agda-spa/Lattice.agda

module Lattice where

import Data.Nat.Properties as NatProps
open import Relation.Binary.PropositionalEquality as Eq using (_≡_; sym)
open import Relation.Binary.Definitions
open import Data.Nat as Nat using (ℕ; _≤_)
open import Data.Product using (_×_; _,_)
open import Data.Sum using (_⊎_; inj₁; inj₂)
open import Agda.Primitive using (lsuc; Level)

open import NatMap using (NatMap)

record IsPreorder {a} (A : Set a) (_≼_ : A → A → Set a) : Set a where
    field
        ≼-refl : Reflexive (_≼_)
        ≼-trans : Transitive (_≼_)
        ≼-antisym : Antisymmetric (_≡_) (_≼_)

isPreorderFlip : {a : Level} → {A : Set a} → {_≼_ : A → A → Set a} → IsPreorder A _≼_ → IsPreorder A (λ x y → y ≼ x)
isPreorderFlip isPreorder = record
    { ≼-refl = IsPreorder.≼-refl isPreorder
    ; ≼-trans = λ {x} {y} {z} x≽y y≽z → IsPreorder.≼-trans isPreorder y≽z x≽y
    ; ≼-antisym = λ {x} {y} x≽y y≽x → IsPreorder.≼-antisym isPreorder y≽x x≽y
    }

record Preorder {a} (A : Set a) : Set (lsuc a) where
    field
        _≼_ : A → A → Set a

        isPreorder : IsPreorder A _≼_

    open IsPreorder isPreorder public

record IsSemilattice {a} (A : Set a) (_≼_ : A → A → Set a) (_⊔_ : A → A → A) : Set a where
    field
        isPreorder : IsPreorder A _≼_

        ⊔-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

        ⊔-bound : (x y z : A) → x ⊔ y ≡ z → (x ≼ z × y ≼ z)
        ⊔-least : (x y z : A) → x ⊔ y ≡ z → ∀ (z' : A) → (x ≼ z' × y ≼ z') → z ≼ z'

    open IsPreorder isPreorder public

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

    _≽_ : A → A → Set a
    a ≽ b = b ≼ a

    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 renaming
        ( ⊔-assoc to ⊓-assoc
        ; ⊔-comm to ⊓-comm
        ; ⊔-idemp to ⊓-idemp
        ; ⊔-bound to ⊓-bound
        ; ⊔-least to ⊓-least
        )


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 PreorderInstances where
    module ForNat where
        open NatProps

        NatPreorder : Preorder ℕ
        NatPreorder = record
            { _≼_ = _≤_
            ; isPreorder = record
                { ≼-refl = ≤-refl
                ; ≼-trans = ≤-trans
                ; ≼-antisym = ≤-antisym
                }
            }

    module ForProd {a} {A B : Set a} (pA : Preorder A) (pB : Preorder B) where
        open Eq

        private
            _≼_ : A × B → A × B → Set a
            (a₁ ,  b₁) ≼ (a₂ , b₂) = Preorder._≼_ pA a₁ a₂ × Preorder._≼_ pB b₁ b₂

            ≼-refl : {p : A × B} → p ≼ p
            ≼-refl {(a , b)} = (Preorder.≼-refl pA {a}, Preorder.≼-refl pB {b})

            ≼-trans : {p₁ p₂ p₃ : A × B} → p₁ ≼ p₂ → p₂ ≼ p₃ → p₁ ≼ p₃
            ≼-trans (a₁≼a₂ , b₁≼b₂) (a₂≼a₃ , b₂≼b₃) =
                ( Preorder.≼-trans pA a₁≼a₂ a₂≼a₃
                , Preorder.≼-trans pB b₁≼b₂ b₂≼b₃
                )

            ≼-antisym : {p₁ p₂ : A × B} → p₁ ≼ p₂ → p₂ ≼ p₁ → p₁ ≡ p₂
            ≼-antisym (a₁≼a₂ , b₁≼b₂) (a₂≼a₁ , b₂≼b₁) = cong₂ (_,_) (Preorder.≼-antisym pA a₁≼a₂ a₂≼a₁) (Preorder.≼-antisym pB b₁≼b₂ b₂≼b₁)

