------------------------------------------------------------------------
-- The Agda standard library
--
-- Some properties about subsets
------------------------------------------------------------------------

{-# OPTIONS --cubical-compatible --safe #-}

module Data.Fin.Subset.Properties where

import Algebra.Definitions as AlgebraicDefinitions
import Algebra.Structures as AlgebraicStructures
import Algebra.Lattice.Structures as AlgebraicLatticeStructures
open import Algebra.Bundles using (Magma; Semigroup; Monoid; Band;
  CommutativeMonoid; IdempotentCommutativeMonoid)
open import Algebra.Lattice.Bundles using (Semilattice; Lattice;
  DistributiveLattice; BooleanAlgebra)
import Algebra.Lattice.Properties.Lattice as L
import Algebra.Lattice.Properties.DistributiveLattice as DL
import Algebra.Lattice.Properties.BooleanAlgebra as BA
open import Data.Bool.Base using (not)
open import Data.Bool.Properties
open import Data.Fin.Base using (Fin; suc; zero)
open import Data.Fin.Subset
open import Data.Fin.Properties using (any?; decFinSubset)
open import Data.Nat.Base hiding (∣_-_∣)
import Data.Nat.Properties as ℕ
open import Data.Product as Product using (∃; ∄; _×_; _,_; proj₁)
open import Data.Sum.Base as Sum using (_⊎_; inj₁; inj₂; [_,_]′)
open import Data.Vec.Base using (Vec; []; _∷_; here; there)
open import Data.Vec.Properties
open import Function.Base using (_∘_; const; id; case_of_)
open import Function.Bundles using (_⇔_; mk⇔)
open import Level using (Level)
open import Relation.Binary.Core using (_⇒_)
open import Relation.Binary.Structures
  using (IsPreorder; IsPartialOrder; IsStrictPartialOrder; IsDecStrictPartialOrder)
open import Relation.Binary.Bundles
  using (Preorder; Poset; StrictPartialOrder; DecStrictPartialOrder)
open import Relation.Binary.Definitions as B hiding (Decidable; Empty)
open import Relation.Binary.PropositionalEquality.Core
  using (_≡_; refl; cong; cong₂; subst; _≢_; sym)
open import Relation.Binary.PropositionalEquality.Properties
  using (module ≡-Reasoning; isEquivalence)
open import Relation.Nullary.Decidable as Dec using (Dec; yes; no; _⊎-dec_)
open import Relation.Nullary.Negation using (contradiction)
open import Relation.Unary using (Pred; Decidable; Satisfiable)

private
  variable
    ℓ : Level
    n : ℕ
    s t : Side
    x y : Fin n
    p q : Subset n

------------------------------------------------------------------------
-- Constructor mangling

drop-there : suc x ∈ s ∷ p → x ∈ p
drop-there (there x∈p) = x∈p

drop-not-there : suc x ∉ s ∷ p → x ∉ p
drop-not-there x∉sp x∈p = contradiction (there x∈p) x∉sp

drop-∷-⊆ : s ∷ p ⊆ t ∷ q → p ⊆ q
drop-∷-⊆ sp⊆tq x∈p = drop-there (sp⊆tq (there x∈p))

drop-∷-⊂ : s ∷ p ⊂ s ∷ q → p ⊂ q
drop-∷-⊂ {s = inside} (_     , zero  , _         , x∉sp) = contradiction here x∉sp
drop-∷-⊂ {s}          (sp⊆sq , suc x , there x∈q , x∉sp) = drop-∷-⊆ sp⊆sq , x , x∈q , drop-not-there x∉sp

out⊆ : p ⊆ q → outside ∷ p ⊆ s ∷ q
out⊆ p⊆q (there ∈p) = there (p⊆q ∈p)

out⊆-⇔ : p ⊆ q ⇔ outside ∷ p ⊆ s ∷ q
out⊆-⇔ = mk⇔ out⊆ drop-∷-⊆

in⊆in : p ⊆ q → inside ∷ p ⊆ inside ∷ q
in⊆in p⊆q here = here
in⊆in p⊆q (there ∈p) = there (p⊆q ∈p)

in⊆in-⇔ : p ⊆ q ⇔ inside ∷ p ⊆ inside ∷ q
in⊆in-⇔ = mk⇔ in⊆in drop-∷-⊆

s⊆s : p ⊆ q → s ∷ p ⊆ s ∷ q
s⊆s p⊆q here = here
s⊆s p⊆q (there ∈p) = there (p⊆q ∈p)

s⊂s : p ⊂ q → s ∷ p ⊂ s ∷ q
s⊂s (p⊆q , v , v∈p , v∉q) = s⊆s p⊆q , suc v , there v∈p , v∉q ∘ drop-there

out⊂ : p ⊂ q → outside ∷ p ⊂ s ∷ q
out⊂ (p⊆q , x , x∈q , x∉p) = out⊆ p⊆q , suc x , there x∈q , x∉p ∘ drop-there

in⊂in : p ⊂ q → inside ∷ p ⊂ inside ∷ q
in⊂in = s⊂s

out⊂in : p ⊆ q → outside ∷ p ⊂ inside ∷ q
out⊂in p⊆q = out⊆ p⊆q , zero , here , λ ()

out⊂in-⇔ : p ⊆ q ⇔ outside ∷ p ⊂ inside ∷ q
out⊂in-⇔ = mk⇔ out⊂in (drop-∷-⊆ ∘ proj₁)

out⊂out-⇔ : p ⊂ q ⇔ outside ∷ p ⊂ outside ∷ q
out⊂out-⇔ = mk⇔ out⊂ drop-∷-⊂

in⊂in-⇔ : p ⊂ q ⇔ inside ∷ p ⊂ inside ∷ q
in⊂in-⇔ = mk⇔ in⊂in drop-∷-⊂

------------------------------------------------------------------------
-- _∈_

infix 4 _∈?_
_∈?_ : ∀ x (p : Subset n) → Dec (x ∈ p)
zero  ∈? inside  ∷ p = yes here
zero  ∈? outside ∷ p = no  λ()
suc n ∈? s       ∷ p = Dec.map′ there drop-there (n ∈? p)

