module Relation.Binary.Sum where
open import Data.Function
open import Data.Sum as Sum
open import Data.Product
open import Data.Unit using (⊤)
open import Data.Empty
open import Level
open import Relation.Nullary
open import Relation.Binary
private
module Dummy {A₁ A₂ : Set} where
infixr 1 _⊎-Rel_ _⊎-<_
data ⊎ʳ (P : Set) (_∼₁_ : Rel A₁ zero) (_∼₂_ : Rel A₂ zero) :
A₁ ⊎ A₂ → A₁ ⊎ A₂ → Set where
₁∼₂ : ∀ {x y} (p : P) → ⊎ʳ P _∼₁_ _∼₂_ (inj₁ x) (inj₂ y)
₁∼₁ : ∀ {x y} (x∼₁y : x ∼₁ y) → ⊎ʳ P _∼₁_ _∼₂_ (inj₁ x) (inj₁ y)
₂∼₂ : ∀ {x y} (x∼₂y : x ∼₂ y) → ⊎ʳ P _∼₁_ _∼₂_ (inj₂ x) (inj₂ y)
_⊎-Rel_ : Rel A₁ zero → Rel A₂ zero → Rel (A₁ ⊎ A₂) zero
_⊎-Rel_ = ⊎ʳ ⊥
_⊎-<_ : Rel A₁ zero → Rel A₂ zero → Rel (A₁ ⊎ A₂) zero
_⊎-<_ = ⊎ʳ ⊤
private
₁≁₂ : {∼₁ : Rel A₁ zero} {∼₂ : Rel A₂ zero} →
∀ {x y} → ¬ (inj₁ x ⟨ ∼₁ ⊎-Rel ∼₂ ⟩ inj₂ y)
₁≁₂ (₁∼₂ ())
drop-inj₁ : {∼₁ : Rel A₁ zero} {∼₂ : Rel A₂ zero} →
∀ {P x y} → inj₁ x ⟨ ⊎ʳ P ∼₁ ∼₂ ⟩ inj₁ y → ∼₁ x y
drop-inj₁ (₁∼₁ x∼y) = x∼y
drop-inj₂ : {∼₁ : Rel A₁ zero} {∼₂ : Rel A₂ zero} →
∀ {P x y} → inj₂ x ⟨ ⊎ʳ P ∼₁ ∼₂ ⟩ inj₂ y → ∼₂ x y
drop-inj₂ (₂∼₂ x∼y) = x∼y
_⊎-reflexive_ : {≈₁ ∼₁ : Rel A₁ zero} → ≈₁ ⇒ ∼₁ →
{≈₂ ∼₂ : Rel A₂ zero} → ≈₂ ⇒ ∼₂ →
∀ {P} → (≈₁ ⊎-Rel ≈₂) ⇒ (⊎ʳ P ∼₁ ∼₂)
refl₁ ⊎-reflexive refl₂ = refl
where
refl : (_ ⊎-Rel _) ⇒ (⊎ʳ _ _ _)
refl (₁∼₁ x₁≈y₁) = ₁∼₁ (refl₁ x₁≈y₁)
refl (₂∼₂ x₂≈y₂) = ₂∼₂ (refl₂ x₂≈y₂)
refl (₁∼₂ ())
_⊎-refl_ : {∼₁ : Rel A₁ zero} → Reflexive ∼₁ →
{∼₂ : Rel A₂ zero} → Reflexive ∼₂ →
Reflexive (∼₁ ⊎-Rel ∼₂)
refl₁ ⊎-refl refl₂ = refl
where
refl : Reflexive (_ ⊎-Rel _)
refl {x = inj₁ _} = ₁∼₁ refl₁
refl {x = inj₂ _} = ₂∼₂ refl₂
_⊎-irreflexive_ : {≈₁ <₁ : Rel A₁ zero} → Irreflexive ≈₁ <₁ →
{≈₂ <₂ : Rel A₂ zero} → Irreflexive ≈₂ <₂ →
∀ {P} → Irreflexive (≈₁ ⊎-Rel ≈₂) (⊎ʳ P <₁ <₂)
irrefl₁ ⊎-irreflexive irrefl₂ = irrefl
where
irrefl : Irreflexive (_ ⊎-Rel _) (⊎ʳ _ _ _)
irrefl (₁∼₁ x₁≈y₁) (₁∼₁ x₁<y₁) = irrefl₁ x₁≈y₁ x₁<y₁
irrefl (₂∼₂ x₂≈y₂) (₂∼₂ x₂<y₂) = irrefl₂ x₂≈y₂ x₂<y₂
irrefl (₁∼₂ ()) _
_⊎-symmetric_ : {∼₁ : Rel A₁ zero} → Symmetric ∼₁ →
{∼₂ : Rel A₂ zero} → Symmetric ∼₂ →
Symmetric (∼₁ ⊎-Rel ∼₂)
