------------------------------------------------------------------------
-- The Agda standard library
--
-- Choosing between elements based on the result of applying a function
------------------------------------------------------------------------

{-# OPTIONS --without-K --safe #-}

open import Algebra

module Algebra.Construct.LiftedChoice where

open import Algebra.Consequences.Base
open import Relation.Binary
open import Relation.Nullary using (¬_; yes; no)
open import Data.Sum.Base as Sum using (_⊎_; inj₁; inj₂; [_,_])
open import Data.Product using (_×_; _,_)
open import Level using (Level; _⊔_)
open import Function.Base using (id; _on_)
open import Function.Injection using (Injection)
open import Function.Equality using (Π)
open import Relation.Binary using (Setoid; _Preserves_⟶_)
open import Relation.Binary.PropositionalEquality as P using (_≡_)
open import Relation.Unary using (Pred)
open import Relation.Nullary.Negation using (contradiction)

import Relation.Binary.Reasoning.Setoid as EqReasoning

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

------------------------------------------------------------------------
-- Definition

module _ (_≈_ : Rel B ) (_•_ : Op₂ B) where

  Lift : Selective _≈_ _•_  (A  B)  Op₂ A
  Lift ∙-sel f x y with ∙-sel (f x) (f y)
  ... | inj₁ _ = x
  ... | inj₂ _ = y

------------------------------------------------------------------------
-- Algebraic properties

module _ {_≈_ : Rel B } {_∙_ : Op₂ B}
         (∙-isSelectiveMagma : IsSelectiveMagma _≈_ _∙_) where

  private module M = IsSelectiveMagma ∙-isSelectiveMagma
  open M hiding (sel; isMagma)
  open EqReasoning setoid

  module _ (f : A  B) where

    private
      _◦_ = Lift _≈_ _∙_ M.sel f

    sel-≡ : Selective _≡_ _◦_
    sel-≡ x y with M.sel (f x) (f y)
    ... | inj₁ _ = inj₁ P.refl
    ... | inj₂ _ = inj₂ P.refl

    distrib :  x y  ((f x)  (f y))  f (x  y)
    distrib x y with M.sel (f x) (f y)
    ... | inj₁ fx∙fy≈fx = fx∙fy≈fx
    ... | inj₂ fx∙fy≈fy = fx∙fy≈fy

  module _ (f : A  B) {_≈′_ : Rel A }
           (≈-reflexive : _≡_  _≈′_) where

    private
      _◦_ = Lift _≈_ _∙_ M.sel f

    sel : Selective _≈′_ _◦_
    sel x y = Sum.map ≈-reflexive ≈-reflexive (sel-≡ f x y)

    idem : Idempotent _≈′_ _◦_
    idem = sel⇒idem _≈′_ sel

  module _ {f : A  B} {_≈′_ : Rel A }
           (f-injective :  {x y}  f x  f y  x ≈′ y)
           where

    private
      _◦_ = Lift _≈_ _∙_ M.sel f

    cong : f Preserves _≈′_  _≈_  Congruent₂ _≈′_ _◦_
    cong f-cong {x} {y} {u} {v} x≈y u≈v
      with M.sel (f x) (f u) | M.sel (f y) (f v)
    ... | inj₁ fx∙fu≈fx | inj₁ fy∙fv≈fy = x≈y
    ... | inj₂ fx∙fu≈fu | inj₂ fy∙fv≈fv = u≈v
    ... | inj₁ fx∙fu≈fx | inj₂ fy∙fv≈fv = f-injective (begin
      f x       ≈⟨ sym fx∙fu≈fx 
      f x  f u ≈⟨ ∙-cong (f-cong x≈y) (f-cong u≈v) 
      f y  f v ≈⟨ fy∙fv≈fv 
      f v       )
    ... | inj₂ fx∙fu≈fu | inj₁ fy∙fv≈fy = f-injective (begin
      f u       ≈⟨ sym fx∙fu≈fu 
      f x  f u ≈⟨ ∙-cong (f-cong x≈y) (f-cong u≈v) 
      f y  f v ≈⟨ fy∙fv≈fy 
      f y       )

