a₂ (≈-trans a₂≈a₄ (≈-sym a₃≈a₄)))
        )

record IsLattice {a} (A : Set a)
    (_≈_ : A → A → Set a)
    (_⊔_ : A → A → A)
    (_⊓_ : A → A → A) : Set a where

    field
        joinSemilattice : IsSemilattice A _≈_ _⊔_
        meetSemilattice : IsSemilattice A _≈_ _⊓_

        absorb-⊔-⊓ : (x y : A) → (x ⊔ (x ⊓ y)) ≈ x
        absorb-⊓-⊔ : (x y : A) → (x ⊓ (x ⊔ y)) ≈ x

    open IsSemilattice joinSemilattice public
    open IsSemilattice meetSemilattice public using () renaming
        ( ⊔-assoc to ⊓-assoc
        ; ⊔-comm to ⊓-comm
        ; ⊔-idemp to ⊓-idemp
        ; ≈-⊔-cong to ≈-⊓-cong
        ; _≼_ to _≽_
        ; _≺_ to _≻_
        ; ≼-refl to ≽-refl
        )

    FixedHeight : ∀ (h : ℕ) → Set a
    FixedHeight h = Chain.Height (_≈_) ≈-equiv _≺_ ≺-cong h

record IsFiniteHeightLattice {a} (A : Set a)
    (h : ℕ)
    (_≈_ : A → A → Set a)
    (_⊔_ : A → A → A)
    (_⊓_ : A → A → A) : Set (lsuc a) where

    field
        isLattice : IsLattice A _≈_ _⊔_ _⊓_

    open IsLattice isLattice public

    field
        fixedHeight : FixedHeight h

module _ {a b} {A : Set a} {B : Set b}
    (_≼₁_ : A → A → Set a) (_≼₂_ : B → B → Set b) where

    Monotonic : (A → B) → Set (a ⊔ℓ b)
    Monotonic f = ∀ {a₁ a₂ : A} → a₁ ≼₁ a₂ → f a₁ ≼₂ f a₂

module ChainMapping {a b} {A : Set a} {B : Set b}
    {_≈₁_ : A → A → Set a} {_≈₂_ : B → B → Set b}
    {_⊔₁_ : A → A → A} {_⊔₂_ : B → B → B}
    (slA : IsSemilattice A _≈₁_ _⊔₁_) (slB : IsSemilattice B _≈₂_ _⊔₂_) where

    open IsSemilattice slA renaming (_≼_ to _≼₁_; _≺_ to _≺₁_; ≈-equiv to ≈₁-equiv; ≺-cong to ≺₁-cong)
    open IsSemilattice slB renaming (_≼_ to _≼₂_; _≺_ to _≺₂_; ≈-equiv to ≈₂-equiv; ≺-cong to ≺₂-cong)

    open Chain _≈₁_ ≈₁-equiv _≺₁_ ≺₁-cong using () renaming (Chain to Chain₁; step to step₁; done to done₁)
    open Chain _≈₂_ ≈₂-equiv _≺₂_ ≺₂-cong using () renaming (Chain to Chain₂; step to step₂; done to done₂)

    Chain-map : ∀ (f : A → B) →
                Monotonic _≼₁_ _≼₂_ f →
                Injective _≈₁_ _≈₂_ f →
                f Preserves _≈₁_ ⟶  _≈₂_ →
                ∀ {a₁ a₂ : A} {n : ℕ} → Chain₁ a₁ a₂ n → Chain₂ (f a₁) (f a₂) n
    Chain-map f Monotonicᶠ Injectiveᶠ Preservesᶠ (done₁ a₁≈a₂) =
        done₂ (Preservesᶠ a₁≈a₂)
    Chain-map f Monotonicᶠ Injectiveᶠ Preservesᶠ (step₁ (a₁≼₁a , a₁̷≈₁a) a≈₁a' a'a₂) =
        let fa₁≺₂fa = (Monotonicᶠ a₁≼₁a , λ fa₁≈₂fa → a₁̷≈₁a (Injectiveᶠ fa₁≈₂fa))
            fa≈fa' = Preservesᶠ a≈₁a'
        in step₂ fa₁≺₂fa fa≈fa' (Chain-map f Monotonicᶠ Injectiveᶠ Preservesᶠ a'a₂)

record Semilattice {a} (A : Set a) : Set (lsuc a) where
    field
        _≈_ : A → A → Set a
        _⊔_ : A → A → A