sym₁ ⊎-symmetric sym₂ = sym
where
sym : Symmetric (_ ⊎-Rel _)
sym (₁∼₁ x₁∼y₁) = ₁∼₁ (sym₁ x₁∼y₁)
sym (₂∼₂ x₂∼y₂) = ₂∼₂ (sym₂ x₂∼y₂)
sym (₁∼₂ ())
_⊎-transitive_ : {∼₁ : Rel A₁ zero} → Transitive ∼₁ →
{∼₂ : Rel A₂ zero} → Transitive ∼₂ →
∀ {P} → Transitive (⊎ʳ P ∼₁ ∼₂)
trans₁ ⊎-transitive trans₂ = trans
where
trans : Transitive (⊎ʳ _ _ _)
trans (₁∼₁ x∼y) (₁∼₁ y∼z) = ₁∼₁ (trans₁ x∼y y∼z)
trans (₂∼₂ x∼y) (₂∼₂ y∼z) = ₂∼₂ (trans₂ x∼y y∼z)
trans (₁∼₂ p) (₂∼₂ _) = ₁∼₂ p
trans (₁∼₁ _) (₁∼₂ p) = ₁∼₂ p
_⊎-antisymmetric_ : {≈₁ ≤₁ : Rel A₁ zero} → Antisymmetric ≈₁ ≤₁ →
{≈₂ ≤₂ : Rel A₂ zero} → Antisymmetric ≈₂ ≤₂ →
∀ {P} → Antisymmetric (≈₁ ⊎-Rel ≈₂) (⊎ʳ P ≤₁ ≤₂)
antisym₁ ⊎-antisymmetric antisym₂ = antisym
where
antisym : Antisymmetric (_ ⊎-Rel _) (⊎ʳ _ _ _)
antisym (₁∼₁ x≤y) (₁∼₁ y≤x) = ₁∼₁ (antisym₁ x≤y y≤x)
antisym (₂∼₂ x≤y) (₂∼₂ y≤x) = ₂∼₂ (antisym₂ x≤y y≤x)
antisym (₁∼₂ _) ()
_⊎-asymmetric_ : {<₁ : Rel A₁ zero} → Asymmetric <₁ →
{<₂ : Rel A₂ zero} → Asymmetric <₂ →
∀ {P} → Asymmetric (⊎ʳ P <₁ <₂)
asym₁ ⊎-asymmetric asym₂ = asym
where
asym : Asymmetric (⊎ʳ _ _ _)
asym (₁∼₁ x<y) (₁∼₁ y<x) = asym₁ x<y y<x
asym (₂∼₂ x<y) (₂∼₂ y<x) = asym₂ x<y y<x
asym (₁∼₂ _) ()
_⊎-≈-respects₂_ : {≈₁ ∼₁ : Rel A₁ zero} → ∼₁ Respects₂ ≈₁ →
{≈₂ ∼₂ : Rel A₂ zero} → ∼₂ Respects₂ ≈₂ →
∀ {P} → (⊎ʳ P ∼₁ ∼₂) Respects₂ (≈₁ ⊎-Rel ≈₂)
_⊎-≈-respects₂_ {≈₁ = ≈₁} {∼₁ = ∼₁} resp₁
{≈₂ = ≈₂} {∼₂ = ∼₂} resp₂ {P} =
(λ {_ _ _} → resp¹) ,
(λ {_ _ _} → resp²)
where
resp¹ : ∀ {x} → ((⊎ʳ P ∼₁ ∼₂) x) Respects (≈₁ ⊎-Rel ≈₂)
resp¹ (₁∼₁ y≈y') (₁∼₁ x∼y) = ₁∼₁ (proj₁ resp₁ y≈y' x∼y)
resp¹ (₂∼₂ y≈y') (₂∼₂ x∼y) = ₂∼₂ (proj₁ resp₂ y≈y' x∼y)
resp¹ (₂∼₂ y≈y') (₁∼₂ p) = (₁∼₂ p)
resp¹ (₁∼₂ ()) _
resp² : ∀ {y}
→ (flip (⊎ʳ P ∼₁ ∼₂) y) Respects (≈₁ ⊎-Rel ≈₂)
resp² (₁∼₁ x≈x') (₁∼₁ x∼y) = ₁∼₁ (proj₂ resp₁ x≈x' x∼y)
resp² (₂∼₂ x≈x') (₂∼₂ x∼y) = ₂∼₂ (proj₂ resp₂ x≈x' x∼y)
resp² (₁∼₁ x≈x') (₁∼₂ p) = (₁∼₂ p)
resp² (₁∼₂ ()) _
_⊎-substitutive_ : {∼₁ : Rel A₁ zero} → Substitutive ∼₁ zero →
{∼₂ : Rel A₂ zero} → Substitutive ∼₂ zero →
Substitutive (∼₁ ⊎-Rel ∼₂) zero
subst₁ ⊎-substitutive subst₂ = subst
where
subst : Substitutive (_ ⊎-Rel _) zero
