------------------------------------------------------------------------
-- Bag equivalence for lists
------------------------------------------------------------------------

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

-- Note that this module is not parametrised by a definition of
-- equality; it uses ordinary propositional equality.

module Bag-equivalence where

open import Equality.Propositional hiding (trans)
open import Fin
open import Logical-equivalence hiding (id; _∘_; inverse)
open import Prelude as P hiding (id)

open import Bijection equality-with-J using (_↔_; module _↔_; Σ-≡,≡↔≡)
open import Equality.Decision-procedures equality-with-J
open import Function-universe equality-with-J as Function-universe
  hiding (_∘_; Kind; module Kind; bijection)
open import Injection equality-with-J using (_↣_)

------------------------------------------------------------------------
-- Any

-- Any.

Any : ∀ {a p} {A : Set a} (P : A → Set p) (xs : List A) → Set p
Any P []       = ⊥
Any P (x ∷ xs) = P x ⊎ Any P xs

-- Alternative definition of Any.

data Any′ {a p} {A : Set a}
          (P : A → Set p) : List A → Set (a ⊔ p) where
  here  : ∀ {x xs} → P x       → Any′ P (x ∷ xs)
  there : ∀ {x xs} → Any′ P xs → Any′ P (x ∷ xs)

-- The two definitions of Any are isomorphic.

Any′-[] : ∀ {a p ℓ} {A : Set a} {P : A → Set p} →
          Any′ P [] ↔ ⊥ {ℓ = ℓ}
Any′-[] {ℓ = ℓ} {P = P} = record
  { surjection = record
    { logical-equivalence = record
      { to   = to
      ; from = from
      }
    ; right-inverse-of = λ p → ⊥-elim p
    }
  ; left-inverse-of = from∘to
  }
  where
  to′ : ∀ {xs} → Any′ P xs → [] ≡ xs → ⊥ {ℓ = ℓ}
  to′ (here  p) = ⊥-elim ∘ List.[]≢∷
  to′ (there p) = ⊥-elim ∘ List.[]≢∷

  to : Any′ P [] → ⊥
  to p = to′ p refl

  from = ⊥-elim

  from∘to′ : ∀ {xs} (p : Any′ P xs) ([]≡xs : [] ≡ xs) →
             subst (Any′ P) []≡xs (from (to′ p []≡xs)) ≡ p
  from∘to′ (here  p) = ⊥-elim ∘ List.[]≢∷
  from∘to′ (there p) = ⊥-elim ∘ List.[]≢∷

  from∘to : ∀ p → from (to p) ≡ p
  from∘to p = from∘to′ p refl

Any′-∷ : ∀ {a p} {A : Set a} {P : A → Set p} {x xs} →
        Any′ P (x ∷ xs) ↔ P x ⊎ Any′ P xs
Any′-∷ {P = P} {x} {xs} = record
  { surjection = record
    { logical-equivalence = record
      { to   = to
      ; from = from
      }
    ; right-inverse-of = [ (λ _ → refl) , (λ _ → refl) ]
    }
  ; left-inverse-of = from∘to
  }
  where
  to′ : ∀ {ys} → Any′ P ys → ys ≡ x ∷ xs → P x ⊎ Any′ P xs
  to′ (here  p) ≡∷ = inj₁ (subst P (List.cancel-∷-head ≡∷) p)
  to′ (there p) ≡∷ = inj₂ (subst (Any′ P) (List.cancel-∷-tail ≡∷) p)

  to : Any′ P (x ∷ xs) → P x ⊎ Any′ P xs
  to p = to′ p refl

  from = [ here , there ]

  from∘to′ : ∀ {ys} (p : Any′ P ys) ≡∷ →
             from (to′ p ≡∷) ≡ subst (Any′ P) ≡∷ p
  from∘to′ (here p)  ≡∷ with List.cancel-∷-head ≡∷
                           | List.cancel-∷-tail ≡∷
                           | sym (List.unfold-∷ ≡∷)
  from∘to′ (here p)  .refl | refl | refl | refl = refl
  from∘to′ (there p) ≡∷ with List.cancel-∷-head ≡∷
                           | List.cancel-∷-tail ≡∷
                           | sym (List.unfold-∷ ≡∷)
  from∘to′ (there p) .refl | refl | refl | refl = refl

  from∘to : ∀ p → from (to p) ≡ p
  from∘to p = from∘to′ p refl

Any↔Any′ : ∀ {a p} {A : Set a} {P : A → Set p} {xs} →
           Any P xs ↔ Any′ P xs
Any↔Any′ {P = P} {[]}     =
  ⊥          ↔⟨ inverse Any′-[] ⟩
  Any′ P []  □
Any↔Any′ {P = P} {x ∷ xs} =
  P x ⊎ Any P xs   ↔⟨ id ⊎-cong Any↔Any′ {P = P} ⟩
  P x ⊎ Any′ P xs  ↔⟨ inverse Any′-∷ ⟩
  Any′ P (x ∷ xs)  □