        isSemilattice : IsSemilattice A _≈_ _⊔_

    open IsSemilattice isSemilattice public


record Lattice {a} (A : Set a) : Set (lsuc a) where
    field
        _≈_ : A → A → Set a
        _⊔_ : A → A → A
        _⊓_ : A → A → A

        isLattice : IsLattice A _≈_ _⊔_ _⊓_

    open IsLattice isLattice public

module IsSemilatticeInstances where
    module ForProd {a} {A B : Set a}
        (_≈₁_ : A → A → Set a) (_≈₂_ : B → B → Set a)
        (_⊔₁_ : A → A → A) (_⊔₂_ : B → B → B)
        (sA : IsSemilattice A _≈₁_ _⊔₁_) (sB : IsSemilattice B _≈₂_ _⊔₂_) where

        open Eq
        open Data.Product

        module ProdEquiv = IsEquivalenceInstances.ForProd _≈₁_ _≈₂_ (IsSemilattice.≈-equiv sA) (IsSemilattice.≈-equiv sB)
        open ProdEquiv using (_≈_) public

        infixl 20 _⊔_

        _⊔_ : A × B → A × B → A × B
        (a₁ , b₁) ⊔ (a₂ , b₂) = (a₁ ⊔₁ a₂ , b₁ ⊔₂ b₂)

        ProdIsSemilattice : IsSemilattice (A × B) _≈_ _⊔_
        ProdIsSemilattice = record
            { ≈-equiv = ProdEquiv.ProdEquivalence
            ; ≈-⊔-cong = λ (a₁≈a₂ , b₁≈b₂) (a₃≈a₄ , b₃≈b₄) →
                ( IsSemilattice.≈-⊔-cong sA a₁≈a₂ a₃≈a₄
                , IsSemilattice.≈-⊔-cong sB b₁≈b₂ b₃≈b₄
                )
            ; ⊔-assoc = λ (a₁ , b₁) (a₂ , b₂) (a₃ , b₃) →
                ( IsSemilattice.⊔-assoc sA a₁ a₂ a₃
                , IsSemilattice.⊔-assoc sB b₁ b₂ b₃
                )
            ; ⊔-comm = λ (a₁ , b₁) (a₂ , b₂) →
                ( IsSemilattice.⊔-comm sA a₁ a₂
                , IsSemilattice.⊔-comm sB b₁ b₂
                )
            ; ⊔-idemp = λ (a , b) →
                ( IsSemilattice.⊔-idemp sA a
                , IsSemilattice.⊔-idemp sB b
                )
            }

module IsLatticeInstances where
    module ForProd {a} {A B : Set a}
        (_≈₁_ : A → A → Set a) (_≈₂_ : B → B → Set a)
        (_⊔₁_ : A → A → A) (_⊓₁_ : A → A → A)
        (_⊔₂_ : B → B → B) (_⊓₂_ : B → B → B)
        (lA : IsLattice A _≈₁_ _⊔₁_ _⊓₁_) (lB : IsLattice B _≈₂_ _⊔₂_ _⊓₂_) where

        module ProdJoin = IsSemilatticeInstances.ForProd _≈₁_ _≈₂_ _⊔₁_ _⊔₂_ (IsLattice.joinSemilattice lA) (IsLattice.joinSemilattice lB)
        open ProdJoin using (_⊔_; _≈_) public

        module ProdMeet = IsSemilatticeInstances.ForProd _≈₁_ _≈₂_ _⊓₁_ _⊓₂_ (IsLattice.meetSemilattice lA) (IsLattice.meetSemilattice lB)
        open ProdMeet using () renaming (_⊔_ to _⊓_) public

        ProdIsLattice : IsLattice (A × B) _≈_ _⊔_ _⊓_
        ProdIsLattice = record
            { joinSemilattice = ProdJoin.ProdIsSemilattice
            ; meetSemilattice = ProdMeet.ProdIsSemilattice
            ; absorb-⊔-⊓ = λ (a₁ , b₁) (a₂ , b₂) →
                ( IsLattice.absorb-⊔-⊓ lA a₁ a₂
                , IsLattice.absorb-⊔-⊓ lB b₁ b₂
                )
            ; absorb-⊓-⊔ = λ (a₁ , b₁) (a₂ , b₂) →
                ( IsLattice.absorb-⊓-⊔ lA a₁ a₂
                , IsLattice.absorb-⊓-⊔ lB b₁ b₂
                )
            }


