```------------------------------------------------------------------------
-- The Agda standard library
--
-- A bunch of properties
------------------------------------------------------------------------

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

module Data.Bool.Properties where

open import Algebra.Bundles
open import Algebra.Lattice.Bundles
import Algebra.Lattice.Properties.BooleanAlgebra as BooleanAlgebraProperties
open import Data.Bool.Base
open import Data.Empty
open import Data.Product
open import Data.Sum.Base
open import Function.Base
open import Function.Equality using (_⟨\$⟩_)
open import Function.Equivalence
using (_⇔_; equivalence; module Equivalence)
open import Induction.WellFounded using (WellFounded; Acc; acc)
open import Level using (Level; 0ℓ)
open import Relation.Binary hiding (_⇔_)
open import Relation.Binary.PropositionalEquality hiding ([_])
open import Relation.Nullary using (ofʸ; ofⁿ; does; proof; yes; no)
open import Relation.Nullary.Decidable using (True)
import Relation.Unary as U

open import Algebra.Definitions {A = Bool} _≡_
open import Algebra.Structures {A = Bool} _≡_
open import Algebra.Lattice.Structures {A = Bool} _≡_

open ≡-Reasoning

private
variable
a b : Level
A : Set a
B : Set b

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

infix 4 _≟_

_≟_ : Decidable {A = Bool} _≡_
true  ≟ true  = yes refl
false ≟ false = yes refl
true  ≟ false = no λ()
false ≟ true  = no λ()

≡-setoid : Setoid 0ℓ 0ℓ
≡-setoid = setoid Bool

≡-decSetoid : DecSetoid 0ℓ 0ℓ
≡-decSetoid = decSetoid _≟_

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

-- Relational properties

≤-reflexive : _≡_ ⇒ _≤_
≤-reflexive refl = b≤b

≤-refl : Reflexive _≤_
≤-refl = ≤-reflexive refl

≤-trans : Transitive _≤_
≤-trans b≤b p   = p
≤-trans f≤t b≤b = f≤t

≤-antisym : Antisymmetric _≡_ _≤_
≤-antisym b≤b _ = refl

≤-minimum : Minimum _≤_ false
≤-minimum false = b≤b
≤-minimum true  = f≤t

≤-maximum : Maximum _≤_ true
≤-maximum false = f≤t
≤-maximum true  = b≤b

≤-total : Total _≤_
≤-total false b = inj₁ (≤-minimum b)
≤-total true  b = inj₂ (≤-maximum b)

infix 4 _≤?_

_≤?_ : Decidable _≤_
false ≤? b     = yes (≤-minimum b)
true  ≤? false = no λ ()
true  ≤? true  = yes b≤b

≤-irrelevant : Irrelevant _≤_
≤-irrelevant {_}     f≤t f≤t = refl
≤-irrelevant {false} b≤b b≤b = refl
≤-irrelevant {true}  b≤b b≤b = refl

-- Structures

≤-isPreorder : IsPreorder _≡_ _≤_
≤-isPreorder = record
{ isEquivalence = isEquivalence
; reflexive     = ≤-reflexive
; trans         = ≤-trans
}

≤-isPartialOrder : IsPartialOrder _≡_ _≤_
≤-isPartialOrder = record
{ isPreorder = ≤-isPreorder
; antisym    = ≤-antisym
}

≤-isTotalOrder : IsTotalOrder _≡_ _≤_
≤-isTotalOrder = record
{ isPartialOrder = ≤-isPartialOrder
; total          = ≤-total
}

≤-isDecTotalOrder : IsDecTotalOrder _≡_ _≤_
≤-isDecTotalOrder = record
{ isTotalOrder = ≤-isTotalOrder
; _≟_          = _≟_
; _≤?_         = _≤?_
}

-- Bundles

≤-poset : Poset 0ℓ 0ℓ 0ℓ
≤-poset = record