------------------------------------------------------------------------
-- Lemmas relating Any to some basic list functions

Any-++ : ∀ {a p} {A : Set a} (P : A → Set p) (xs ys : List A) →
         Any P (xs ++ ys) ↔ Any P xs ⊎ Any P ys
Any-++ P [] ys =
  Any P ys      ↔⟨ inverse ⊎-left-identity ⟩
  ⊥ ⊎ Any P ys  □
Any-++ P (x ∷ xs) ys =
  P x ⊎ Any P (xs ++ ys)       ↔⟨ id ⊎-cong Any-++ P xs ys ⟩
  P x ⊎ (Any P xs ⊎ Any P ys)  ↔⟨ ⊎-assoc ⟩
  (P x ⊎ Any P xs) ⊎ Any P ys  □

Any-concat : ∀ {a p} {A : Set a}
             (P : A → Set p) (xss : List (List A)) →
             Any P (concat xss) ↔ Any (Any P) xss
Any-concat P []         = id
Any-concat P (xs ∷ xss) =
  Any P (xs ++ concat xss)       ↔⟨ Any-++ P xs (concat xss) ⟩
  Any P xs ⊎ Any P (concat xss)  ↔⟨ id ⊎-cong Any-concat P xss ⟩
  Any P xs ⊎ Any (Any P) xss     □

Any-map : ∀ {a b p} {A : Set a} {B : Set b}
          (P : B → Set p) (f : A → B) (xs : List A) →
          Any P (map f xs) ↔ Any (P ∘ f) xs
Any-map P f []       = id
Any-map P f (x ∷ xs) =
  P (f x)   ⊎ Any P (map f xs)  ↔⟨ id ⊎-cong Any-map P f xs ⟩
  (P ∘ f) x ⊎ Any (P ∘ f) xs    □

Any->>= : ∀ {a b p} {A : Set a} {B : Set b}
          (P : B → Set p) (xs : List A) (f : A → List B) →
          Any P (xs >>= f) ↔ Any (Any P ∘ f) xs
Any->>= P xs f =
  Any P (concat (map f xs))  ↔⟨ Any-concat P (map f xs) ⟩
  Any (Any P) (map f xs)     ↔⟨ Any-map (Any P) f xs ⟩
  Any (Any P ∘ f) xs         □

Any-filter : ∀ {a p} {A : Set a}
             (P : A → Set p) (p : A → Bool) (xs : List A) →
             Any P (filter p xs) ↔ Any (λ x → P x × T (p x)) xs
Any-filter P p []       = ⊥ □
Any-filter P p (x ∷ xs) with p x
... | true  =
   P x      ⊎ Any P (filter p xs)           ↔⟨ inverse ×-right-identity ⊎-cong Any-filter P p xs ⟩
  (P x × ⊤) ⊎ Any (λ x → P x × T (p x)) xs  □
... | false =
  Any P (filter p xs)                       ↔⟨ Any-filter P p xs ⟩
              Any (λ x → P x × T (p x)) xs  ↔⟨ inverse ⊎-left-identity ⟩
  ⊥₀        ⊎ Any (λ x → P x × T (p x)) xs  ↔⟨ inverse ×-right-zero ⊎-cong (_ □) ⟩
  (P x × ⊥) ⊎ Any (λ x → P x × T (p x)) xs  □

------------------------------------------------------------------------
-- List membership

infix  _∈_

_∈_ : ∀ {a} {A : Set a} → A → List A → Set _
x ∈ xs = Any (λ y → x ≡ y) xs

-- Any can be expressed using _∈_.

Any-∈ : ∀ {a p} {A : Set a} (P : A → Set p) (xs : List A) →
        Any P xs ↔ ∃ λ x → P x × x ∈ xs
Any-∈ P [] =
  ⊥                  ↔⟨ inverse ×-right-zero ⟩
  (∃ λ x → ⊥₀)       ↔⟨ ∃-cong (λ x → inverse ×-right-zero) ⟩
  (∃ λ x → P x × ⊥)  □
Any-∈ P (x ∷ xs) =
  P x                   ⊎ Any P xs                ↔⟨ ∃-intro P x ⊎-cong Any-∈ P xs ⟩
  (∃ λ y → P y × y ≡ x) ⊎ (∃ λ y → P y × y ∈ xs)  ↔⟨ inverse ∃-⊎-distrib-left ⟩
  (∃ λ y → P y × y ≡ x ⊎ P y × y ∈ xs)            ↔⟨ ∃-cong (λ y → inverse ×-⊎-distrib-left) ⟩
  (∃ λ y → P y × (y ≡ x ⊎ y ∈ xs))                □

-- Using this property we can prove that Any and _⊎_ commute.