------------------------------------------------------------------------
-- Empty

drop-∷-Empty : Empty (s ∷ p) → Empty p
drop-∷-Empty ¬∃∈ (x , x∈p) = ¬∃∈ (suc x , there x∈p)

Empty-unique : Empty p → p ≡ ⊥
Empty-unique {p = []}          ¬∃∈ = refl
Empty-unique {p = inside  ∷ p} ¬∃∈ = contradiction (zero , here) ¬∃∈
Empty-unique {p = outside ∷ p} ¬∃∈ =
  cong (outside ∷_) (Empty-unique (drop-∷-Empty ¬∃∈))

nonempty? : Decidable {A = Subset n} Nonempty
nonempty? p = any? (_∈? p)

------------------------------------------------------------------------
-- ∣_∣

∣p∣≤n : ∀ (p : Subset n) → ∣ p ∣ ≤ n
∣p∣≤n = count≤n (_≟ inside)

∣p∣≤∣x∷p∣ : ∀ x (p : Subset n)  → ∣ p ∣ ≤ ∣ x ∷ p ∣
∣p∣≤∣x∷p∣ outside p = ℕ.≤-refl
∣p∣≤∣x∷p∣ inside  p = ℕ.n≤1+n ∣ p ∣

------------------------------------------------------------------------
-- ⊥

∉⊥ : x ∉ ⊥
∉⊥ (there p) = ∉⊥ p

⊥⊆ : ⊥ ⊆ p
⊥⊆ x∈⊥ = contradiction x∈⊥ ∉⊥

∣⊥∣≡0 : ∀ n → ∣ ⊥ {n = n} ∣ ≡ 0
∣⊥∣≡0 zero    = refl
∣⊥∣≡0 (suc n) = ∣⊥∣≡0 n

------------------------------------------------------------------------
-- ⊤

∈⊤ : x ∈ ⊤
∈⊤ {x = zero}  = here
∈⊤ {x = suc x} = there ∈⊤

⊆⊤ : p ⊆ ⊤
⊆⊤ = const ∈⊤

∣⊤∣≡n : ∀ n → ∣ ⊤ {n} ∣ ≡ n
∣⊤∣≡n zero    = refl
∣⊤∣≡n (suc n) = cong suc (∣⊤∣≡n n)

∣p∣≡n⇒p≡⊤ : ∣ p ∣ ≡ n → p ≡ ⊤ {n}
∣p∣≡n⇒p≡⊤ {p = []}          _     = refl
∣p∣≡n⇒p≡⊤ {p = outside ∷ p} |p|≡n = contradiction |p|≡n (ℕ.<⇒≢ (s≤s (∣p∣≤n p)))
∣p∣≡n⇒p≡⊤ {p = inside  ∷ p} |p|≡n = cong (inside ∷_) (∣p∣≡n⇒p≡⊤ (ℕ.suc-injective |p|≡n))

------------------------------------------------------------------------
-- ⁅_⁆

x∈⁅x⁆ : ∀ (x : Fin n) → x ∈ ⁅ x ⁆
x∈⁅x⁆ zero    = here
x∈⁅x⁆ (suc x) = there (x∈⁅x⁆ x)

x∈⁅y⁆⇒x≡y : ∀ {x} (y : Fin n) → x ∈ ⁅ y ⁆ → x ≡ y
x∈⁅y⁆⇒x≡y zero    here      = refl
x∈⁅y⁆⇒x≡y zero    (there p) = contradiction p ∉⊥
x∈⁅y⁆⇒x≡y (suc y) (there p) = cong suc (x∈⁅y⁆⇒x≡y y p)

x∈⁅y⁆⇔x≡y : x ∈ ⁅ y ⁆ ⇔ x ≡ y
x∈⁅y⁆⇔x≡y {x = x} {y = y} = mk⇔
  (x∈⁅y⁆⇒x≡y y)
  (λ x≡y → subst (λ y → x ∈ ⁅ y ⁆) x≡y (x∈⁅x⁆ x))

x≢y⇒x∉⁅y⁆ : x ≢ y → x ∉ ⁅ y ⁆
x≢y⇒x∉⁅y⁆ x≢y = x≢y ∘ x∈⁅y⁆⇒x≡y _

x∉⁅y⁆⇒x≢y : x ∉ ⁅ y ⁆ → x ≢ y
x∉⁅y⁆⇒x≢y x∉⁅x⁆ refl = x∉⁅x⁆ (x∈⁅x⁆ _)

∣⁅x⁆∣≡1 : ∀ (i : Fin n) → ∣ ⁅ i ⁆ ∣ ≡ 1
∣⁅x⁆∣≡1 {suc n} zero    = cong suc (∣⊥∣≡0 n)
∣⁅x⁆∣≡1 {_}     (suc i) = ∣⁅x⁆∣≡1 i

------------------------------------------------------------------------
-- _⊆_

⊆-refl : Reflexive {A = Subset n} _⊆_
⊆-refl = id

⊆-reflexive : _≡_ {A = Subset n} ⇒ _⊆_
⊆-reflexive refl = ⊆-refl

⊆-trans : Transitive {A = Subset n} _⊆_
⊆-trans p⊆q q⊆r x∈p = q⊆r (p⊆q x∈p)

⊆-antisym : Antisymmetric {A = Subset n} _≡_ _⊆_
⊆-antisym {i = []}     {[]}     p⊆q q⊆p = refl
⊆-antisym {i = x ∷ xs} {y ∷ ys} p⊆q q⊆p with x | y
... | inside  | inside  = cong₂ _∷_ refl (⊆-antisym (drop-∷-⊆ p⊆q) (drop-∷-⊆ q⊆p))
... | inside  | outside = contradiction (p⊆q here) λ()
... | outside | inside  = contradiction (q⊆p here) λ()
... | outside | outside = cong₂ _∷_ refl (⊆-antisym (drop-∷-⊆ p⊆q) (drop-∷-⊆ q⊆p))