{ isPartialOrder = ≤-isPartialOrder
}

≤-preorder : Preorder 0ℓ 0ℓ 0ℓ
≤-preorder = record
{ isPreorder = ≤-isPreorder
}

≤-totalOrder : TotalOrder 0ℓ 0ℓ 0ℓ
≤-totalOrder = record
{ isTotalOrder = ≤-isTotalOrder
}

≤-decTotalOrder : DecTotalOrder 0ℓ 0ℓ 0ℓ
≤-decTotalOrder = record
{ isDecTotalOrder = ≤-isDecTotalOrder
}

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

-- Relational properties

<-irrefl : Irreflexive _≡_ _<_
<-irrefl refl ()

<-asym : Asymmetric _<_
<-asym f<t ()

<-trans : Transitive _<_
<-trans f<t ()

<-transʳ : Trans _≤_ _<_ _<_
<-transʳ b≤b f<t = f<t

<-transˡ : Trans _<_ _≤_ _<_
<-transˡ f<t b≤b = f<t

<-cmp : Trichotomous _≡_ _<_
<-cmp false false = tri≈ (λ()) refl  (λ())
<-cmp false true  = tri< f<t   (λ()) (λ())
<-cmp true  false = tri> (λ()) (λ()) f<t
<-cmp true  true  = tri≈ (λ()) refl  (λ())

infix 4 _<?_

_<?_ : Decidable _<_
false <? false = no  (λ())
false <? true  = yes f<t
true  <? _     = no  (λ())

<-resp₂-≡ : _<_ Respects₂ _≡_
<-resp₂-≡ = subst (_ <_) , subst (_< _)

<-irrelevant : Irrelevant _<_
<-irrelevant f<t f<t = refl

<-wellFounded : WellFounded _<_
<-wellFounded _ = acc <-acc
where
<-acc : ∀ {x} y → y < x → Acc _<_ y
<-acc false f<t = acc (λ _ → λ())

-- Structures

<-isStrictPartialOrder : IsStrictPartialOrder _≡_ _<_
<-isStrictPartialOrder = record
{ isEquivalence = isEquivalence
; irrefl        = <-irrefl
; trans         = <-trans
; <-resp-≈      = <-resp₂-≡
}

<-isStrictTotalOrder : IsStrictTotalOrder _≡_ _<_
<-isStrictTotalOrder = record
{ isEquivalence = isEquivalence
; trans         = <-trans
; compare       = <-cmp
}

-- Bundles

<-strictPartialOrder : StrictPartialOrder 0ℓ 0ℓ 0ℓ
<-strictPartialOrder = record
{ isStrictPartialOrder = <-isStrictPartialOrder
}

<-strictTotalOrder : StrictTotalOrder 0ℓ 0ℓ 0ℓ
<-strictTotalOrder = record
{ isStrictTotalOrder = <-isStrictTotalOrder
}

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

∨-assoc : Associative _∨_
∨-assoc true  y z = refl
∨-assoc false y z = refl

∨-comm : Commutative _∨_
∨-comm true  true  = refl
∨-comm true  false = refl
∨-comm false true  = refl
∨-comm false false = refl

∨-identityˡ : LeftIdentity false _∨_
∨-identityˡ _ = refl

∨-identityʳ : RightIdentity false _∨_
∨-identityʳ false = refl
∨-identityʳ true  = refl

∨-identity : Identity false _∨_
∨-identity = ∨-identityˡ , ∨-identityʳ

∨-zeroˡ : LeftZero true _∨_
∨-zeroˡ _ = refl

∨-zeroʳ : RightZero true _∨_
∨-zeroʳ false = refl
∨-zeroʳ true  = refl

∨-zero : Zero true _∨_
∨-zero = ∨-zeroˡ , ∨-zeroʳ

∨-inverseˡ : LeftInverse true not _∨_
∨-inverseˡ false = refl
∨-inverseˡ true  = refl

∨-inverseʳ : RightInverse true not _∨_
∨-inverseʳ x = ∨-comm x (not x) ⟨ trans ⟩ ∨-inverseˡ x

∨-inverse : Inverse true not _∨_
∨-inverse = ∨-inverseˡ , ∨-inverseʳ

∨-idem : Idempotent _∨_
∨-idem false = refl
∨-idem true  = refl

∨-sel : Selective _∨_
∨-sel false y = inj₂ refl
∨-sel true y  = inj₁ refl

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

∨-magma : Magma 0ℓ 0ℓ
∨-magma = record