        ProdPreorder : Preorder (A × B)
        ProdPreorder = record
            { _≼_ = _≼_
            ; isPreorder = record
                { ≼-refl = ≼-refl
                ; ≼-trans = ≼-trans
                ; ≼-antisym = ≼-antisym
                }
            }

module SemilatticeInstances where
    module ForNat where
        open Nat
        open NatProps
        open Eq
        open PreorderInstances.ForNat

        private
            max-bound₁ : {x y z : ℕ} → x ⊔ y ≡ z → x ≤ z
            max-bound₁ {x} {y} {z} x⊔y≡z rewrite sym x⊔y≡z rewrite ⊔-comm x y = m≤n⇒m≤o⊔n y (≤-refl)

            max-bound₂ : {x y z : ℕ} → x ⊔ y ≡ z → y ≤ z
            max-bound₂ {x} {y} {z} x⊔y≡z rewrite sym x⊔y≡z = m≤n⇒m≤o⊔n x (≤-refl)

            max-least : (x y z : ℕ) → x ⊔ y ≡ z → ∀ (z' : ℕ) → (x ≤ z' × y ≤ z') → z ≤ z'
            max-least x y z x⊔y≡z z' (x≤z' , y≤z') with (⊔-sel x y)
            ...                                    | inj₁ x⊔y≡x rewrite trans (sym x⊔y≡z) (x⊔y≡x) = x≤z'
            ...                                    | inj₂ x⊔y≡y rewrite trans (sym x⊔y≡z) (x⊔y≡y) = y≤z'

        NatMaxSemilattice : Semilattice ℕ
        NatMaxSemilattice = record
            { _≼_ = _≤_
            ; _⊔_ = _⊔_
            ; isSemilattice = record
                { isPreorder = Preorder.isPreorder NatPreorder
                ; ⊔-assoc = ⊔-assoc
                ; ⊔-comm = ⊔-comm
                ; ⊔-idemp = ⊔-idem
                ; ⊔-bound = λ x y z x⊔y≡z → (max-bound₁ x⊔y≡z , max-bound₂ x⊔y≡z)
                ; ⊔-least = max-least
                }
            }

        private
            min-bound₁ : {x y z : ℕ} → x ⊓ y ≡ z → z ≤ x
            min-bound₁ {x} {y} {z} x⊓y≡z rewrite sym x⊓y≡z = m≤n⇒m⊓o≤n y (≤-refl)

            min-bound₂ : {x y z : ℕ} → x ⊓ y ≡ z → z ≤ y
            min-bound₂ {x} {y} {z} x⊓y≡z rewrite sym x⊓y≡z rewrite ⊓-comm x y = m≤n⇒m⊓o≤n x (≤-refl)

            min-greatest : (x y z : ℕ) → x ⊓ y ≡ z → ∀ (z' : ℕ) → (z' ≤ x × z' ≤ y) → z' ≤ z
            min-greatest x y z x⊓y≡z z' (z'≤x , z'≤y) with (⊓-sel x y)
            ...                                    | inj₁ x⊓y≡x rewrite trans (sym x⊓y≡z) (x⊓y≡x) = z'≤x
            ...                                    | inj₂ x⊓y≡y rewrite trans (sym x⊓y≡z) (x⊓y≡y) = z'≤y


        NatMinSemilattice : Semilattice ℕ
        NatMinSemilattice = record
            { _≼_ = _≥_
            ; _⊔_ = _⊓_
            ; isSemilattice = record
                { isPreorder = isPreorderFlip (Preorder.isPreorder NatPreorder)
                ; ⊔-assoc = ⊓-assoc
                ; ⊔-comm = ⊓-comm
                ; ⊔-idemp = ⊓-idem
                ; ⊔-bound = λ x y z x⊓y≡z → (min-bound₁ x⊓y≡z , min-bound₂ x⊓y≡z)
                ; ⊔-least = min-greatest
                }
            }

    module ForProd {a} {A B : Set a} (sA : Semilattice A) (sB : Semilattice B) where
        private
            _≼₁_ = Semilattice._≼_ sA
            _≼₂_ = Semilattice._≼_ sB

            pA = record { _≼_ = _≼₁_; isPreorder = Semilattice.isPreorder sA }
            pB = record { _≼_ = _≼₂_; isPreorder = Semilattice.isPreorder sB }

        open PreorderInstances.ForProd pA pB
        open Eq
        open Data.Product

        private
            _≼_ = Preorder._≼_ ProdPreorder

            _⊔_ : A × B → A × B → A × B
            (a₁ , b₁) ⊔ (a₂ , b₂) = (Semilattice._⊔_ sA a₁ a₂ , Semilattice._⊔_ sB b₁ b₂)

            ⊔-assoc : (p₁ p₂ p₃ : A × B) → (p₁ ⊔ p₂) ⊔ p₃ ≡ p₁ ⊔ (p₂ ⊔ p₃)
            ⊔-assoc (a₁ , b₁) (a₂ , b₂) (a₃ , b₃)
                rewrite Semilattice.⊔-assoc sA a₁ a₂ a₃
                rewrite Semilattice.⊔-assoc sB b₁ b₂ b₃ = refl