⊆-min : Minimum {A = Subset n} _⊆_ ⊥
⊆-min p = ⊥⊆

⊆-max : Maximum {A = Subset n} _⊆_ ⊤
⊆-max p = ⊆⊤

infix 4 _⊆?_
_⊆?_ : B.Decidable {A = Subset n} _⊆_
[]          ⊆? []          = yes id
outside ∷ p ⊆?       y ∷ q = Dec.map out⊆-⇔ (p ⊆? q)
inside  ∷ p ⊆? outside ∷ q = no (λ p⊆q → case (p⊆q here) of λ())
inside  ∷ p ⊆? inside  ∷ q = Dec.map in⊆in-⇔ (p ⊆? q)

module _ (n : ℕ) where

  ⊆-isPreorder : IsPreorder {A = Subset n} _≡_ _⊆_
  ⊆-isPreorder = record
    { isEquivalence = isEquivalence
    ; reflexive     = ⊆-reflexive
    ; trans         = ⊆-trans
    }

  ⊆-isPartialOrder : IsPartialOrder {A = Subset n} _≡_ _⊆_
  ⊆-isPartialOrder = record
    { isPreorder = ⊆-isPreorder
    ; antisym    = ⊆-antisym
    }

  ⊆-preorder : Preorder _ _ _
  ⊆-preorder = record
    { isPreorder = ⊆-isPreorder
    }

  ⊆-poset : Poset _ _ _
  ⊆-poset = record
    { isPartialOrder = ⊆-isPartialOrder
    }

p⊆q⇒∣p∣≤∣q∣ : p ⊆ q → ∣ p ∣ ≤ ∣ q ∣
p⊆q⇒∣p∣≤∣q∣ {p = []}          {[]}          p⊆q = z≤n
p⊆q⇒∣p∣≤∣q∣ {p = outside ∷ p} {outside ∷ q} p⊆q = p⊆q⇒∣p∣≤∣q∣ (drop-∷-⊆ p⊆q)
p⊆q⇒∣p∣≤∣q∣ {p = outside ∷ p} {inside  ∷ q} p⊆q = ℕ.m≤n⇒m≤1+n (p⊆q⇒∣p∣≤∣q∣ (drop-∷-⊆ p⊆q))
p⊆q⇒∣p∣≤∣q∣ {p = inside  ∷ p} {outside ∷ q} p⊆q = contradiction (p⊆q here) λ()
p⊆q⇒∣p∣≤∣q∣ {p = inside  ∷ p} {inside  ∷ q} p⊆q = s≤s (p⊆q⇒∣p∣≤∣q∣ (drop-∷-⊆ p⊆q))


------------------------------------------------------------------------
-- _⊂_

p⊂q⇒p⊆q : p ⊂ q → p ⊆ q
p⊂q⇒p⊆q = proj₁

⊂-trans : Transitive {A = Subset n} _⊂_
⊂-trans (p⊆q , x , x∈q , x∉p) (q⊆r , _ , _ , _) = ⊆-trans p⊆q q⊆r , x , q⊆r x∈q , x∉p

⊂-⊆-trans : Trans {A = Subset n} _⊂_ _⊆_ _⊂_
⊂-⊆-trans (p⊆q , x , x∈q , x∉p) q⊆r = ⊆-trans p⊆q q⊆r , x , q⊆r x∈q , x∉p

⊆-⊂-trans : Trans {A = Subset n} _⊆_ _⊂_ _⊂_
⊆-⊂-trans p⊆q (q⊆r , x , x∈r , x∉q) = ⊆-trans p⊆q q⊆r , x , x∈r , x∉q ∘ p⊆q

⊂-irref : Irreflexive {A = Subset n} _≡_ _⊂_
⊂-irref refl (_ , x , x∈p , x∉q) = contradiction x∈p x∉q

⊂-antisym : Antisymmetric {A = Subset n} _≡_ _⊂_
⊂-antisym (p⊆q , _) (q⊆p , _) = ⊆-antisym p⊆q q⊆p

⊂-asymmetric : Asymmetric {A = Subset n} _⊂_
⊂-asymmetric (p⊆q , _) (_ , x , x∈p , x∉q) = contradiction (p⊆q x∈p) x∉q

infix 4 _⊂?_

_⊂?_ : B.Decidable {A = Subset n} _⊂_
[]          ⊂? []          = no λ ()
outside ∷ p ⊂? outside ∷ q = Dec.map out⊂out-⇔ (p ⊂? q)
outside ∷ p ⊂? inside  ∷ q = Dec.map out⊂in-⇔ (p ⊆? q)
inside  ∷ p ⊂? outside ∷ q = no (λ {(p⊆q , _) → case (p⊆q here) of λ ()})
inside  ∷ p ⊂? inside  ∷ q = Dec.map in⊂in-⇔ (p ⊂? q)

module _ (n : ℕ) where

  ⊂-isStrictPartialOrder : IsStrictPartialOrder {A = Subset n} _≡_ _⊂_
  ⊂-isStrictPartialOrder = record
    { isEquivalence = isEquivalence
    ; irrefl = ⊂-irref
    ; trans = ⊂-trans
    ; <-resp-≈ = (λ {refl → id}) , (λ {refl → id})
    }

  ⊂-isDecStrictPartialOrder : IsDecStrictPartialOrder {A = Subset n} _≡_ _⊂_
  ⊂-isDecStrictPartialOrder = record
    { isStrictPartialOrder = ⊂-isStrictPartialOrder
    ; _≟_ = ≡-dec _≟_
    ; _<?_ = _⊂?_
    }

  ⊂-strictPartialOrder : StrictPartialOrder _ _ _
  ⊂-strictPartialOrder = record
    { isStrictPartialOrder = ⊂-isStrictPartialOrder
    }