subst P (₁∼₁ x∼y) Px = subst₁ (λ z → P (inj₁ z)) x∼y Px
subst P (₂∼₂ x∼y) Px = subst₂ (λ z → P (inj₂ z)) x∼y Px
subst P (₁∼₂ ()) Px
⊎-decidable : {∼₁ : Rel A₁ zero} → Decidable ∼₁ →
{∼₂ : Rel A₂ zero} → Decidable ∼₂ →
∀ {P} → (∀ {x y} → Dec (inj₁ x ⟨ ⊎ʳ P ∼₁ ∼₂ ⟩ inj₂ y)) →
Decidable (⊎ʳ P ∼₁ ∼₂)
⊎-decidable {∼₁ = ∼₁} dec₁ {∼₂ = ∼₂} dec₂ {P} dec₁₂ = dec
where
dec : Decidable (⊎ʳ P ∼₁ ∼₂)
dec (inj₁ x) (inj₁ y) with dec₁ x y
... | yes x∼y = yes (₁∼₁ x∼y)
... | no x≁y = no (x≁y ∘ drop-inj₁)
dec (inj₂ x) (inj₂ y) with dec₂ x y
... | yes x∼y = yes (₂∼₂ x∼y)
... | no x≁y = no (x≁y ∘ drop-inj₂)
dec (inj₁ x) (inj₂ y) = dec₁₂
dec (inj₂ x) (inj₁ y) = no (λ())
_⊎-<-total_ : {≤₁ : Rel A₁ zero} → Total ≤₁ →
{≤₂ : Rel A₂ zero} → Total ≤₂ →
Total (≤₁ ⊎-< ≤₂)
total₁ ⊎-<-total total₂ = total
where
total : Total (_ ⊎-< _)
total (inj₁ x) (inj₁ y) = Sum.map ₁∼₁ ₁∼₁ $ total₁ x y
total (inj₂ x) (inj₂ y) = Sum.map ₂∼₂ ₂∼₂ $ total₂ x y
total (inj₁ x) (inj₂ y) = inj₁ (₁∼₂ _)
total (inj₂ x) (inj₁ y) = inj₂ (₁∼₂ _)
_⊎-<-trichotomous_ : {≈₁ <₁ : Rel A₁ zero} → Trichotomous ≈₁ <₁ →
{≈₂ <₂ : Rel A₂ zero} → Trichotomous ≈₂ <₂ →
Trichotomous (≈₁ ⊎-Rel ≈₂) (<₁ ⊎-< <₂)
_⊎-<-trichotomous_ {≈₁ = ≈₁} {<₁ = <₁} tri₁
{≈₂ = ≈₂} {<₂ = <₂} tri₂ = tri
where
tri : Trichotomous (≈₁ ⊎-Rel ≈₂) (<₁ ⊎-< <₂)
tri (inj₁ x) (inj₂ y) = tri< (₁∼₂ _) ₁≁₂ (λ())
tri (inj₂ x) (inj₁ y) = tri> (λ()) (λ()) (₁∼₂ _)
tri (inj₁ x) (inj₁ y) with tri₁ x y
... | tri< x<y x≉y x≯y =
tri< (₁∼₁ x<y) (x≉y ∘ drop-inj₁) (x≯y ∘ drop-inj₁)
... | tri≈ x≮y x≈y x≯y =
tri≈ (x≮y ∘ drop-inj₁) (₁∼₁ x≈y) (x≯y ∘ drop-inj₁)
... | tri> x≮y x≉y x>y =
tri> (x≮y ∘ drop-inj₁) (x≉y ∘ drop-inj₁) (₁∼₁ x>y)
tri (inj₂ x) (inj₂ y) with tri₂ x y
... | tri< x<y x≉y x≯y =
tri< (₂∼₂ x<y) (x≉y ∘ drop-inj₂) (x≯y ∘ drop-inj₂)
... | tri≈ x≮y x≈y x≯y =
tri≈ (x≮y ∘ drop-inj₂) (₂∼₂ x≈y) (x≯y ∘ drop-inj₂)
... | tri> x≮y x≉y x>y =
tri> (x≮y ∘ drop-inj₂) (x≉y ∘ drop-inj₂) (₂∼₂ x>y)
_⊎-isEquivalence_ : {≈₁ : Rel A₁ zero} → IsEquivalence ≈₁ →
{≈₂ : Rel A₂ zero} → IsEquivalence ≈₂ →
IsEquivalence (≈₁ ⊎-Rel ≈₂)
eq₁ ⊎-isEquivalence eq₂ = record