            ⊔-comm : (p₁ p₂ : A × B) → p₁ ⊔ p₂ ≡ p₂ ⊔ p₁
            ⊔-comm (a₁ , b₁) (a₂ , b₂)
                rewrite Semilattice.⊔-comm sA a₁ a₂
                rewrite Semilattice.⊔-comm sB b₁ b₂ = refl

            ⊔-idemp : (p : A × B) → p ⊔ p ≡ p
            ⊔-idemp (a , b)
                rewrite Semilattice.⊔-idemp sA a
                rewrite Semilattice.⊔-idemp sB b = refl

            ⊔-bound₁ : {p₁ p₂ p₃ : A × B} → p₁ ⊔ p₂ ≡ p₃ → p₁ ≼ p₃
            ⊔-bound₁ {(a₁ , b₁)} {(a₂ , b₂)} {(a₃ , b₃)} p₁⊔p₂≡p₃ = (⊔-bound-a , ⊔-bound-b)
                where
                    ⊔-bound-a = proj₁ (Semilattice.⊔-bound sA a₁ a₂ a₃ (cong proj₁ p₁⊔p₂≡p₃))
                    ⊔-bound-b = proj₁ (Semilattice.⊔-bound sB b₁ b₂ b₃ (cong proj₂ p₁⊔p₂≡p₃))

            ⊔-bound₂ : {p₁ p₂ p₃ : A × B} → p₁ ⊔ p₂ ≡ p₃ → p₂ ≼ p₃
            ⊔-bound₂ {(a₁ , b₁)} {(a₂ , b₂)} {(a₃ , b₃)} p₁⊔p₂≡p₃ = (⊔-bound-a , ⊔-bound-b)
                where
                    ⊔-bound-a = proj₂ (Semilattice.⊔-bound sA a₁ a₂ a₃ (cong proj₁ p₁⊔p₂≡p₃))
                    ⊔-bound-b = proj₂ (Semilattice.⊔-bound sB b₁ b₂ b₃ (cong proj₂ p₁⊔p₂≡p₃))

            ⊔-least : (p₁ p₂ p₃ : A × B) → p₁ ⊔ p₂ ≡ p₃ → ∀ (p₃' : A × B) → (p₁ ≼ p₃' × p₂ ≼ p₃') → p₃ ≼ p₃'
            ⊔-least (a₁ , b₁) (a₂ , b₂) (a₃ , b₃) p₁⊔p₂≡p₃ (a₃' , b₃') (p₁≼p₃' , p₂≼p₃') = (⊔-least-a , ⊔-least-b)
                where
                    ⊔-least-a : a₃ ≼₁ a₃'
                    ⊔-least-a = Semilattice.⊔-least sA a₁ a₂ a₃ (cong proj₁ p₁⊔p₂≡p₃) a₃' (proj₁ p₁≼p₃' , proj₁ p₂≼p₃')

                    ⊔-least-b : b₃ ≼₂ b₃'
                    ⊔-least-b = Semilattice.⊔-least sB b₁ b₂ b₃ (cong proj₂ p₁⊔p₂≡p₃) b₃' (proj₂ p₁≼p₃' , proj₂ p₂≼p₃')

        ProdSemilattice : Semilattice (A × B)
        ProdSemilattice = record
            { _≼_ = _≼_
            ; _⊔_ = _⊔_
            ; isSemilattice = record
                { isPreorder = Preorder.isPreorder ProdPreorder
                ; ⊔-assoc = ⊔-assoc
                ; ⊔-comm = ⊔-comm
                ; ⊔-idemp = ⊔-idemp
                ; ⊔-bound = λ x y z x⊓y≡z → (⊔-bound₁ x⊓y≡z , ⊔-bound₂ x⊓y≡z)
                ; ⊔-least = ⊔-least
                }
            }

module LatticeInstances where
    module ForNat where
        open Nat
        open NatProps
        open Eq
        open SemilatticeInstances.ForNat
        open Data.Product


        private
            minmax-absorb : {x y : ℕ} → x ⊓ (x ⊔ y) ≡ x
            minmax-absorb {x} {y} = ≤-antisym x⊓x⊔y≤x (helper x⊓x≤x⊓x⊔y (⊓-idem x))
                where
                    x⊓x⊔y≤x = proj₁ (Semilattice.⊔-bound NatMinSemilattice x (x ⊔ y) (x ⊓ (x ⊔ y)) refl)
                    x⊓x≤x⊓x⊔y = ⊓-mono-≤ {x} {x} ≤-refl (proj₁ (Semilattice.⊔-bound NatMaxSemilattice x y (x ⊔ y) refl))