  ⊂-decStrictPartialOrder : DecStrictPartialOrder _ _ _
  ⊂-decStrictPartialOrder = record
    { isDecStrictPartialOrder = ⊂-isDecStrictPartialOrder
    }

p⊂q⇒∣p∣<∣q∣ : p ⊂ q → ∣ p ∣ < ∣ q ∣
p⊂q⇒∣p∣<∣q∣ {p = outside ∷ p} {outside ∷ q} op⊂oq@(_     , _     , _ , _)    = p⊂q⇒∣p∣<∣q∣ (drop-∷-⊂ op⊂oq)
p⊂q⇒∣p∣<∣q∣ {p = outside ∷ p} {inside  ∷ q}       (op⊆iq , _     , _ , _)    = s≤s (p⊆q⇒∣p∣≤∣q∣ (drop-∷-⊆ op⊆iq))
p⊂q⇒∣p∣<∣q∣ {p = inside  ∷ p} {outside ∷ q}       (ip⊆oq , _     , _ , _)    = contradiction (ip⊆oq here) λ()
p⊂q⇒∣p∣<∣q∣ {p = inside  ∷ p} {inside  ∷ q}       (_     , zero  , _ , x∉ip) = contradiction here x∉ip
p⊂q⇒∣p∣<∣q∣ {p = inside  ∷ p} {inside  ∷ q} ip⊂iq@(_     , suc x , _ , _)    = s≤s (p⊂q⇒∣p∣<∣q∣ (drop-∷-⊂ ip⊂iq))

------------------------------------------------------------------------
-- ∁

x∈p⇒x∉∁p : x ∈ p → x ∉ ∁ p
x∈p⇒x∉∁p (there x∈p) (there x∈∁p) = x∈p⇒x∉∁p x∈p x∈∁p

x∈∁p⇒x∉p : x ∈ ∁ p → x ∉ p
x∈∁p⇒x∉p (there x∈∁p) (there x∈p) = x∈∁p⇒x∉p x∈∁p x∈p

x∉∁p⇒x∈p : x ∉ ∁ p → x ∈ p
x∉∁p⇒x∈p {x = zero}  {outside ∷ p} x∉∁p = contradiction here x∉∁p
x∉∁p⇒x∈p {x = zero}  {inside  ∷ p} x∉∁p = here
x∉∁p⇒x∈p {x = suc x} {_       ∷ p} x∉∁p = there (x∉∁p⇒x∈p (x∉∁p ∘ there))

x∉p⇒x∈∁p : x ∉ p → x ∈ ∁ p
x∉p⇒x∈∁p {x = zero}  {outside ∷ p} x∉p = here
x∉p⇒x∈∁p {x = zero}  {inside  ∷ p} x∉p = contradiction here x∉p
x∉p⇒x∈∁p {x = suc x} {_       ∷ p} x∉p = there (x∉p⇒x∈∁p (x∉p ∘ there))

p∪∁p≡⊤ : ∀ (p : Subset n) → p ∪ ∁ p ≡ ⊤
p∪∁p≡⊤ []            = refl
p∪∁p≡⊤ (outside ∷ p) = cong (inside ∷_) (p∪∁p≡⊤ p)
p∪∁p≡⊤ (inside  ∷ p) = cong (inside ∷_) (p∪∁p≡⊤ p)

∣∁p∣≡n∸∣p∣ : ∀ (p : Subset n) → ∣ ∁ p ∣ ≡ n ∸ ∣ p ∣
∣∁p∣≡n∸∣p∣ []            = refl
∣∁p∣≡n∸∣p∣ (inside  ∷ p) = ∣∁p∣≡n∸∣p∣ p
∣∁p∣≡n∸∣p∣ (outside ∷ p) = begin
  suc ∣ ∁ p ∣     ≡⟨ cong suc (∣∁p∣≡n∸∣p∣ p) ⟩
  suc (_ ∸ ∣ p ∣) ≡⟨ sym (ℕ.+-∸-assoc 1 (∣p∣≤n p)) ⟩
  suc  _ ∸ ∣ p ∣  ∎
  where open ≡-Reasoning

------------------------------------------------------------------------
-- _∩_

module _ {n : ℕ} where

  open AlgebraicDefinitions {A = Subset n} _≡_

  ∩-assoc : Associative _∩_
  ∩-assoc = zipWith-assoc ∧-assoc

  ∩-comm : Commutative _∩_
  ∩-comm = zipWith-comm ∧-comm

  ∩-idem : Idempotent _∩_
  ∩-idem = zipWith-idem ∧-idem

  ∩-identityˡ : LeftIdentity ⊤ _∩_
  ∩-identityˡ = zipWith-identityˡ ∧-identityˡ

  ∩-identityʳ : RightIdentity ⊤ _∩_
  ∩-identityʳ = zipWith-identityʳ ∧-identityʳ

  ∩-identity : Identity ⊤ _∩_
  ∩-identity = ∩-identityˡ , ∩-identityʳ

  ∩-zeroˡ : LeftZero ⊥ _∩_
  ∩-zeroˡ = zipWith-zeroˡ ∧-zeroˡ

  ∩-zeroʳ : RightZero ⊥ _∩_
  ∩-zeroʳ = zipWith-zeroʳ ∧-zeroʳ

  ∩-zero : Zero ⊥ _∩_
  ∩-zero = ∩-zeroˡ , ∩-zeroʳ

  ∩-inverseˡ : LeftInverse ⊥ ∁ _∩_
  ∩-inverseˡ = zipWith-inverseˡ ∧-inverseˡ

  ∩-inverseʳ : RightInverse ⊥ ∁ _∩_
  ∩-inverseʳ = zipWith-inverseʳ ∧-inverseʳ

  ∩-inverse : Inverse ⊥ ∁ _∩_
  ∩-inverse = ∩-inverseˡ , ∩-inverseʳ

module _ (n : ℕ) where

  open AlgebraicStructures        {A = Subset n} _≡_
  open AlgebraicLatticeStructures {A = Subset n} _≡_

  ∩-isMagma : IsMagma _∩_
  ∩-isMagma = record
    { isEquivalence = isEquivalence
    ; ∙-cong        = cong₂ _∩_
    }

  ∩-isSemigroup : IsSemigroup _∩_
  ∩-isSemigroup = record
    { isMagma = ∩-isMagma
    ; assoc   = ∩-assoc
    }

  ∩-isBand : IsBand _∩_
  ∩-isBand = record
    { isSemigroup = ∩-isSemigroup
    ; idem        = ∩-idem
    }

  ∩-isSemilattice : IsSemilattice _∩_
  ∩-isSemilattice = record
    { isBand = ∩-isBand
    ; comm   = ∩-comm
    }

