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HasOrder follow the same structure as everything else
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| {-# OPTIONS --safe #-} | ||
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| module Class.HasOrder.Core where | ||
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| open import Class.Decidable | ||
| open import Data.Empty | ||
| open import Data.Product | ||
| open import Data.Sum | ||
| open import Function | ||
| open import Level | ||
| open import Relation.Binary | ||
| open import Relation.Binary using () renaming (Decidable to Decidable²) | ||
| open import Relation.Binary.PropositionalEquality | ||
| open import Relation.Nullary | ||
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| open Equivalence | ||
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| module _ {a} {A : Set a} where | ||
| module _ {_≈_ : Rel A a} where | ||
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| record HasPreorder : Set (suc a) where | ||
| infix 4 _≤_ _<_ _≥_ _>_ | ||
| field | ||
| _≤_ _<_ : Rel A a | ||
| ≤-isPreorder : IsPreorder _≈_ _≤_ | ||
| <-irrefl : Irreflexive _≈_ _<_ | ||
| ≤⇔<∨≈ : ∀ {x y} → x ≤ y ⇔ (x < y ⊎ x ≈ y) | ||
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| _≥_ = flip _≤_ | ||
| _>_ = flip _<_ | ||
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| open IsPreorder ≤-isPreorder public | ||
| using () | ||
| renaming (isEquivalence to ≈-isEquivalence; refl to ≤-refl; trans to ≤-trans) | ||
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| _≤?_ : ⦃ _≤_ ⁇² ⦄ → Decidable _≤_ | ||
| _≤?_ = dec² | ||
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| _<?_ : ⦃ _<_ ⁇² ⦄ → Decidable _<_ | ||
| _<?_ = dec² | ||
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| infix 4 _<?_ _≤?_ | ||
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| <⇒≤∧≉ : ∀ {x y} → x < y → x ≤ y × ¬ (x ≈ y) | ||
| <⇒≤∧≉ x<y = from ≤⇔<∨≈ (inj₁ x<y) , λ x≈y → <-irrefl x≈y x<y | ||
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| ≤∧≉⇒< : ∀ {x y} → x ≤ y × ¬ (x ≈ y) → x < y | ||
| ≤∧≉⇒< {x} {y} (x≤y , ¬x≈y) with to ≤⇔<∨≈ x≤y | ||
| ... | inj₁ x<y = x<y | ||
| ... | inj₂ x≈y = ⊥-elim (¬x≈y x≈y) | ||
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| ≤-antisym⇒<-asym : Antisymmetric _≈_ _≤_ → Asymmetric _<_ | ||
| ≤-antisym⇒<-asym antisym x<y y<x = | ||
| proj₂ (<⇒≤∧≉ x<y) $ antisym (proj₁ $ <⇒≤∧≉ x<y) (proj₁ $ <⇒≤∧≉ y<x) | ||
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| open HasPreorder ⦃...⦄ | ||
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| record HasDecPreorder : Set (suc a) where | ||
| field ⦃ hasPreorder ⦄ : HasPreorder | ||
| ⦃ dec-≤ ⦄ : _≤_ ⁇² | ||
| ⦃ dec-< ⦄ : _<_ ⁇² | ||
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| record HasPartialOrder : Set (suc a) where | ||
| field | ||
| ⦃ hasPreorder ⦄ : HasPreorder | ||
| ≤-antisym : Antisymmetric _≈_ _≤_ | ||
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| ≤-isPartialOrder : IsPartialOrder _≈_ _≤_ | ||
| ≤-isPartialOrder = record { isPreorder = ≤-isPreorder ; antisym = ≤-antisym } | ||
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| <-asymmetric : Asymmetric _<_ | ||
| <-asymmetric = ≤-antisym⇒<-asym ≤-antisym | ||
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| open IsEquivalence ≈-isEquivalence renaming (sym to ≈-sym) | ||
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| <-trans : Transitive _<_ | ||
| <-trans i<j j<k = | ||
| let | ||
| j≤k = proj₁ $ <⇒≤∧≉ j<k; i≤j = proj₁ $ <⇒≤∧≉ i<j | ||
| i≈k⇒j≈k = λ i≈k → ≤-antisym j≤k $ ≤-trans (from ≤⇔<∨≈ $ inj₂ (≈-sym i≈k)) i≤j | ||
| in | ||
| ≤∧≉⇒< (≤-trans i≤j j≤k , (proj₂ $ <⇒≤∧≉ j<k) ∘ i≈k⇒j≈k) | ||
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| <⇒¬>⊎≈ : ∀ {x y} → x < y → ¬ (y < x ⊎ y ≈ x) | ||
