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 Relation.Nullary using (Dec; ¬_)
open import Data.Nat as Nat using (ℕ; _≤_; _+_)
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 Chain using (Chain; Height; done; step; concat)
open import Function.Definitions using (Injective)

record IsEquivalence {a} (A : Set a) (_≈_ : A → A → Set a) : Set a where
    field
        ≈-refl : {a : A} → a ≈ a
        ≈-sym : {a b : A} → a ≈ b → b ≈ a
        ≈-trans : {a b c : A} → a ≈ b → b ≈ c → a ≈ c

record IsDecidable {a} (A : Set a) (R : A → A → Set a) : Set a where
    field
        R-dec : ∀ (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 _≈_

        ⊔-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

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

    open IsEquivalence ≈-equiv public

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
        ; _≼_ to _≽_
        ; _≺_ to _≻_
        ; ≼-refl to ≽-refl
        )

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 _≈_ _⊔_ _⊓_
        fixedHeight : Height (IsLattice._≺_ isLattice) h

    open IsLattice isLattice public

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_)
    open IsSemilattice slB renaming (_≼_ to _≼₂_; _≺_ to _≺₂_)

    Chain-map : ∀ (f : A → B) → Monotonic _≼₁_ _≼₂_ f → Injective _≈₁_ _≈₂_ f →
                ∀ {a₁ a₂ : A} {n : ℕ} → Chain _≺₁_ a₁ a₂ n → Chain _≺₂_ (f a₁) (f a₂) n
    Chain-map f Monotonicᶠ Injectiveᶠ done = done
    Chain-map f Monotonicᶠ Injectiveᶠ (step (a₁≼₁a , a₁̷≈₁a) aa₂) =
        let fa₁≺₂fa = (Monotonicᶠ a₁≼₁a , λ fa₁≈₂fa → a₁̷≈₁a (Injectiveᶠ fa₁≈₂fa))
        in step fa₁≺₂fa (Chain-map f Monotonicᶠ Injectiveᶠ aa₂)

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 IsEquivalenceInstances where
    module ForProd {a} {A B : Set a}
        (_≈₁_ : A → A → Set a) (_≈₂_ : B → B → Set a)
        (eA : IsEquivalence A _≈₁_) (eB : IsEquivalence B _≈₂_) where

        infix 4 _≈_

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

        ProdEquivalence : IsEquivalence (A × B) _≈_
        ProdEquivalence = record
            { ≈-refl = λ {p} →
                ( IsEquivalence.≈-refl eA
                , IsEquivalence.≈-refl eB
                )
            ; ≈-sym = λ {p₁} {p₂} (a₁≈a₂ , b₁≈b₂) →
                ( IsEquivalence.≈-sym eA a₁≈a₂
                , IsEquivalence.≈-sym eB b₁≈b₂
                )
            ; ≈-trans = λ {p₁} {p₂} {p₃} (a₁≈a₂ , b₁≈b₂) (a₂≈a₃ , b₂≈b₃) →
                ( IsEquivalence.≈-trans eA a₁≈a₂ a₂≈a₃
                , IsEquivalence.≈-trans eB b₁≈b₂ b₂≈b₃
                )
            }

    module ForMap {a b} (A : Set a) (B : Set b)
        (≡-dec-A : Decidable (_≡_ {a} {A}))
        (_≈₂_ : B → B → Set b)
        (eB : IsEquivalence B _≈₂_) where

        open import Map A B ≡-dec-A using (Map; lift; subset)
        open import Data.List using (_∷_; []) -- TODO: re-export these with nicer names from map

        open IsEquivalence eB renaming
            ( ≈-refl to ≈₂-refl
            ; ≈-sym to ≈₂-sym
            ; ≈-trans to ≈₂-trans
            )

        _≈_ : Map → Map → Set (Agda.Primitive._⊔_ a b)
        _≈_ = lift _≈₂_

        _⊆_ : Map → Map → Set (Agda.Primitive._⊔_ a b)
        _⊆_ = subset _≈₂_

        private
            ⊆-refl : (m : Map) →  m ⊆ m
            ⊆-refl _ k v k,v∈m = (v , (≈₂-refl , k,v∈m))