{ isMagma = ∨-isMagma
}

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

∨-semigroup : Semigroup 0ℓ 0ℓ
∨-semigroup = record
{ isSemigroup = ∨-isSemigroup
}

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

∨-band : Band 0ℓ 0ℓ
∨-band = record
{ isBand = ∨-isBand
}

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

∨-semilattice : Semilattice 0ℓ 0ℓ
∨-semilattice = record
{ isSemilattice = ∨-isSemilattice
}

∨-isMonoid : IsMonoid _∨_ false
∨-isMonoid = record
{ isSemigroup = ∨-isSemigroup
; identity = ∨-identity
}

∨-isCommutativeMonoid : IsCommutativeMonoid _∨_ false
∨-isCommutativeMonoid = record
{ isMonoid = ∨-isMonoid
; comm = ∨-comm
}

∨-commutativeMonoid : CommutativeMonoid 0ℓ 0ℓ
∨-commutativeMonoid = record
{ isCommutativeMonoid = ∨-isCommutativeMonoid
}

∨-isIdempotentCommutativeMonoid :
IsIdempotentCommutativeMonoid _∨_ false
∨-isIdempotentCommutativeMonoid = record
{ isCommutativeMonoid = ∨-isCommutativeMonoid
; idem                = ∨-idem
}

∨-idempotentCommutativeMonoid : IdempotentCommutativeMonoid 0ℓ 0ℓ
∨-idempotentCommutativeMonoid = record
{ isIdempotentCommutativeMonoid = ∨-isIdempotentCommutativeMonoid
}

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

∧-assoc : Associative _∧_
∧-assoc true  y z = refl
∧-assoc false y z = refl

∧-comm : Commutative _∧_
∧-comm true  true  = refl
∧-comm true  false = refl
∧-comm false true  = refl
∧-comm false false = refl

∧-identityˡ : LeftIdentity true _∧_
∧-identityˡ _ = refl

∧-identityʳ : RightIdentity true _∧_
∧-identityʳ false = refl
∧-identityʳ true  = refl

∧-identity : Identity true _∧_
∧-identity = ∧-identityˡ , ∧-identityʳ

∧-zeroˡ : LeftZero false _∧_
∧-zeroˡ _ = refl

∧-zeroʳ : RightZero false _∧_
∧-zeroʳ false = refl
∧-zeroʳ true  = refl

∧-zero : Zero false _∧_
∧-zero = ∧-zeroˡ , ∧-zeroʳ

∧-inverseˡ : LeftInverse false not _∧_
∧-inverseˡ false = refl
∧-inverseˡ true = refl

∧-inverseʳ : RightInverse false not _∧_
∧-inverseʳ x = ∧-comm x (not x) ⟨ trans ⟩ ∧-inverseˡ x

∧-inverse : Inverse false not _∧_
∧-inverse = ∧-inverseˡ , ∧-inverseʳ

∧-idem : Idempotent _∧_
∧-idem false = refl
∧-idem true  = refl

∧-sel : Selective _∧_
∧-sel false y = inj₁ refl
∧-sel true y  = inj₂ refl

∧-distribˡ-∨ : _∧_ DistributesOverˡ _∨_
∧-distribˡ-∨ true  y z = refl
∧-distribˡ-∨ false y z = refl

∧-distribʳ-∨ : _∧_ DistributesOverʳ _∨_
∧-distribʳ-∨ x y z = begin
(y ∨ z) ∧ x     ≡⟨ ∧-comm (y ∨ z) x ⟩
x ∧ (y ∨ z)     ≡⟨ ∧-distribˡ-∨ x y z ⟩
x ∧ y ∨ x ∧ z   ≡⟨ cong₂ _∨_ (∧-comm x y) (∧-comm x z) ⟩
y ∧ x ∨ z ∧ x   ∎

∧-distrib-∨ : _∧_ DistributesOver _∨_
∧-distrib-∨ = ∧-distribˡ-∨ , ∧-distribʳ-∨

∨-distribˡ-∧ : _∨_ DistributesOverˡ _∧_
∨-distribˡ-∧ true  y z = refl
∨-distribˡ-∧ false y z = refl

∨-distribʳ-∧ : _∨_ DistributesOverʳ _∧_
∨-distribʳ-∧ x y z = begin
(y ∧ z) ∨ x        ≡⟨ ∨-comm (y ∧ z) x ⟩
x ∨ (y ∧ z)        ≡⟨ ∨-distribˡ-∧ x y z ⟩
(x ∨ y) ∧ (x ∨ z)  ≡⟨ cong₂ _∧_ (∨-comm x y) (∨-comm x z) ⟩
(y ∨ x) ∧ (z ∨ x)  ∎

∨-distrib-∧ : _∨_ DistributesOver _∧_
∨-distrib-∧ = ∨-distribˡ-∧ , ∨-distribʳ-∧

