------------------------------------------------------------------------
-- 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
using (IsMagma; IsSemigroup; IsMonoid; IsBand; IsCommutativeMonoid
; IsIdempotentCommutativeMonoid)
import Algebra.Lattice.Structures as AlgebraicLatticeStructures
using (IsSemilattice; IsLattice; IsDistributiveLattice; IsBooleanAlgebra)
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.∸-suc (∣p∣≤n p)) ⟩
suc _ ∸ ∣ p ∣ ∎
where open ≡-Reasoning
p⊆q⇒∁p⊇∁q : p ⊆ q → ∁ p ⊇ ∁ q
p⊆q⇒∁p⊇∁q p⊆q x∈∁q = x∉p⇒x∈∁p (x∈∁p⇒x∉p x∈∁q ∘ p⊆q)
∁p⊆∁q⇒p⊇q : ∁ p ⊆ ∁ q → p ⊇ q
∁p⊆∁q⇒p⊇q ∁p⊆∁q x∈q = x∉∁p⇒x∈p (x∈p⇒x∉∁p x∈q ∘ ∁p⊆∁q)
p⊂q⇒∁p⊃∁q : p ⊂ q → ∁ p ⊃ ∁ q
p⊂q⇒∁p⊃∁q (p⊆q , x , x∈q , x∉p) = p⊆q⇒∁p⊇∁q p⊆q , x , x∉p⇒x∈∁p x∉p , x∈p⇒x∉∁p x∈q
∁p⊂∁q⇒p⊃q : ∁ p ⊂ ∁ q → p ⊃ q
∁p⊂∁q⇒p⊃q (∁p⊆∁q , x , x∈∁q , x∉∁p) = ∁p⊆∁q⇒p⊇q ∁p⊆∁q , x , x∉∁p⇒x∈p x∉∁p , x∈∁p⇒x∉p x∈∁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
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."
#-}
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