  ∩-isMonoid : IsMonoid _∩_ ⊤
  ∩-isMonoid = record
    { isSemigroup = ∩-isSemigroup
    ; identity    = ∩-identity
    }

  ∩-isCommutativeMonoid : IsCommutativeMonoid _∩_ ⊤
  ∩-isCommutativeMonoid = record
    { isMonoid = ∩-isMonoid
    ; comm     = ∩-comm
    }

  ∩-isIdempotentCommutativeMonoid : IsIdempotentCommutativeMonoid _∩_ ⊤
  ∩-isIdempotentCommutativeMonoid = record
    { isCommutativeMonoid = ∩-isCommutativeMonoid
    ; idem                = ∩-idem
    }

  ∩-magma : Magma _ _
  ∩-magma = record
    { isMagma = ∩-isMagma
    }

  ∩-semigroup : Semigroup _ _
  ∩-semigroup = record
    { isSemigroup = ∩-isSemigroup
    }

  ∩-band : Band _ _
  ∩-band = record
    { isBand = ∩-isBand
    }

  ∩-semilattice : Semilattice _ _
  ∩-semilattice = record
    { isSemilattice = ∩-isSemilattice
    }

  ∩-monoid : Monoid _ _
  ∩-monoid = record
    { isMonoid = ∩-isMonoid
    }

  ∩-commutativeMonoid : CommutativeMonoid _ _
  ∩-commutativeMonoid = record
    { isCommutativeMonoid = ∩-isCommutativeMonoid
    }

  ∩-idempotentCommutativeMonoid : IdempotentCommutativeMonoid _ _
  ∩-idempotentCommutativeMonoid = record
    { isIdempotentCommutativeMonoid = ∩-isIdempotentCommutativeMonoid
    }

p∩q⊆p : ∀ (p q : Subset n) → p ∩ q ⊆ p
p∩q⊆p []            []           x∈p∩q        = x∈p∩q
p∩q⊆p (inside  ∷ p) (inside ∷ q) here         = here
p∩q⊆p (inside  ∷ p) (_      ∷ q) (there ∈p∩q) = there (p∩q⊆p p q ∈p∩q)
p∩q⊆p (outside ∷ p) (_      ∷ q) (there ∈p∩q) = there (p∩q⊆p p q ∈p∩q)

p∩q⊆q : ∀ (p q : Subset n) → p ∩ q ⊆ q
p∩q⊆q p q rewrite ∩-comm p q = p∩q⊆p q p

x∈p∩q⁺ : x ∈ p × x ∈ q → x ∈ p ∩ q
x∈p∩q⁺ (here      , here)      = here
x∈p∩q⁺ (there x∈p , there x∈q) = there (x∈p∩q⁺ (x∈p , x∈q))

x∈p∩q⁻ : ∀ (p q : Subset n) → x ∈ p ∩ q → x ∈ p × x ∈ q
x∈p∩q⁻ (inside ∷ p) (inside ∷ q) here          = here , here
x∈p∩q⁻ (s      ∷ p) (t      ∷ q) (there x∈p∩q) =
  Product.map there there (x∈p∩q⁻ p q x∈p∩q)

∩⇔× : x ∈ p ∩ q ⇔ (x ∈ p × x ∈ q)
∩⇔× = mk⇔ (x∈p∩q⁻ _ _) x∈p∩q⁺

∣p∩q∣≤∣p∣ : ∀ (p q : Subset n) → ∣ p ∩ q ∣ ≤ ∣ p ∣
∣p∩q∣≤∣p∣ p q = p⊆q⇒∣p∣≤∣q∣ (p∩q⊆p p q)

∣p∩q∣≤∣q∣ : ∀ (p q : Subset n) → ∣ p ∩ q ∣ ≤ ∣ q ∣
∣p∩q∣≤∣q∣ p q = p⊆q⇒∣p∣≤∣q∣ (p∩q⊆q p q)

∣p∩q∣≤∣p∣⊓∣q∣ : ∀ (p q : Subset n) → ∣ p ∩ q ∣ ≤ ∣ p ∣ ⊓ ∣ q ∣
∣p∩q∣≤∣p∣⊓∣q∣ p q = ℕ.⊓-glb (∣p∩q∣≤∣p∣ p q) (∣p∩q∣≤∣q∣ p q)

------------------------------------------------------------------------
-- _∪_

module _ {n : ℕ} where

  open AlgebraicDefinitions {A = Subset n} _≡_

  ∪-assoc : Associative _∪_
  ∪-assoc = zipWith-assoc ∨-assoc

  ∪-comm : Commutative _∪_
  ∪-comm = zipWith-comm ∨-comm

  ∪-idem : Idempotent _∪_
  ∪-idem = zipWith-idem ∨-idem

  ∪-identityˡ : LeftIdentity ⊥ _∪_
  ∪-identityˡ = zipWith-identityˡ ∨-identityˡ

  ∪-identityʳ : RightIdentity ⊥ _∪_
  ∪-identityʳ = zipWith-identityʳ ∨-identityʳ

  ∪-identity : Identity ⊥ _∪_
  ∪-identity = ∪-identityˡ , ∪-identityʳ

  ∪-zeroˡ : LeftZero ⊤ _∪_
  ∪-zeroˡ = zipWith-zeroˡ ∨-zeroˡ

  ∪-zeroʳ : RightZero ⊤ _∪_
  ∪-zeroʳ = zipWith-zeroʳ ∨-zeroʳ

  ∪-zero : Zero ⊤ _∪_
  ∪-zero = ∪-zeroˡ , ∪-zeroʳ

  ∪-inverseˡ : LeftInverse ⊤ ∁ _∪_
  ∪-inverseˡ = zipWith-inverseˡ ∨-inverseˡ

  ∪-inverseʳ : RightInverse ⊤ ∁ _∪_
  ∪-inverseʳ = zipWith-inverseʳ ∨-inverseʳ

  ∪-inverse : Inverse ⊤ ∁ _∪_
  ∪-inverse = ∪-inverseˡ , ∪-inverseʳ

  ∪-distribˡ-∩ : _∪_ DistributesOverˡ _∩_
  ∪-distribˡ-∩ = zipWith-distribˡ ∨-distribˡ-∧