∧-abs-∨ : _∧_ Absorbs _∨_
∧-abs-∨ true  y = refl
∧-abs-∨ false y = refl

∨-abs-∧ : _∨_ Absorbs _∧_
∨-abs-∧ true  y = refl
∨-abs-∧ false y = refl

∨-∧-absorptive : Absorptive _∨_ _∧_
∨-∧-absorptive = ∨-abs-∧ , ∧-abs-∨

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

∧-magma : Magma 0ℓ 0ℓ
∧-magma = record
{ isMagma = ∧-isMagma
}

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

∧-semigroup : Semigroup 0ℓ 0ℓ
∧-semigroup = record
{ isSemigroup = ∧-isSemigroup
}

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

∧-band : Band 0ℓ 0ℓ
∧-band = record
{ isBand = ∧-isBand
}

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

∧-semilattice : Semilattice 0ℓ 0ℓ
∧-semilattice = record
{ isSemilattice = ∧-isSemilattice
}

∧-isMonoid : IsMonoid _∧_ true
∧-isMonoid = record
{ isSemigroup = ∧-isSemigroup
; identity = ∧-identity
}

∧-isCommutativeMonoid : IsCommutativeMonoid _∧_ true
∧-isCommutativeMonoid = record
{ isMonoid = ∧-isMonoid
; comm = ∧-comm
}

∧-commutativeMonoid : CommutativeMonoid 0ℓ 0ℓ
∧-commutativeMonoid = record
{ isCommutativeMonoid = ∧-isCommutativeMonoid
}

∧-isIdempotentCommutativeMonoid :
IsIdempotentCommutativeMonoid _∧_ true
∧-isIdempotentCommutativeMonoid = record
{ isCommutativeMonoid = ∧-isCommutativeMonoid
; idem = ∧-idem
}

∧-idempotentCommutativeMonoid : IdempotentCommutativeMonoid 0ℓ 0ℓ
∧-idempotentCommutativeMonoid = record
{ isIdempotentCommutativeMonoid = ∧-isIdempotentCommutativeMonoid
}

∨-∧-isSemiring : IsSemiring _∨_ _∧_ false true
∨-∧-isSemiring = record
{ isSemiringWithoutAnnihilatingZero = record
{ +-isCommutativeMonoid = ∨-isCommutativeMonoid
; *-cong = cong₂ _∧_
; *-assoc = ∧-assoc
; *-identity = ∧-identity
; distrib = ∧-distrib-∨
}
; zero = ∧-zero
}

∨-∧-isCommutativeSemiring
: IsCommutativeSemiring _∨_ _∧_ false true
∨-∧-isCommutativeSemiring = record
{ isSemiring = ∨-∧-isSemiring
; *-comm = ∧-comm
}

∨-∧-commutativeSemiring : CommutativeSemiring 0ℓ 0ℓ
∨-∧-commutativeSemiring = record
{ _+_                   = _∨_
; _*_                   = _∧_
; 0#                    = false
; 1#                    = true
; isCommutativeSemiring = ∨-∧-isCommutativeSemiring
}

∧-∨-isSemiring : IsSemiring _∧_ _∨_ true false
∧-∨-isSemiring = record
{ isSemiringWithoutAnnihilatingZero = record
{ +-isCommutativeMonoid = ∧-isCommutativeMonoid
; *-cong = cong₂ _∨_
; *-assoc = ∨-assoc
; *-identity = ∨-identity
; distrib = ∨-distrib-∧
}
; zero = ∨-zero
}

∧-∨-isCommutativeSemiring
: IsCommutativeSemiring _∧_ _∨_ true false
∧-∨-isCommutativeSemiring = record
{ isSemiring = ∧-∨-isSemiring
; *-comm = ∨-comm
}

∧-∨-commutativeSemiring : CommutativeSemiring 0ℓ 0ℓ
∧-∨-commutativeSemiring = record
{ _+_                   = _∧_
; _*_                   = _∨_
; 0#                    = true
; 1#                    = false
; isCommutativeSemiring = ∧-∨-isCommutativeSemiring
}

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

∨-∧-lattice : Lattice 0ℓ 0ℓ
∨-∧-lattice = record
{ isLattice = ∨-∧-isLattice
}