| <⇒¬>⊎≈ x<y (inj₁ y<x) = <-asymmetric x<y y<x | ||
| <⇒¬>⊎≈ x<y (inj₂ x≈y) = <-irrefl (≈-sym x≈y) x<y | ||
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| record HasDecPartialOrder : Set (suc a) where | ||
| field | ||
| ⦃ hasPartialOrder ⦄ : HasPartialOrder | ||
| ⦃ dec-≤ ⦄ : _≤_ ⁇² | ||
| ⦃ dec-< ⦄ : _<_ ⁇² | ||
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| HasPreorder≡ = HasPreorder {_≈_ = _≡_} | ||
| HasDecPreorder≡ = HasDecPreorder {_≈_ = _≡_} | ||
| HasPartialOrder≡ = HasPartialOrder {_≈_ = _≡_} | ||
| HasDecPartialOrder≡ = HasDecPartialOrder {_≈_ = _≡_} | ||
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| open HasPreorder ⦃...⦄ public | ||
| open HasPartialOrder ⦃...⦄ public hiding (hasPreorder) | ||
| open HasDecPartialOrder ⦃...⦄ public hiding (hasPartialOrder) | ||
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| module _ {a} {A : Set a} {_≈_ : Rel A a} where | ||
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| module _ {_≤_ : Rel A a} where | ||
| import Relation.Binary.Construct.NonStrictToStrict _≈_ _≤_ as SNS | ||
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| module _ (≤-isPreorder : IsPreorder _≈_ _≤_) | ||
| (_≈?_ : Decidable² _≈_) where | ||
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| hasPreorderFromNonStrict : HasPreorder | ||
| hasPreorderFromNonStrict = record | ||
| { _≤_ = _≤_ | ||
| ; _<_ = SNS._<_ | ||
| ; ≤-isPreorder = ≤-isPreorder | ||
| ; <-irrefl = SNS.<-irrefl | ||
| ; ≤⇔<∨≈ = λ {a b} → mk⇔ | ||
| (λ a≤b → case (a ≈? b) of λ where (yes p) → inj₂ p ; (no ¬p) → inj₁ (a≤b , ¬p)) | ||
| λ where (inj₁ a<b) → proj₁ a<b ; (inj₂ a≈b) → IsPreorder.reflexive ≤-isPreorder a≈b | ||
| } | ||
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| hasPartialOrderFromNonStrict : Antisymmetric _≈_ _≤_ → HasPartialOrder | ||
| hasPartialOrderFromNonStrict antsym = record | ||
| { hasPreorder = hasPreorderFromNonStrict | ||
| ; ≤-antisym = antsym | ||
| } | ||
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| module _ {_<_ : Rel A a} where | ||
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| import Relation.Binary.Construct.StrictToNonStrict _≈_ _<_ as SNS | ||
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| module _ (<-isStrictPartialOrder : IsStrictPartialOrder _≈_ _<_) where | ||
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| hasPreorderFromStrictPartialOrder : HasPreorder | ||
| hasPreorderFromStrictPartialOrder = record | ||
| { _≤_ = SNS._≤_ | ||
| ; _<_ = _<_ | ||
| ; ≤-isPreorder = SNS.isPreorder₂ <-isStrictPartialOrder | ||
| ; <-irrefl = <-isStrictPartialOrder .IsStrictPartialOrder.irrefl | ||
| ; ≤⇔<∨≈ = mk⇔ id id | ||
| } | ||
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| instance _ = hasPreorderFromStrictPartialOrder | ||
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| hasPartialOrderFromStrictPartialOrder : HasPartialOrder | ||
| hasPartialOrderFromStrictPartialOrder = record | ||
| { hasPreorder = hasPreorderFromStrictPartialOrder | ||
| ; ≤-antisym = SNS.isPartialOrder <-isStrictPartialOrder .IsPartialOrder.antisym | ||
| } | ||
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| module _ (<-isStrictTotalOrder : IsStrictTotalOrder _≈_ _<_) where | ||
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| private spo = IsStrictTotalOrder.isStrictPartialOrder <-isStrictTotalOrder | ||
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| hasPreorderFromStrictTotalOrder : HasPreorder | ||
| hasPreorderFromStrictTotalOrder = hasPreorderFromStrictPartialOrder spo | ||
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| hasPartialOrderFromStrictTotalOrder : HasPartialOrder | ||
| hasPartialOrderFromStrictTotalOrder = hasPartialOrderFromStrictPartialOrder spo |
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