  ∪-distribʳ-∩ : _∪_ DistributesOverʳ _∩_
  ∪-distribʳ-∩ = zipWith-distribʳ ∨-distribʳ-∧

  ∪-distrib-∩ : _∪_ DistributesOver _∩_
  ∪-distrib-∩ = ∪-distribˡ-∩ , ∪-distribʳ-∩

  ∩-distribˡ-∪ : _∩_ DistributesOverˡ _∪_
  ∩-distribˡ-∪ = zipWith-distribˡ ∧-distribˡ-∨

  ∩-distribʳ-∪ : _∩_ DistributesOverʳ _∪_
  ∩-distribʳ-∪ = zipWith-distribʳ ∧-distribʳ-∨

  ∩-distrib-∪ : _∩_ DistributesOver _∪_
  ∩-distrib-∪ = ∩-distribˡ-∪ , ∩-distribʳ-∪

  ∪-abs-∩ : _∪_ Absorbs _∩_
  ∪-abs-∩ = zipWith-absorbs ∨-abs-∧

  ∩-abs-∪ : _∩_ Absorbs _∪_
  ∩-abs-∪ = zipWith-absorbs ∧-abs-∨

module _ (n : ℕ) where

  open AlgebraicStructures        {A = Subset n} _≡_
  open AlgebraicLatticeStructures {A = Subset n} _≡_

  ∪-isMagma : IsMagma _∪_
  ∪-isMagma = record
    { isEquivalence = isEquivalence
    ; ∙-cong        = cong₂ _∪_
    }

  ∪-magma : Magma _ _
  ∪-magma = record
    { isMagma = ∪-isMagma
    }

  ∪-isSemigroup : IsSemigroup _∪_
  ∪-isSemigroup = record
    { isMagma = ∪-isMagma
    ; assoc   = ∪-assoc
    }

  ∪-semigroup : Semigroup _ _
  ∪-semigroup = record
    { isSemigroup = ∪-isSemigroup
    }

  ∪-isBand : IsBand _∪_
  ∪-isBand = record
    { isSemigroup = ∪-isSemigroup
    ; idem        = ∪-idem
    }

  ∪-band : Band _ _
  ∪-band = record
    { isBand = ∪-isBand
    }

  ∪-isSemilattice : IsSemilattice _∪_
  ∪-isSemilattice = record
    { isBand = ∪-isBand
    ; comm   = ∪-comm
    }

  ∪-semilattice : Semilattice _ _
  ∪-semilattice = record
    { isSemilattice = ∪-isSemilattice
    }

  ∪-isMonoid : IsMonoid _∪_ ⊥
  ∪-isMonoid = record
    { isSemigroup = ∪-isSemigroup
    ; identity    = ∪-identity
    }

  ∪-monoid : Monoid _ _
  ∪-monoid = record
    { isMonoid = ∪-isMonoid
    }

  ∪-isCommutativeMonoid : IsCommutativeMonoid _∪_ ⊥
  ∪-isCommutativeMonoid = record
    { isMonoid = ∪-isMonoid
    ; comm     = ∪-comm
    }

  ∪-commutativeMonoid : CommutativeMonoid _ _
  ∪-commutativeMonoid = record
    { isCommutativeMonoid = ∪-isCommutativeMonoid
    }

  ∪-isIdempotentCommutativeMonoid : IsIdempotentCommutativeMonoid _∪_ ⊥
  ∪-isIdempotentCommutativeMonoid = record
    { isCommutativeMonoid = ∪-isCommutativeMonoid
    ; idem                = ∪-idem
    }

  ∪-idempotentCommutativeMonoid : IdempotentCommutativeMonoid _ _
  ∪-idempotentCommutativeMonoid = record
    { isIdempotentCommutativeMonoid = ∪-isIdempotentCommutativeMonoid
    }

  ∪-∩-isLattice : IsLattice _∪_ _∩_
  ∪-∩-isLattice = record
    { isEquivalence = isEquivalence
    ; ∨-comm        = ∪-comm
    ; ∨-assoc       = ∪-assoc
    ; ∨-cong        = cong₂ _∪_
    ; ∧-comm        = ∩-comm
    ; ∧-assoc       = ∩-assoc
    ; ∧-cong        = cong₂ _∩_
    ; absorptive    = ∪-abs-∩ , ∩-abs-∪
    }

  ∪-∩-lattice : Lattice _ _
  ∪-∩-lattice = record
    { isLattice = ∪-∩-isLattice
    }

  ∪-∩-isDistributiveLattice : IsDistributiveLattice _∪_ _∩_
  ∪-∩-isDistributiveLattice = record
    { isLattice   = ∪-∩-isLattice
    ; ∨-distrib-∧ = ∪-distrib-∩
    ; ∧-distrib-∨ = ∩-distrib-∪
    }

  ∪-∩-distributiveLattice : DistributiveLattice _ _
  ∪-∩-distributiveLattice = record
    { isDistributiveLattice = ∪-∩-isDistributiveLattice
    }

  ∪-∩-isBooleanAlgebra : IsBooleanAlgebra _∪_ _∩_ ∁ ⊤ ⊥
  ∪-∩-isBooleanAlgebra = record
    { isDistributiveLattice = ∪-∩-isDistributiveLattice
    ; ∨-complement          = ∪-inverse
    ; ∧-complement          = ∩-inverse
    ; ¬-cong                = cong ∁
    }

  ∪-∩-booleanAlgebra : BooleanAlgebra _ _
  ∪-∩-booleanAlgebra = record
    { isBooleanAlgebra = ∪-∩-isBooleanAlgebra
    }

  ∩-∪-isLattice : IsLattice _∩_ _∪_
  ∩-∪-isLattice = L.∧-∨-isLattice ∪-∩-lattice

  ∩-∪-lattice : Lattice _ _
  ∩-∪-lattice = L.∧-∨-lattice ∪-∩-lattice

  ∩-∪-isDistributiveLattice : IsDistributiveLattice _∩_ _∪_
  ∩-∪-isDistributiveLattice = DL.∧-∨-isDistributiveLattice ∪-∩-distributiveLattice