{ refl = refl eq₁ ⊎-refl refl eq₂
; sym = sym eq₁ ⊎-symmetric sym eq₂
; trans = trans eq₁ ⊎-transitive trans eq₂
}
where open IsEquivalence
_⊎-isPreorder_ : {≈₁ ∼₁ : Rel A₁ zero} → IsPreorder ≈₁ ∼₁ →
{≈₂ ∼₂ : Rel A₂ zero} → IsPreorder ≈₂ ∼₂ →
∀ {P} → IsPreorder (≈₁ ⊎-Rel ≈₂) (⊎ʳ P ∼₁ ∼₂)
pre₁ ⊎-isPreorder pre₂ = record
{ isEquivalence = isEquivalence pre₁ ⊎-isEquivalence
isEquivalence pre₂
; reflexive = reflexive pre₁ ⊎-reflexive reflexive pre₂
; trans = trans pre₁ ⊎-transitive trans pre₂
; ∼-resp-≈ = ∼-resp-≈ pre₁ ⊎-≈-respects₂ ∼-resp-≈ pre₂
}
where open IsPreorder
_⊎-isDecEquivalence_ : {≈₁ : Rel A₁ zero} → IsDecEquivalence ≈₁ →
{≈₂ : Rel A₂ zero} → IsDecEquivalence ≈₂ →
IsDecEquivalence (≈₁ ⊎-Rel ≈₂)
eq₁ ⊎-isDecEquivalence eq₂ = record
{ isEquivalence = isEquivalence eq₁ ⊎-isEquivalence
isEquivalence eq₂
; _≟_ = ⊎-decidable (_≟_ eq₁) (_≟_ eq₂) (no ₁≁₂)
}
where open IsDecEquivalence
_⊎-isPartialOrder_ : {≈₁ ≤₁ : Rel A₁ zero} → IsPartialOrder ≈₁ ≤₁ →
{≈₂ ≤₂ : Rel A₂ zero} → IsPartialOrder ≈₂ ≤₂ →
∀ {P} → IsPartialOrder (≈₁ ⊎-Rel ≈₂) (⊎ʳ P ≤₁ ≤₂)
po₁ ⊎-isPartialOrder po₂ = record
{ isPreorder = isPreorder po₁ ⊎-isPreorder isPreorder po₂
; antisym = antisym po₁ ⊎-antisymmetric antisym po₂
}
where open IsPartialOrder
_⊎-isStrictPartialOrder_ :
{≈₁ <₁ : Rel A₁ zero} → IsStrictPartialOrder ≈₁ <₁ →
{≈₂ <₂ : Rel A₂ zero} → IsStrictPartialOrder ≈₂ <₂ →
∀ {P} → IsStrictPartialOrder (≈₁ ⊎-Rel ≈₂) (⊎ʳ P <₁ <₂)
spo₁ ⊎-isStrictPartialOrder spo₂ = record
{ isEquivalence = isEquivalence spo₁ ⊎-isEquivalence
isEquivalence spo₂
; irrefl = irrefl spo₁ ⊎-irreflexive irrefl spo₂
; trans = trans spo₁ ⊎-transitive trans spo₂
; <-resp-≈ = <-resp-≈ spo₁ ⊎-≈-respects₂ <-resp-≈ spo₂
}
where open IsStrictPartialOrder
_⊎-<-isTotalOrder_ : {≈₁ ≤₁ : Rel A₁ zero} → IsTotalOrder ≈₁ ≤₁ →
{≈₂ ≤₂ : Rel A₂ zero} → IsTotalOrder ≈₂ ≤₂ →
IsTotalOrder (≈₁ ⊎-Rel ≈₂) (≤₁ ⊎-< ≤₂)
to₁ ⊎-<-isTotalOrder to₂ = record
{ isPartialOrder = isPartialOrder to₁ ⊎-isPartialOrder
isPartialOrder to₂
; total = total to₁ ⊎-<-total total to₂
}
where open IsTotalOrder
_⊎-<-isDecTotalOrder_ :