Any-⊎ : ∀ {a p q} {A : Set a}
        (P : A → Set p) (Q : A → Set q) (xs : List A) →
        Any (λ x → P x ⊎ Q x) xs ↔ Any P xs ⊎ Any Q xs
Any-⊎ P Q xs =
  Any (λ x → P x ⊎ Q x) xs                         ↔⟨ Any-∈ (λ x → P x ⊎ Q x) xs ⟩
  (∃ λ x → (P x ⊎ Q x) × x ∈ xs)                   ↔⟨ ∃-cong (λ x → ×-⊎-distrib-right) ⟩
  (∃ λ x → P x × x ∈ xs ⊎ Q x × x ∈ xs)            ↔⟨ ∃-⊎-distrib-left ⟩
  (∃ λ x → P x × x ∈ xs) ⊎ (∃ λ x → Q x × x ∈ xs)  ↔⟨ inverse $ Any-∈ P xs ⊎-cong Any-∈ Q xs ⟩
  Any P xs ⊎ Any Q xs                              □

------------------------------------------------------------------------
-- Bag and set equivalence and the subset and subbag orders

-- Various kinds of relatedness.

open Function-universe public using (Kind)

module Kind where

  open Function-universe public
    using ()
    renaming ( implication         to subset
             ; logical-equivalence to set
             ; injection           to subbag
             ; bijection           to bag
             ; equivalence         to bag-with-equivalence
             )

open Kind public

-- A general definition of "relatedness" for lists.

infix  _∼[_]_

_∼[_]_ : ∀ {a} {A : Set a} → List A → Kind → List A → Set a
xs ∼[ k ] ys = ∀ z → z ∈ xs ↝[ k ] z ∈ ys

-- Bag equivalence.

infix  _≈-bag_

_≈-bag_ : ∀ {a} {A : Set a} → List A → List A → Set a
xs ≈-bag ys = xs ∼[ bag ] ys

-- Alternative definition of bag equivalence.

infix  _≈-bag′_

record _≈-bag′_ {a} {A : Set a} (xs ys : List A) : Set a where
  field
    bijection : Fin (length xs) ↔ Fin (length ys)
    related   : xs And ys Are-related-by bijection

-- Yet another definition of bag equivalence. This definition is taken
-- from Coq's standard library.

infixr  _∷_
infix   _≈-bag″_

data _≈-bag″_ {a} {A : Set a} : List A → List A → Set a where
  []    : [] ≈-bag″ []
  _∷_   : ∀ x {xs ys} (xs≈ys : xs ≈-bag″ ys) → x ∷ xs ≈-bag″ x ∷ ys
  swap  : ∀ {x y xs} → x ∷ y ∷ xs ≈-bag″ y ∷ x ∷ xs
  trans : ∀ {xs ys zs}
          (xs≈ys : xs ≈-bag″ ys) (ys≈zs : ys ≈-bag″ zs) → xs ≈-bag″ zs

------------------------------------------------------------------------
-- Some congruence lemmas

Any-cong : ∀ {k a p q} {A : Set a}
             {P : A → Set p} {Q : A → Set q} {xs ys : List A} →
           (∀ x → P x ↝[ k ] Q x) → xs ∼[ k ] ys →
           Any P xs ↝[ k ] Any Q ys
Any-cong {P = P} {Q} {xs} {ys} P↔Q xs≈ys =
  Any P xs                ↔⟨ Any-∈ P xs ⟩
  (∃ λ z → P z × z ∈ xs)  ↝⟨ ∃-cong (λ z → P↔Q z ×-cong xs≈ys z) ⟩
  (∃ λ z → Q z × z ∈ ys)  ↔⟨ inverse (Any-∈ Q ys) ⟩
  Any Q ys                □