                    -- >:(
                    helper : x ⊓ x ≤ x ⊓ (x ⊔ y) → x ⊓ x ≡ x → x ≤ x ⊓ (x ⊔ y)
                    helper x⊓x≤x⊓x⊔y x⊓x≡x rewrite x⊓x≡x = x⊓x≤x⊓x⊔y

            maxmin-absorb : {x y : ℕ} → x ⊔ (x ⊓ y) ≡ x
            maxmin-absorb {x} {y} = ≤-antisym (helper x⊔x⊓y≤x⊔x (⊔-idem x)) x≤x⊔x⊓y
                where
                    x≤x⊔x⊓y = proj₁ (Semilattice.⊔-bound NatMaxSemilattice x (x ⊓ y) (x ⊔ (x ⊓ y)) refl)
                    x⊔x⊓y≤x⊔x = ⊔-mono-≤ {x} {x} ≤-refl (proj₁ (Semilattice.⊔-bound NatMinSemilattice x y (x ⊓ y) refl))

                    -- >:(
                    helper : x ⊔ (x ⊓ y) ≤ x ⊔ x  → x ⊔ x ≡ x → x ⊔ (x ⊓ y) ≤ x
                    helper x⊔x⊓y≤x⊔x x⊔x≡x rewrite x⊔x≡x = x⊔x⊓y≤x⊔x

        NatLattice : Lattice ℕ
        NatLattice = record
            { _≼_ = _≤_
            ; _⊔_ = _⊔_
            ; _⊓_ = _⊓_
            ; isLattice = record
                { joinSemilattice = Semilattice.isSemilattice NatMaxSemilattice
                ; meetSemilattice = Semilattice.isSemilattice NatMinSemilattice
                ; absorb-⊔-⊓ = λ x y → maxmin-absorb {x} {y}
                ; absorb-⊓-⊔ = λ x y → minmax-absorb {x} {y}
                }
            }

    module ForProd {a} {A B : Set a} (lA : Lattice A) (lB : Lattice B) where
        private
            _≼₁_ = Lattice._≼_ lA
            _≼₂_ = Lattice._≼_ lB

            _⊔₁_ = Lattice._⊔_ lA
            _⊔₂_ = Lattice._⊔_ lB

            _⊓₁_ = Lattice._⊓_ lA
            _⊓₂_ = Lattice._⊓_ lB

            joinA = record { _≼_ = _≼₁_; _⊔_ = _⊔₁_; isSemilattice = Lattice.joinSemilattice lA }
            joinB = record { _≼_ = _≼₂_; _⊔_ = _⊔₂_; isSemilattice = Lattice.joinSemilattice lB }


            meetA = record { _≼_ = λ a b → b ≼₁ a; _⊔_ = _⊓₁_; isSemilattice = Lattice.meetSemilattice lA }
            meetB = record { _≼_ = λ a b → b ≼₂ a; _⊔_ = _⊓₂_; isSemilattice = Lattice.meetSemilattice lB }

            module ProdJoin = SemilatticeInstances.ForProd joinA joinB
            module ProdMeet = SemilatticeInstances.ForProd meetA meetB


            _≼_ = Semilattice._≼_ ProdJoin.ProdSemilattice
            _⊔_ = Semilattice._⊔_ ProdJoin.ProdSemilattice
            _⊓_ = Semilattice._⊔_ ProdMeet.ProdSemilattice

        open Eq
        open Data.Product

        private
            absorb-⊔-⊓ : (p₁ p₂ : A × B) → p₁ ⊔ (p₁ ⊓ p₂) ≡ p₁
            absorb-⊔-⊓ (a₁ , b₁) (a₂ , b₂)
                rewrite Lattice.absorb-⊔-⊓ lA a₁ a₂
                rewrite Lattice.absorb-⊔-⊓ lB b₁ b₂ = refl

            absorb-⊓-⊔ : (p₁ p₂ : A × B) → p₁ ⊓ (p₁ ⊔ p₂) ≡ p₁
            absorb-⊓-⊔ (a₁ , b₁) (a₂ , b₂)
                rewrite Lattice.absorb-⊓-⊔ lA a₁ a₂
                rewrite Lattice.absorb-⊓-⊔ lB b₁ b₂ = refl

        ProdLattice : Lattice (A × B)
        ProdLattice = record
            { _≼_ = _≼_
            ; _⊔_ = _⊔_
            ; _⊓_ = _⊓_
            ; isLattice = record
                { joinSemilattice = Semilattice.isSemilattice ProdJoin.ProdSemilattice
                ; meetSemilattice = Semilattice.isSemilattice ProdMeet.ProdSemilattice
                ; absorb-⊔-⊓ = absorb-⊔-⊓
                ; absorb-⊓-⊔ = absorb-⊓-⊔
                }
            }