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

∨-∧-distributiveLattice : DistributiveLattice 0ℓ 0ℓ
∨-∧-distributiveLattice = record
{ isDistributiveLattice = ∨-∧-isDistributiveLattice
}

∨-∧-isBooleanAlgebra : IsBooleanAlgebra _∨_ _∧_ not true false
∨-∧-isBooleanAlgebra = record
{ isDistributiveLattice = ∨-∧-isDistributiveLattice
; ∨-complement          = ∨-inverse
; ∧-complement          = ∧-inverse
; ¬-cong                = cong not
}

∨-∧-booleanAlgebra : BooleanAlgebra 0ℓ 0ℓ
∨-∧-booleanAlgebra = record
{ isBooleanAlgebra = ∨-∧-isBooleanAlgebra
}

------------------------------------------------------------------------
-- Properties of _xor_

xor-is-ok : ∀ x y → x xor y ≡ (x ∨ y) ∧ not (x ∧ y)
xor-is-ok true  y = refl
xor-is-ok false y = sym (∧-identityʳ _)

xor-∧-commutativeRing : CommutativeRing 0ℓ 0ℓ
xor-∧-commutativeRing = ⊕-∧-commutativeRing
where
open BooleanAlgebraProperties ∨-∧-booleanAlgebra
open XorRing _xor_ xor-is-ok

------------------------------------------------------------------------
-- Miscellaneous other properties

not-involutive : Involutive not
not-involutive true  = refl
not-involutive false = refl

not-injective : ∀ {x y} → not x ≡ not y → x ≡ y
not-injective {false} {false} nx≢ny = refl
not-injective {true}  {true}  nx≢ny = refl

not-¬ : ∀ {x y} → x ≡ y → x ≢ not y
not-¬ {true}  refl ()
not-¬ {false} refl ()

¬-not : ∀ {x y} → x ≢ y → x ≡ not y
¬-not {true}  {true}  x≢y = ⊥-elim (x≢y refl)
¬-not {true}  {false} _   = refl
¬-not {false} {true}  _   = refl
¬-not {false} {false} x≢y = ⊥-elim (x≢y refl)

⇔→≡ : {x y z : Bool} → x ≡ z ⇔ y ≡ z → x ≡ y
⇔→≡ {true } {true }         hyp = refl
⇔→≡ {true } {false} {true } hyp = sym (Equivalence.to hyp ⟨\$⟩ refl)
⇔→≡ {true } {false} {false} hyp = Equivalence.from hyp ⟨\$⟩ refl
⇔→≡ {false} {true } {true } hyp = Equivalence.from hyp ⟨\$⟩ refl
⇔→≡ {false} {true } {false} hyp = sym (Equivalence.to hyp ⟨\$⟩ refl)
⇔→≡ {false} {false}         hyp = refl

T-≡ : ∀ {x} → T x ⇔ x ≡ true
T-≡ {false} = equivalence (λ ())       (λ ())
T-≡ {true}  = equivalence (const refl) (const _)

T-not-≡ : ∀ {x} → T (not x) ⇔ x ≡ false
T-not-≡ {false} = equivalence (const refl) (const _)
T-not-≡ {true}  = equivalence (λ ())       (λ ())

T-∧ : ∀ {x y} → T (x ∧ y) ⇔ (T x × T y)
T-∧ {true}  {true}  = equivalence (const (_ , _)) (const _)
T-∧ {true}  {false} = equivalence (λ ())          proj₂
T-∧ {false} {_}     = equivalence (λ ())          proj₁

T-∨ : ∀ {x y} → T (x ∨ y) ⇔ (T x ⊎ T y)
T-∨ {true}  {_}     = equivalence inj₁ (const _)
T-∨ {false} {true}  = equivalence inj₂ (const _)
T-∨ {false} {false} = equivalence inj₁ [ id , id ]

T-irrelevant : U.Irrelevant T
T-irrelevant {true}  _  _  = refl

T? : U.Decidable T
does  (T? b) = b
proof (T? true ) = ofʸ _
proof (T? false) = ofⁿ λ()

T?-diag : ∀ b → T b → True (T? b)
T?-diag true  _ = _

push-function-into-if : ∀ (f : A → B) x {y z} →
f (if x then y else z) ≡ (if x then f y else f z)
push-function-into-if _ true  = refl
push-function-into-if _ false = refl
```