    assoc : Associative _≈_ _∙_  Associative _≈′_ _◦_
    assoc ∙-assoc x y z = f-injective (begin
      f ((x  y)  z)   ≈˘⟨ distrib f (x  y) z 
      f (x  y)  f z   ≈˘⟨ ∙-congʳ (distrib f x y) 
      (f x  f y)  f z ≈⟨  ∙-assoc (f x) (f y) (f z) 
      f x  (f y  f z) ≈⟨  ∙-congˡ (distrib f y z) 
      f x  f (y  z)   ≈⟨  distrib f x (y  z) 
      f (x  (y  z))   )

    comm : Commutative _≈_ _∙_  Commutative _≈′_ _◦_
    comm ∙-comm x y = f-injective (begin
      f (x  y) ≈˘⟨ distrib f x y 
      f x  f y ≈⟨  ∙-comm (f x) (f y) 
      f y  f x ≈⟨  distrib f y x 
      f (y  x) )

------------------------------------------------------------------------
-- Algebraic structures

  module _ {_≈′_ : Rel A } {f : A  B}
           (f-injective :  {x y}  f x  f y  x ≈′ y)
           (f-cong : f Preserves _≈′_  _≈_)
           (≈′-isEquivalence : IsEquivalence _≈′_)
           where

    private
      module E = IsEquivalence ≈′-isEquivalence
      _◦_ = Lift _≈_ _∙_ M.sel f

    isMagma : IsMagma _≈′_ _◦_
    isMagma = record
      { isEquivalence = ≈′-isEquivalence
      ; ∙-cong        = cong  {x y}  f-injective {x} {y}) f-cong
      }

    isSemigroup : Associative _≈_ _∙_  IsSemigroup _≈′_ _◦_
    isSemigroup ∙-assoc = record
      { isMagma = isMagma
      ; assoc   = assoc  {x y}  f-injective {x} {y}) ∙-assoc
      }

    isBand : Associative _≈_ _∙_  IsBand _≈′_ _◦_
    isBand ∙-assoc = record
      { isSemigroup = isSemigroup ∙-assoc
      ; idem        = idem f E.reflexive
      }

    isSemilattice : Associative _≈_ _∙_  Commutative _≈_ _∙_ 
                    IsSemilattice _≈′_ _◦_
    isSemilattice ∙-assoc ∙-comm = record
      { isBand = isBand ∙-assoc
      ; comm   = comm  {x y}  f-injective {x} {y}) ∙-comm
      }

    isSelectiveMagma : IsSelectiveMagma _≈′_ _◦_
    isSelectiveMagma = record
      { isMagma = isMagma
      ; sel     = sel f E.reflexive
      }

------------------------------------------------------------------------
-- Other properties

  module _ {P : Pred A p} (f : A  B) where

    private
      _◦_ = Lift _≈_ _∙_ M.sel f

    preservesᵒ : (∀ {x y}  P x  (f x  f y)  f y  P y) 
                 (∀ {x y}  P y  (f x  f y)  f x  P x) 
                  x y  P x  P y  P (x  y)
    preservesᵒ left right x y (inj₁ px) with M.sel (f x) (f y)
    ... | inj₁ _        = px
    ... | inj₂ fx∙fy≈fx = left px fx∙fy≈fx
    preservesᵒ left right x y (inj₂ py) with M.sel (f x) (f y)
    ... | inj₁ fx∙fy≈fy = right py fx∙fy≈fy
    ... | inj₂ _        = py

    preservesʳ : (∀ {x y}  P y  (f x  f y)  f x  P x) 
                  x {y}  P y  P (x  y)
    preservesʳ right x {y} Py with M.sel (f x) (f y)
    ... | inj₁ fx∙fy≈fx = right Py fx∙fy≈fx
    ... | inj₂ fx∙fy≈fy = Py

    preservesᵇ :  {x y}  P x  P y  P (x  y)
    preservesᵇ {x} {y} Px Py with M.sel (f x) (f y)
    ... | inj₁ _ = Px
    ... | inj₂ _ = Py

    forcesᵇ : (∀ {x y}  P x  (f x  f y)  f x  P y) 
              (∀ {x y}  P y  (f x  f y)  f y  P x) 
               x y  P (x  y)  P x × P y
    forcesᵇ presˡ presʳ x y P[x∙y] with M.sel (f x) (f y)
    ... | inj₁ fx∙fy≈fx = P[x∙y] , presˡ P[x∙y] fx∙fy≈fx
    ... | inj₂ fx∙fy≈fy = presʳ P[x∙y] fx∙fy≈fy , P[x∙y]