module IsFiniteHeightLatticeInstances where
    module ForProd {a} {A B : Set a}
        (_≈₁_ : A → A → Set a) (_≈₂_ : B → B → Set a)
        (≈₁-dec : IsDecidable _≈₁_) (≈₂-dec : IsDecidable _≈₂_)
        (_⊔₁_ : A → A → A) (_⊓₁_ : A → A → A)
        (_⊔₂_ : B → B → B) (_⊓₂_ : B → B → B)
        (h₁ h₂ : ℕ)
        (lA : IsFiniteHeightLattice A h₁ _≈₁_ _⊔₁_ _⊓₁_) (lB : IsFiniteHeightLattice B h₂ _≈₂_ _⊔₂_ _⊓₂_) where

        open NatProps
        module ProdLattice = IsLatticeInstances.ForProd _≈₁_ _≈₂_ _⊔₁_ _⊓₁_ _⊔₂_ _⊓₂_ (IsFiniteHeightLattice.isLattice lA) (IsFiniteHeightLattice.isLattice lB)
        open ProdLattice using (_⊔_; _⊓_; _≈_) public
        open IsLattice ProdLattice.ProdIsLattice using (_≼_; _≺_; ≺-cong; ≈-equiv)
        open IsFiniteHeightLattice lA using () renaming (⊔-idemp to ⊔₁-idemp; _≼_ to _≼₁_; ≈-equiv to ≈₁-equiv; ≈-refl to ≈₁-refl; ≈-trans to ≈₁-trans; ≈-sym to ≈₁-sym; _≺_ to _≺₁_; ≺-cong to ≺₁-cong)
        open IsFiniteHeightLattice lB using () renaming (⊔-idemp to ⊔₂-idemp; _≼_ to _≼₂_; ≈-equiv to ≈₂-equiv; ≈-refl to ≈₂-refl; ≈-trans to ≈₂-trans; ≈-sym to ≈₂-sym; _≺_ to _≺₂_; ≺-cong to ≺₂-cong)

        module ChainMapping₁ = ChainMapping (IsFiniteHeightLattice.joinSemilattice lA) (IsLattice.joinSemilattice ProdLattice.ProdIsLattice)
        module ChainMapping₂ = ChainMapping (IsFiniteHeightLattice.joinSemilattice lB) (IsLattice.joinSemilattice ProdLattice.ProdIsLattice)

        module ChainA = Chain _≈₁_ ≈₁-equiv _≺₁_ ≺₁-cong
        module ChainB = Chain _≈₂_ ≈₂-equiv _≺₂_ ≺₂-cong
        module ProdChain = Chain _≈_ ≈-equiv _≺_ ≺-cong

        open ChainA using () renaming (Chain to Chain₁; done to done₁; step to step₁; Chain-≈-cong₁ to Chain₁-≈-cong₁)
        open ChainB using () renaming (Chain to Chain₂; done to done₂; step to step₂; Chain-≈-cong₁ to Chain₂-≈-cong₁)
        open ProdChain using (Chain; concat; done; step)

        private
            a,∙-Monotonic : ∀ (a : A) → Monotonic _≼₂_ _≼_ (λ b → (a , b))
            a,∙-Monotonic a {b₁} {b₂} (b , b₁⊔b≈b₂) = ((a , b) , (⊔₁-idemp a , b₁⊔b≈b₂))

            a,∙-Preserves-≈₂ : ∀ (a : A) → (λ b → (a , b)) Preserves _≈₂_ ⟶  _≈_
            a,∙-Preserves-≈₂ a {b₁} {b₂} b₁≈b₂ = (≈₁-refl , b₁≈b₂)

            ∙,b-Monotonic : ∀ (b : B) → Monotonic _≼₁_ _≼_ (λ a → (a , b))
            ∙,b-Monotonic b {a₁} {a₂} (a , a₁⊔a≈a₂) = ((a , b) , (a₁⊔a≈a₂ , ⊔₂-idemp b))