++-cong : ∀ {k a} {A : Set a} {xs₁ xs₂ ys₁ ys₂ : List A} →
          xs₁ ∼[ k ] ys₁ → xs₂ ∼[ k ] ys₂ →
          xs₁ ++ xs₂ ∼[ k ] ys₁ ++ ys₂
++-cong {xs₁ = xs₁} {xs₂} {ys₁} {ys₂} xs₁∼ys₁ xs₂∼ys₂ = λ z →
  z ∈ xs₁ ++ xs₂          ↔⟨ Any-++ _ xs₁ xs₂ ⟩
  z ∈ xs₁ ⊎ z ∈ xs₂       ↝⟨ xs₁∼ys₁ z ⊎-cong xs₂∼ys₂ z ⟩
  z ∈ ys₁ ⊎ z ∈ ys₂       ↔⟨ inverse (Any-++ _ ys₁ ys₂) ⟩
  z ∈ ys₁ ++ ys₂          □

map-cong : ∀ {k a b} {A : Set a} {B : Set b}
           (f : A → B) {xs ys : List A} →
           xs ∼[ k ] ys → map f xs ∼[ k ] map f ys
map-cong f {xs} {ys} xs∼ys = λ z →
  z ∈ map f xs            ↔⟨ Any-map _ f xs ⟩
  Any (λ x → z ≡ f x) xs  ↝⟨ Any-cong (λ x → z ≡ f x □) xs∼ys ⟩
  Any (λ x → z ≡ f x) ys  ↔⟨ inverse (Any-map _ f ys) ⟩
  z ∈ map f ys            □

concat-cong : ∀ {k a} {A : Set a} {xss yss : List (List A)} →
              xss ∼[ k ] yss → concat xss ∼[ k ] concat yss
concat-cong {xss = xss} {yss} xss∼yss = λ z →
  z ∈ concat xss           ↔⟨ Any-concat _ xss ⟩
  Any (λ zs → z ∈ zs) xss  ↝⟨ Any-cong (λ zs → z ∈ zs □) xss∼yss ⟩
  Any (λ zs → z ∈ zs) yss  ↔⟨ inverse (Any-concat _ yss) ⟩
  z ∈ concat yss           □

>>=-cong : ∀ {k a b} {A : Set a} {B : Set b}
           {xs ys : List A} {f g : A → List B} →
           xs ∼[ k ] ys → (∀ x → f x ∼[ k ] g x) →
           (xs >>= f) ∼[ k ] (ys >>= g)
>>=-cong {xs = xs} {ys} {f} {g} xs∼ys f∼g = λ z →
  z ∈ xs >>= f            ↔⟨ Any->>= _ xs f ⟩
  Any (λ x → z ∈ f x) xs  ↝⟨ Any-cong (λ x → f∼g x z) xs∼ys ⟩
  Any (λ x → z ∈ g x) ys  ↔⟨ inverse (Any->>= _ ys g) ⟩
  z ∈ ys >>= g            □

filter-cong : ∀ {k a} {A : Set a} (p : A → Bool) (xs ys : List A) →
              xs ∼[ k ] ys → filter p xs ∼[ k ] filter p ys
filter-cong p xs ys xs∼ys = λ z →
  z ∈ filter p xs                 ↔⟨ Any-filter _ p xs ⟩
  Any (λ x → z ≡ x × T (p x)) xs  ↝⟨ Any-cong (λ _ → _ □) xs∼ys ⟩
  Any (λ x → z ≡ x × T (p x)) ys  ↔⟨ inverse (Any-filter _ p ys) ⟩
  z ∈ filter p ys                 □

------------------------------------------------------------------------
-- More properties

-- Bind distributes from the left over append.

>>=-left-distributive :
  ∀ {a b} {A : Set a} {B : Set b} (xs : List A) (f g : A → List B) →
  xs >>= (λ x → f x ++ g x) ≈-bag (xs >>= f) ++ (xs >>= g)
>>=-left-distributive xs f g = λ z →
  z ∈ xs >>= (λ x → f x ++ g x)                    ↔⟨ Any->>= (_≡_ z) xs (λ x → f x ++ g x) ⟩
  Any (λ x → z ∈ f x ++ g x) xs                    ↔⟨ Any-cong (λ x → Any-++ (_≡_ z) (f x) (g x)) (λ _ → id) ⟩
  Any (λ x → z ∈ f x ⊎ z ∈ g x) xs                 ↔⟨ Any-⊎ (λ x → z ∈ f x) (λ x → z ∈ g x) xs ⟩
  Any (λ x → z ∈ f x) xs ⊎ Any (λ x → z ∈ g x) xs  ↔⟨ inverse (Any->>= (_≡_ z) xs f ⊎-cong Any->>= (_≡_ z) xs g) ⟩
  z ∈ xs >>= f ⊎ z ∈ xs >>= g                      ↔⟨ inverse (Any-++ (_≡_ z) (xs >>= f) (xs >>= g)) ⟩
  z ∈ (xs >>= f) ++ (xs >>= g)                     □

-- This property does not hold for ordinary list equality.