            ⊆-trans : (m₁ m₂ m₃ : Map) →  m₁ ⊆ m₂ → m₂ ⊆ m₃ → m₁ ⊆ m₃
            ⊆-trans _ _ _ m₁⊆m₂ m₂⊆m₃ k v k,v∈m₁ =
                let
                    (v' , (v≈v' , k,v'∈m₂)) = m₁⊆m₂ k v k,v∈m₁
                    (v'' , (v'≈v'' , k,v''∈m₃)) = m₂⊆m₃ k v' k,v'∈m₂
                in (v'' , (≈₂-trans v≈v' v'≈v'' , k,v''∈m₃))

        LiftEquivalence : IsEquivalence Map _≈_
        LiftEquivalence = record
            { ≈-refl = λ {m} → (⊆-refl m , ⊆-refl m)
            ; ≈-sym = λ {m₁} {m₂} (m₁⊆m₂ , m₂⊆m₁) → (m₂⊆m₁ , m₁⊆m₂)
            ; ≈-trans = λ {m₁} {m₂} {m₃} (m₁⊆m₂ , m₂⊆m₁) (m₂⊆m₃ , m₃⊆m₂) →
                ( ⊆-trans m₁ m₂ m₃ m₁⊆m₂ m₂⊆m₃
                , ⊆-trans m₃ m₂ m₁ m₃⊆m₂ m₂⊆m₁
                )
            }

module IsSemilatticeInstances where
    module ForNat where
        open Nat
        open NatProps
        open Eq

        NatIsMaxSemilattice : IsSemilattice ℕ _≡_ _⊔_
        NatIsMaxSemilattice = record
            { ≈-equiv = record
                { ≈-refl = refl
                ; ≈-sym = sym
                ; ≈-trans = trans
                }
            ; ⊔-assoc = ⊔-assoc
            ; ⊔-comm = ⊔-comm
            ; ⊔-idemp = ⊔-idem
            }

        NatIsMinSemilattice : IsSemilattice ℕ _≡_ _⊓_
        NatIsMinSemilattice = record
            { ≈-equiv = record
                { ≈-refl = refl
                ; ≈-sym = sym
                ; ≈-trans = trans
                }
            ; ⊔-assoc = ⊓-assoc
            ; ⊔-comm = ⊓-comm
            ; ⊔-idemp = ⊓-idem
            }

    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
            ; ⊔-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 ForMap {a} {A B : Set a}
        (≡-dec-A : Decidable (_≡_ {a} {A}))
        (_≈₂_ : B → B → Set a)
        (_⊔₂_ : B → B → B)
        (sB : IsSemilattice B _≈₂_ _⊔₂_) where

        open import Map A B ≡-dec-A
        open IsSemilattice sB renaming
            ( ≈-refl to ≈₂-refl; ≈-sym to ≈₂-sym
            ; ⊔-assoc to ⊔₂-assoc; ⊔-comm to ⊔₂-comm; ⊔-idemp to ⊔₂-idemp
            )

        module MapEquiv = IsEquivalenceInstances.ForMap A B ≡-dec-A _≈₂_ (IsSemilattice.≈-equiv sB)
        open MapEquiv using (_≈_) public

        infixl 20 _⊔_
        infixl 20 _⊓_

        _⊔_ : Map → Map → Map
        m₁ ⊔ m₂ = union _⊔₂_ m₁ m₂

        _⊓_ : Map → Map → Map
        m₁ ⊓ m₂ = intersect _⊔₂_ m₁ m₂

        MapIsUnionSemilattice : IsSemilattice Map _≈_ _⊔_
        MapIsUnionSemilattice = record
            { ≈-equiv = MapEquiv.LiftEquivalence
            ; ⊔-assoc = union-assoc _≈₂_ ≈₂-refl ≈₂-sym _⊔₂_ ⊔₂-assoc
            ; ⊔-comm = union-comm _≈₂_ ≈₂-refl ≈₂-sym _⊔₂_ ⊔₂-comm
            ; ⊔-idemp = union-idemp _≈₂_ ≈₂-refl ≈₂-sym _⊔₂_ ⊔₂-idemp
            }

        MapIsIntersectSemilattice : IsSemilattice Map _≈_ _⊓_
        MapIsIntersectSemilattice = record
            { ≈-equiv = MapEquiv.LiftEquivalence
            ; ⊔-assoc = intersect-assoc _≈₂_ ≈₂-refl ≈₂-sym _⊔₂_ ⊔₂-assoc
            ; ⊔-comm = intersect-comm _≈₂_ ≈₂-refl ≈₂-sym _⊔₂_ ⊔₂-comm
            ; ⊔-idemp = intersect-idemp _≈₂_ ≈₂-refl ≈₂-sym _⊔₂_ ⊔₂-idemp
            }

module IsLatticeInstances where
    module ForNat where
        open Nat
        open NatProps
        open Eq
        open IsSemilatticeInstances.ForNat
        open Data.Product