  ∩-∪-distributiveLattice : DistributiveLattice _ _
  ∩-∪-distributiveLattice = DL.∧-∨-distributiveLattice ∪-∩-distributiveLattice

  ∩-∪-isBooleanAlgebra : IsBooleanAlgebra _∩_ _∪_ ∁ ⊥ ⊤
  ∩-∪-isBooleanAlgebra = BA.∧-∨-isBooleanAlgebra ∪-∩-booleanAlgebra

  ∩-∪-booleanAlgebra : BooleanAlgebra _ _
  ∩-∪-booleanAlgebra = BA.∧-∨-booleanAlgebra ∪-∩-booleanAlgebra

p⊆p∪q : ∀ (q : Subset n) → p ⊆ p ∪ q
p⊆p∪q (s ∷ q) here        = here
p⊆p∪q (s ∷ q) (there x∈p) = there (p⊆p∪q q x∈p)

q⊆p∪q : ∀ (p q : Subset n) → q ⊆ p ∪ q
q⊆p∪q p q rewrite ∪-comm p q = p⊆p∪q p

x∈p∪q⁻ :  ∀ (p q : Subset n) → x ∈ p ∪ q → x ∈ p ⊎ x ∈ q
x∈p∪q⁻ (inside  ∷ p) (t       ∷ q) here          = inj₁ here
x∈p∪q⁻ (outside ∷ p) (inside  ∷ q) here          = inj₂ here
x∈p∪q⁻ (s       ∷ p) (t       ∷ q) (there x∈p∪q) =
  Sum.map there there (x∈p∪q⁻ p q x∈p∪q)

x∈p∪q⁺ : x ∈ p ⊎ x ∈ q → x ∈ p ∪ q
x∈p∪q⁺ (inj₁ x∈p) = p⊆p∪q _   x∈p
x∈p∪q⁺ (inj₂ x∈q) = q⊆p∪q _ _ x∈q

∪⇔⊎ : x ∈ p ∪ q ⇔ (x ∈ p ⊎ x ∈ q)
∪⇔⊎ = mk⇔ (x∈p∪q⁻ _ _) x∈p∪q⁺

∣p∣≤∣p∪q∣ : ∀ (p q : Subset n) → ∣ p ∣ ≤ ∣ p ∪ q ∣
∣p∣≤∣p∪q∣ p q = p⊆q⇒∣p∣≤∣q∣ (p⊆p∪q {p = p} q)

∣q∣≤∣p∪q∣ : ∀ (p q : Subset n) → ∣ q ∣ ≤ ∣ p ∪ q ∣
∣q∣≤∣p∪q∣ p q = p⊆q⇒∣p∣≤∣q∣ (q⊆p∪q p q)

∣p∣⊔∣q∣≤∣p∪q∣ : ∀ (p q : Subset n) → ∣ p ∣ ⊔ ∣ q ∣ ≤ ∣ p ∪ q ∣
∣p∣⊔∣q∣≤∣p∪q∣ p q = ℕ.⊔-lub (∣p∣≤∣p∪q∣ p q) (∣q∣≤∣p∪q∣ p q)

------------------------------------------------------------------------
-- Properties of _─_

p─⊥≡p : ∀ (p : Subset n) → p ─ ⊥ ≡ p
p─⊥≡p []      = refl
p─⊥≡p (x ∷ p) = cong (x ∷_) (p─⊥≡p p)

p─⊤≡⊥ : ∀ (p : Subset n) → p ─ ⊤ ≡ ⊥
p─⊤≡⊥ []      = refl
p─⊤≡⊥ (x ∷ p) = cong (outside ∷_) (p─⊤≡⊥ p)

p─q─r≡p─q∪r : ∀ (p q r : Subset n) → p ─ q ─ r ≡ p ─ (q ∪ r)
p─q─r≡p─q∪r []      []            []            = refl
p─q─r≡p─q∪r (x ∷ p) (outside ∷ q) (outside ∷ r) = cong (x ∷_) (p─q─r≡p─q∪r p q r)
p─q─r≡p─q∪r (x ∷ p) (inside  ∷ q) (outside ∷ r) = cong (outside ∷_) (p─q─r≡p─q∪r p q r)
p─q─r≡p─q∪r (x ∷ p) (outside ∷ q) (inside  ∷ r) = cong (outside ∷_) (p─q─r≡p─q∪r p q r)
p─q─r≡p─q∪r (x ∷ p) (inside  ∷ q) (inside  ∷ r) = cong (outside ∷_) (p─q─r≡p─q∪r p q r)

p─q─q≡p─q : ∀ (p q : Subset n) → p ─ q ─ q ≡ p ─ q
p─q─q≡p─q p q = begin
  p ─ q ─ q  ≡⟨ p─q─r≡p─q∪r p q q ⟩
  p ─ q ∪ q  ≡⟨ cong (p ─_) (∪-idem q) ⟩
  p ─ q      ∎
  where open ≡-Reasoning

p─q─r≡p─r─q : ∀ (p q r : Subset n) → p ─ q ─ r ≡ p ─ r ─ q
p─q─r≡p─r─q p q r = begin
  (p ─ q) ─ r  ≡⟨  p─q─r≡p─q∪r p q r ⟩
  p ─ (q ∪ r)  ≡⟨  cong (p ─_) (∪-comm q r) ⟩
  p ─ (r ∪ q)  ≡⟨ p─q─r≡p─q∪r p r q ⟨
  (p ─ r) ─ q  ∎
  where open ≡-Reasoning

x∈p∧x∉q⇒x∈p─q : x ∈ p → x ∉ q → x ∈ p ─ q
x∈p∧x∉q⇒x∈p─q {q = outside ∷ q} here        i∉q = here
x∈p∧x∉q⇒x∈p─q {q = inside  ∷ q} here        i∉q = contradiction here i∉q
x∈p∧x∉q⇒x∈p─q {q = outside ∷ q} (there i∈p) i∉q = there (x∈p∧x∉q⇒x∈p─q i∈p (i∉q ∘ there))
x∈p∧x∉q⇒x∈p─q {q = inside  ∷ q} (there i∈p) i∉q = there (x∈p∧x∉q⇒x∈p─q i∈p (i∉q ∘ there))