{≈₁ ≤₁ : Rel A₁ zero} → IsDecTotalOrder ≈₁ ≤₁ →
{≈₂ ≤₂ : Rel A₂ zero} → IsDecTotalOrder ≈₂ ≤₂ →
IsDecTotalOrder (≈₁ ⊎-Rel ≈₂) (≤₁ ⊎-< ≤₂)
to₁ ⊎-<-isDecTotalOrder to₂ = record
{ isTotalOrder = isTotalOrder to₁ ⊎-<-isTotalOrder isTotalOrder to₂
; _≟_ = ⊎-decidable (_≟_ to₁) (_≟_ to₂) (no ₁≁₂)
; _≤?_ = ⊎-decidable (_≤?_ to₁) (_≤?_ to₂) (yes (₁∼₂ _))
}
where open IsDecTotalOrder
open Dummy public
_⊎-setoid_ : Setoid _ _ → Setoid _ _ → Setoid _ _
s₁ ⊎-setoid s₂ = record
{ isEquivalence = isEquivalence s₁ ⊎-isEquivalence isEquivalence s₂
} where open Setoid
_⊎-preorder_ : Preorder _ _ _ → Preorder _ _ _ → Preorder _ _ _
p₁ ⊎-preorder p₂ = record
{ _∼_ = _∼_ p₁ ⊎-Rel _∼_ p₂
; isPreorder = isPreorder p₁ ⊎-isPreorder isPreorder p₂
} where open Preorder
_⊎-decSetoid_ : DecSetoid _ _ → DecSetoid _ _ → DecSetoid _ _
ds₁ ⊎-decSetoid ds₂ = record
{ isDecEquivalence = isDecEquivalence ds₁ ⊎-isDecEquivalence
isDecEquivalence ds₂
} where open DecSetoid
_⊎-poset_ : Poset _ _ _ → Poset _ _ _ → Poset _ _ _
po₁ ⊎-poset po₂ = record
{ _≤_ = _≤_ po₁ ⊎-Rel _≤_ po₂
; isPartialOrder = isPartialOrder po₁ ⊎-isPartialOrder
isPartialOrder po₂
} where open Poset
_⊎-<-poset_ : Poset _ _ _ → Poset _ _ _ → Poset _ _ _
po₁ ⊎-<-poset po₂ = record
{ _≤_ = _≤_ po₁ ⊎-< _≤_ po₂
; isPartialOrder = isPartialOrder po₁ ⊎-isPartialOrder
isPartialOrder po₂
} where open Poset
_⊎-<-strictPartialOrder_ :
StrictPartialOrder _ _ _ → StrictPartialOrder _ _ _ →
StrictPartialOrder _ _ _
spo₁ ⊎-<-strictPartialOrder spo₂ = record
{ _<_ = _<_ spo₁ ⊎-< _<_ spo₂
; isStrictPartialOrder = isStrictPartialOrder spo₁
⊎-isStrictPartialOrder
isStrictPartialOrder spo₂
} where open StrictPartialOrder
_⊎-<-totalOrder_ :
TotalOrder _ _ _ → TotalOrder _ _ _ → TotalOrder _ _ _
to₁ ⊎-<-totalOrder to₂ = record
{ isTotalOrder = isTotalOrder to₁ ⊎-<-isTotalOrder isTotalOrder to₂
} where open TotalOrder
_⊎-<-decTotalOrder_ :
DecTotalOrder _ _ _ → DecTotalOrder _ _ _ → DecTotalOrder _ _ _
to₁ ⊎-<-decTotalOrder to₂ = record
{ isDecTotalOrder = isDecTotalOrder to₁ ⊎-<-isDecTotalOrder
isDecTotalOrder to₂
} where open DecTotalOrder