------------------------------------------------------------------------
-- The Agda standard library
--
-- Properties of min and max operators specified over a total
-- preorder.
------------------------------------------------------------------------

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

open import Algebra.Core
open import Algebra.Bundles
open import Algebra.Construct.NaturalChoice.Base
open import Data.Sum.Base as Sum using (inj₁; inj₂; [_,_])
open import Data.Product using (_,_)
open import Function.Base using (id; _∘_; flip)
open import Relation.Binary
open import Relation.Binary.Consequences

module Algebra.Construct.NaturalChoice.MinMaxOp
{a ℓ₁ ℓ₂} {O : TotalPreorder a ℓ₁ ℓ₂}
(minOp : MinOperator O)
(maxOp : MaxOperator O)
where

open TotalPreorder O renaming
( Carrier   to A
; _≲_       to _≤_
; ≲-resp-≈  to ≤-resp-≈
; ≲-respʳ-≈ to ≤-respʳ-≈
; ≲-respˡ-≈ to ≤-respˡ-≈
)
open MinOperator minOp
open MaxOperator maxOp

open import Algebra.Definitions _≈_
open import Algebra.Structures _≈_
open import Algebra.Consequences.Setoid Eq.setoid
open import Relation.Binary.Reasoning.Preorder preorder

------------------------------------------------------------------------
-- Re-export properties of individual operators

open import Algebra.Construct.NaturalChoice.MinOp minOp public
open import Algebra.Construct.NaturalChoice.MaxOp maxOp public

------------------------------------------------------------------------
-- Joint algebraic structures

⊓-distribˡ-⊔ : _⊓_ DistributesOverˡ _⊔_
⊓-distribˡ-⊔ x y z with total y z
... | inj₁ y≤z = begin-equality
x  (y  z)       ≈⟨  ⊓-congˡ x (x≤y⇒x⊔y≈y y≤z)
x  z             ≈˘⟨ x≤y⇒x⊔y≈y (⊓-monoʳ-≤ x y≤z)
(x  y)  (x  z)
... | inj₂ y≥z = begin-equality
x  (y  z)       ≈⟨  ⊓-congˡ x (x≥y⇒x⊔y≈x y≥z)
x  y             ≈˘⟨ x≥y⇒x⊔y≈x (⊓-monoʳ-≤ x y≥z)
(x  y)  (x  z)

⊓-distribʳ-⊔ : _⊓_ DistributesOverʳ _⊔_
⊓-distribʳ-⊔ = comm+distrˡ⇒distrʳ ⊔-cong ⊓-comm ⊓-distribˡ-⊔

⊓-distrib-⊔ : _⊓_ DistributesOver _⊔_
⊓-distrib-⊔ = ⊓-distribˡ-⊔ , ⊓-distribʳ-⊔

⊔-distribˡ-⊓ : _⊔_ DistributesOverˡ _⊓_
⊔-distribˡ-⊓ x y z with total y z
... | inj₁ y≤z = begin-equality
x  (y  z)       ≈⟨  ⊔-congˡ x (x≤y⇒x⊓y≈x y≤z)
x  y             ≈˘⟨ x≤y⇒x⊓y≈x (⊔-monoʳ-≤ x y≤z)
(x  y)  (x  z)
... | inj₂ y≥z = begin-equality
x  (y  z)       ≈⟨  ⊔-congˡ x (x≥y⇒x⊓y≈y y≥z)
x  z             ≈˘⟨ x≥y⇒x⊓y≈y (⊔-monoʳ-≤ x y≥z)
(x  y)  (x  z)

⊔-distribʳ-⊓ : _⊔_ DistributesOverʳ _⊓_
⊔-distribʳ-⊓ = comm+distrˡ⇒distrʳ ⊓-cong ⊔-comm ⊔-distribˡ-⊓

⊔-distrib-⊓ : _⊔_ DistributesOver _⊓_
⊔-distrib-⊓ = ⊔-distribˡ-⊓ , ⊔-distribʳ-⊓

⊓-absorbs-⊔ : _⊓_ Absorbs _⊔_
⊓-absorbs-⊔ x y with total x y
... | inj₁ x≤y = begin-equality
x  (x  y)  ≈⟨ ⊓-congˡ x (x≤y⇒x⊔y≈y x≤y)
x  y        ≈⟨ x≤y⇒x⊓y≈x x≤y
x
... | inj₂ y≤x = begin-equality
x  (x  y)  ≈⟨ ⊓-congˡ x (x≥y⇒x⊔y≈x y≤x)
x  x        ≈⟨ ⊓-idem x
x

⊔-absorbs-⊓ : _⊔_ Absorbs _⊓_
⊔-absorbs-⊓ x y with total x y
... | inj₁ x≤y = begin-equality
x  (x  y)  ≈⟨ ⊔-congˡ x (x≤y⇒x⊓y≈x x≤y)
x  x        ≈⟨ ⊔-idem x
x
... | inj₂ y≤x = begin-equality
x  (x  y)  ≈⟨ ⊔-congˡ x (x≥y⇒x⊓y≈y y≤x)
x  y        ≈⟨ x≥y⇒x⊔y≈x y≤x
x

⊔-⊓-absorptive : Absorptive _⊔_ _⊓_
⊔-⊓-absorptive = ⊔-absorbs-⊓ , ⊓-absorbs-⊔

⊓-⊔-absorptive : Absorptive _⊓_ _⊔_
⊓-⊔-absorptive = ⊓-absorbs-⊔ , ⊔-absorbs-⊓

------------------------------------------------------------------------
-- Other joint properties

private _≥_ = flip _≤_

antimono-≤-distrib-⊓ :  {f}  f Preserves _≈_  _≈_  f Preserves _≤_  _≥_
x y  f (x  y)  f x  f y
antimono-≤-distrib-⊓ {f} cong antimono x y with total x y
... | inj₁ x≤y = begin-equality
f (x  y)  ≈⟨ cong (x≤y⇒x⊓y≈x x≤y)
f x        ≈˘⟨ x≥y⇒x⊔y≈x (antimono x≤y)
f x  f y
... | inj₂ y≤x = begin-equality
f (x  y)  ≈⟨ cong (x≥y⇒x⊓y≈y y≤x)
f y        ≈˘⟨ x≤y⇒x⊔y≈y (antimono y≤x)
f x  f y

antimono-≤-distrib-⊔ :  {f}  f Preserves _≈_  _≈_  f Preserves _≤_  _≥_
x y  f (x  y)  f x  f y
antimono-≤-distrib-⊔ {f} cong antimono x y with total x y
... | inj₁ x≤y = begin-equality
f (x  y)  ≈⟨ cong (x≤y⇒x⊔y≈y x≤y)
f y        ≈˘⟨ x≥y⇒x⊓y≈y (antimono x≤y)
f x  f y
... | inj₂ y≤x = begin-equality
f (x  y)  ≈⟨ cong (x≥y⇒x⊔y≈x y≤x)
f x        ≈˘⟨ x≤y⇒x⊓y≈x (antimono y≤x)
f x  f y

x⊓y≤x⊔y :  x y  x  y  x  y
x⊓y≤x⊔y x y = begin
x  y ∼⟨ x⊓y≤x x y
x     ∼⟨ x≤x⊔y x y
x  y