p─q⊆p : ∀ (p q : Subset n) → p ─ q ⊆ p
p─q⊆p (inside  ∷ p) (outside ∷ q) here        = here
p─q⊆p (inside  ∷ p) (outside ∷ q) (there x∈p) = there (p─q⊆p p q x∈p)
p─q⊆p (outside ∷ p) (outside ∷ q) (there x∈p) = there (p─q⊆p p q x∈p)
p─q⊆p (_       ∷ p) (inside  ∷ q) (there x∈p) = there (p─q⊆p p q x∈p)

p∩q≢∅⇒p─q⊂p : ∀ (p q : Subset n) → Nonempty (p ∩ q) → p ─ q ⊂ p
p∩q≢∅⇒p─q⊂p (inside  ∷ p) (inside ∷ q)  (zero  , here)        = out⊂in (p─q⊆p p q)
p∩q≢∅⇒p─q⊂p (x       ∷ p) (inside ∷ q)  (suc i , there i∈p∩q) = out⊂ (p∩q≢∅⇒p─q⊂p p q (i , i∈p∩q))
p∩q≢∅⇒p─q⊂p (outside ∷ p) (outside ∷ q) (suc i , there i∈p∩q) = out⊂ (p∩q≢∅⇒p─q⊂p p q (i , i∈p∩q))
p∩q≢∅⇒p─q⊂p (inside  ∷ p) (outside ∷ q) (suc i , there i∈p∩q) = s⊂s  (p∩q≢∅⇒p─q⊂p p q (i , i∈p∩q))

∣p─q∣≤∣p∣ : ∀ (p q : Subset n) → ∣ p ─ q ∣ ≤ ∣ p ∣
∣p─q∣≤∣p∣ p q = p⊆q⇒∣p∣≤∣q∣ (p─q⊆p p q)

p∩q≢∅⇒∣p─q∣<∣p∣ : ∀ (p q : Subset n) → Nonempty (p ∩ q) → ∣ p ─ q ∣ < ∣ p ∣
p∩q≢∅⇒∣p─q∣<∣p∣ p q ne = p⊂q⇒∣p∣<∣q∣ (p∩q≢∅⇒p─q⊂p p q ne)

------------------------------------------------------------------------
-- Properties of _-_

x∈p∧x≢y⇒x∈p-y : x ∈ p → x ≢ y → x ∈ p - y
x∈p∧x≢y⇒x∈p-y x∈p x≢y = x∈p∧x∉q⇒x∈p─q x∈p (x≢y⇒x∉⁅y⁆ x≢y)

p─x─y≡p─y─x : ∀ (p : Subset n) x y → p - x - y ≡ p - y - x
p─x─y≡p─y─x p x y = p─q─r≡p─r─q p ⁅ x ⁆ ⁅ y ⁆

x∈p⇒p-x⊂p : x ∈ p → p - x ⊂ p
x∈p⇒p-x⊂p {n} {x} {p} x∈p = p∩q≢∅⇒p─q⊂p p ⁅ x ⁆ (x , x∈p∩q⁺ (x∈p , x∈⁅x⁆ x))

x∈p⇒∣p-x∣<∣p∣ : x ∈ p → ∣ p - x ∣ < ∣ p ∣
x∈p⇒∣p-x∣<∣p∣ x∈p = p⊂q⇒∣p∣<∣q∣ (x∈p⇒p-x⊂p x∈p)

------------------------------------------------------------------------
-- Lift

Lift? : ∀ {P : Pred (Fin n) ℓ} → Decidable P → Decidable (Lift P)
Lift? P? p = decFinSubset (_∈? p) (λ {x} _ → P? x)

------------------------------------------------------------------------
-- Other

module _ {P : Pred (Subset 0) ℓ} where

  ∃-Subset-zero : ∃⟨ P ⟩ → P []
  ∃-Subset-zero ([] , P[]) = P[]

  ∃-Subset-[]-⇔ : P [] ⇔ ∃⟨ P ⟩
  ∃-Subset-[]-⇔ = mk⇔ ([] ,_) ∃-Subset-zero

module _ {P : Pred (Subset (suc n)) ℓ} where

  ∃-Subset-suc : ∃⟨ P ⟩ → ∃⟨ P ∘ (inside ∷_) ⟩ ⊎ ∃⟨ P ∘ (outside ∷_) ⟩
  ∃-Subset-suc (outside ∷ p , Pop) = inj₂ (p , Pop)
  ∃-Subset-suc ( inside ∷ p , Pip) = inj₁ (p , Pip)

  ∃-Subset-∷-⇔ : (∃⟨ P ∘ (inside ∷_) ⟩ ⊎ ∃⟨ P ∘ (outside ∷_) ⟩) ⇔ ∃⟨ P ⟩
  ∃-Subset-∷-⇔ = mk⇔
    [ Product.map _ id , Product.map _ id ]′
    ∃-Subset-suc

anySubset? : ∀ {P : Pred (Subset n) ℓ} → Decidable P → Dec ∃⟨ P ⟩
anySubset? {n = zero}  P? = Dec.map ∃-Subset-[]-⇔ (P? [])
anySubset? {n = suc n} P? = Dec.map ∃-Subset-∷-⇔
  (anySubset? (P? ∘ (inside ∷_)) ⊎-dec anySubset? (P? ∘ (outside ∷_)))



------------------------------------------------------------------------
-- DEPRECATED NAMES
------------------------------------------------------------------------
-- Please use the new names as continuing support for the old names is
-- not guaranteed.

-- Version 1.3

p⊆q⇒∣p∣<∣q∣ = p⊆q⇒∣p∣≤∣q∣
{-# WARNING_ON_USAGE p⊆q⇒∣p∣<∣q∣
"Warning: p⊆q⇒∣p∣<∣q∣ was deprecated in v1.3.
Please use p⊆q⇒∣p∣≤∣q∣ instead."
#-}