            ∙,b-Preserves-≈₁ : ∀ (b : B) → (λ a → (a , b)) Preserves _≈₁_ ⟶  _≈_
            ∙,b-Preserves-≈₁ b {a₁} {a₂} a₁≈a₂ = (a₁≈a₂ , ≈₂-refl)

            amin : A
            amin = proj₁ (proj₁ (proj₁ (IsFiniteHeightLattice.fixedHeight lA)))

            amax : A
            amax = proj₂ (proj₁ (proj₁ (IsFiniteHeightLattice.fixedHeight lA)))

            bmin : B
            bmin = proj₁ (proj₁ (proj₁ (IsFiniteHeightLattice.fixedHeight lB)))

            bmax : B
            bmax = proj₂ (proj₁ (proj₁ (IsFiniteHeightLattice.fixedHeight lB)))

            unzip : ∀ {a₁ a₂ : A} {b₁ b₂ : B} {n : ℕ} → Chain (a₁ , b₁) (a₂ , b₂) n → Σ (ℕ × ℕ) (λ (n₁ , n₂) → ((Chain₁ a₁ a₂ n₁ × Chain₂ b₁ b₂ n₂) × (n ≤ n₁ + n₂)))
            unzip (done (a₁≈a₂ , b₁≈b₂)) = ((0 , 0) , ((done₁ a₁≈a₂ , done₂ b₁≈b₂) , ≤-refl))
            unzip {a₁} {a₂} {b₁} {b₂} {n} (step {(a₁ , b₁)} {(a , b)} (((d₁ , d₂) , (a₁⊔d₁≈a , b₁⊔d₂≈b)) , a₁b₁̷≈ab) (a≈a' , b≈b') a'b'a₂b₂)
                with ≈₁-dec a₁ a | ≈₂-dec b₁ b | unzip a'b'a₂b₂
            ...   | yes a₁≈a | yes b₁≈b | ((n₁ , n₂) , ((c₁ , c₂) , n≤n₁+n₂)) = absurd (a₁b₁̷≈ab (a₁≈a , b₁≈b))
            ...   | no a₁̷≈a  | yes b₁≈b | ((n₁ , n₂) , ((c₁ , c₂) , n≤n₁+n₂)) =
                    ((suc n₁ , n₂) , ((step₁ ((d₁ , a₁⊔d₁≈a) , a₁̷≈a) a≈a' c₁ , Chain₂-≈-cong₁ (≈₂-sym (≈₂-trans b₁≈b b≈b')) c₂), +-monoʳ-≤ 1 (n≤n₁+n₂)))
            ...   | yes a₁≈a | no b₁̷≈b | ((n₁ , n₂) , ((c₁ , c₂) , n≤n₁+n₂)) =
                    ((n₁ , suc n₂) , ( (Chain₁-≈-cong₁ (≈₁-sym (≈₁-trans a₁≈a a≈a')) c₁ , step₂ ((d₂ , b₁⊔d₂≈b) , b₁̷≈b) b≈b' c₂)
                                     , subst (n ≤_) (sym (+-suc n₁ n₂)) (+-monoʳ-≤ 1 n≤n₁+n₂)
                                     ))
            ...   | no a₁̷≈a  | no b₁̷≈b | ((n₁ , n₂) , ((c₁ , c₂) , n≤n₁+n₂)) =
                    ((suc n₁ , suc n₂) , ( (step₁ ((d₁ , a₁⊔d₁≈a) , a₁̷≈a) a≈a' c₁ , step₂ ((d₂ , b₁⊔d₂≈b) , b₁̷≈b) b≈b' c₂)
                                         , ≤-stepsˡ 1 (subst (n ≤_) (sym (+-suc n₁ n₂)) (+-monoʳ-≤ 1 n≤n₁+n₂))
                                         ))

        ProdIsFiniteHeightLattice : IsFiniteHeightLattice (A × B) (h₁ + h₂) _≈_ _⊔_ _⊓_
        ProdIsFiniteHeightLattice = record
            { isLattice = ProdLattice.ProdIsLattice
            ; fixedHeight =
                ( ( ((amin , bmin) , (amax , bmax))
                  , concat
                      (ChainMapping₁.Chain-map (λ a → (a , bmin)) (∙,b-Monotonic _) proj₁ (∙,b-Preserves-≈₁ _) (proj₂ (proj₁ (IsFiniteHeightLattice.fixedHeight lA))))
                      (ChainMapping₂.Chain-map (λ b → (amax , b)) (a,∙-Monotonic _) proj₂ (a,∙-Preserves-≈₂ _) (proj₂ (proj₁ (IsFiniteHeightLattice.fixedHeight lB))))
                  )
                , λ a₁b₁a₂b₂ → let ((n₁ , n₂) , ((a₁a₂ , b₁b₂) , n≤n₁+n₂)) = unzip a₁b₁a₂b₂
                               in ≤-trans n≤n₁+n₂ (+-mono-≤ (proj₂ (IsFiniteHeightLattice.fixedHeight lA) a₁a₂)
                                                            (proj₂ (IsFiniteHeightLattice.fixedHeight lB) b₁b₂))
                )
            }