        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)

            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)

            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 = min-bound₁ {x} {x ⊔ y} {x ⊓ (x ⊔ y)} refl
                    x⊓x≤x⊓x⊔y = ⊓-mono-≤ {x} {x} ≤-refl (max-bound₁ {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 = max-bound₁ {x} {x ⊓ y} {x ⊔ (x ⊓ y)} refl
                    x⊔x⊓y≤x⊔x = ⊔-mono-≤ {x} {x} ≤-refl (min-bound₁ {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

        NatIsLattice : IsLattice ℕ _≡_ _⊔_ _⊓_
        NatIsLattice = record
            { joinSemilattice = NatIsMaxSemilattice
            ; meetSemilattice = NatIsMinSemilattice
            ; absorb-⊔-⊓ = λ x y → maxmin-absorb {x} {y}
            ; absorb-⊓-⊔ = λ x y → minmax-absorb {x} {y}
            }

    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 ForMap {a} {A B : Set a}
        (≡-dec-A : Decidable (_≡_ {a} {A}))
        (_≈₂_ : B → B → Set a)
        (_⊔₂_ : B → B → B)
        (_⊓₂_ : B → B → B)
        (lB : IsLattice B _≈₂_ _⊔₂_ _⊓₂_) where

        open import Map A B ≡-dec-A
        open IsLattice lB renaming
            ( ≈-refl to ≈₂-refl; ≈-sym to ≈₂-sym
            ; ⊔-idemp to ⊔₂-idemp; ⊓-idemp to ⊓₂-idemp
            ; absorb-⊔-⊓ to absorb-⊔₂-⊓₂; absorb-⊓-⊔ to absorb-⊓₂-⊔₂
            )

        module MapJoin = IsSemilatticeInstances.ForMap ≡-dec-A _≈₂_ _⊔₂_ (IsLattice.joinSemilattice lB)
        open MapJoin using (_⊔_; _≈_) public

        module MapMeet = IsSemilatticeInstances.ForMap ≡-dec-A _≈₂_ _⊓₂_ (IsLattice.meetSemilattice lB)
        open MapMeet using (_⊓_) public

        MapIsLattice : IsLattice Map _≈_ _⊔_ _⊓_
        MapIsLattice = record
            { joinSemilattice = MapJoin.MapIsUnionSemilattice
            ; meetSemilattice = MapMeet.MapIsIntersectSemilattice
            ; absorb-⊔-⊓ = union-intersect-absorb _≈₂_ ≈₂-refl ≈₂-sym _⊔₂_ _⊓₂_ ⊔₂-idemp ⊓₂-idemp absorb-⊔₂-⊓₂ absorb-⊓₂-⊔₂
            ; absorb-⊓-⊔ = intersect-union-absorb _≈₂_ ≈₂-refl ≈₂-sym _⊔₂_ _⊓₂_ ⊔₂-idemp ⊓₂-idemp absorb-⊔₂-⊓₂ absorb-⊓₂-⊔₂
            }

module IsFiniteHeightLatticeInstances 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)
        (h₁ h₂ : ℕ)
        (lA : IsFiniteHeightLattice A h₁ _≈₁_ _⊔₁_ _⊓₁_) (lB : IsFiniteHeightLattice B h₂ _≈₂_ _⊔₂_ _⊓₂_) where

        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 (_≼_; _≺_)
        open IsFiniteHeightLattice lA using () renaming (⊔-idemp to ⊔₁-idemp; _≼_ to _≼₁_)
        open IsFiniteHeightLattice lB using () renaming (⊔-idemp to ⊔₂-idemp; _≼_ to _≼₂_)

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

        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))

            ∙,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))

            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)))

        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 (proj₂ (proj₁ (IsFiniteHeightLattice.fixedHeight lA))))
                      (ChainMapping₂.Chain-map (λ b → (amax , b)) (a,∙-Monotonic _) proj₂ (proj₂ (proj₁ (IsFiniteHeightLattice.fixedHeight lB))))
                  )
                , _
                )
            }