¬->>=-left-distributive :
  ¬ ({A B : Set} (xs : List A) (f g : A → List B) →
     xs >>= (λ x → f x ++ g x) ≡ (xs >>= f) ++ (xs >>= g))
¬->>=-left-distributive distrib = Bool.true≢false true≡false
  where
  xs = true ∷ false ∷ []
  f  = λ x → x ∷ []
  g  = f

  eq : true ∷ true ∷ false ∷ false ∷ [] ≡
       true ∷ false ∷ true ∷ false ∷ []
  eq = distrib xs f g

  true≡false : true ≡ false
  true≡false = List.cancel-∷-head (List.cancel-∷-tail eq)

-- _++_ is commutative.

++-comm : ∀ {a} {A : Set a} (xs ys : List A) → xs ++ ys ≈-bag ys ++ xs
++-comm xs ys = λ z →
  z ∈ xs ++ ys     ↔⟨ Any-++ (_≡_ z) xs ys ⟩
  z ∈ xs ⊎ z ∈ ys  ↔⟨ ⊎-comm ⟩
  z ∈ ys ⊎ z ∈ xs  ↔⟨ inverse (Any-++ (_≡_ z) ys xs) ⟩
  z ∈ ys ++ xs     □

-- _++_ is idempotent (when set equivalence is used).

++-idempotent : ∀ {a} {A : Set a} (xs : List A) → xs ++ xs ∼[ set ] xs
++-idempotent xs = λ z →
  z ∈ xs ++ xs     ↔⟨ Any-++ (_≡_ z) xs xs ⟩
  z ∈ xs ⊎ z ∈ xs  ↝⟨ ⊎-idempotent ⟩
  z ∈ xs           □

-- The so-called "range splitting" property (see, for instance,
-- Hoogendijks "(Relational) Programming Laws in the Boom Hierarchy of
-- Types").

range-splitting : ∀ {a} {A : Set a} (p : A → Bool) (xs : List A) →
                  filter p xs ++ filter (not ∘ p) xs ≈-bag xs
range-splitting p xs = λ z →
  z ∈ filter p xs ++ filter (not ∘ p) xs                  ↔⟨ Any-++ _ _ (filter (not ∘ p) xs) ⟩
  z ∈ filter p xs ⊎ z ∈ filter (not ∘ p) xs               ↔⟨ Any-filter _ p xs ⊎-cong Any-filter _ (not ∘ p) xs ⟩
  Any (λ x → z ≡ x × T (p x)) xs ⊎
    Any (λ x → z ≡ x × T (not (p x))) xs                  ↔⟨ inverse $ Any-⊎ _ _ xs ⟩
  Any (λ x → z ≡ x × T (p x) ⊎ z ≡ x × T (not (p x))) xs  ↔⟨ Any-cong (λ x → lemma (z ≡ x) (p x)) (λ x → x ∈ xs □) ⟩
  z ∈ xs                                                  □
  where
  lemma : ∀ {a} (A : Set a) (b : Bool) → A × T b ⊎ A × T (not b) ↔ A
  lemma A b =
    A × T b ⊎ A × T (not b)  ↔⟨ ×-comm ⊎-cong ×-comm ⟩
    T b × A ⊎ T (not b) × A  ↔⟨ if-lemma (λ _ → A) id id b ⟩
    A                        □

-- The so-called "range disjunction" property, strengthened to use the
-- subbag preorder instead of set equivalence.

range-disjunction :
  ∀ {a} {A : Set a} (p q : A → Bool) (xs : List A) →
  filter (λ x → p x ∨ q x) xs ∼[ subbag ]
  filter p xs ++ filter q xs
range-disjunction p q xs = λ z →
  z ∈ filter (λ x → p x ∨ q x) xs                                  ↔⟨ Any-filter _ (λ x → p x ∨ q x) _ ⟩
  Any (λ x → z ≡ x × T (p x ∨ q x)) xs                             ↝⟨ Any-cong (λ x → lemma (z ≡ x) (p x) (q x)) (λ x → x ∈ xs □) ⟩
  Any (λ x → z ≡ x × T (p x) ⊎ z ≡ x × T (q x)) xs                 ↔⟨ Any-⊎ _ _ _ ⟩
  Any (λ x → z ≡ x × T (p x)) xs ⊎ Any (λ x → z ≡ x × T (q x)) xs  ↔⟨ inverse (Any-filter _ p _ ⊎-cong Any-filter _ q _) ⟩
  z ∈ filter p xs ⊎ z ∈ filter q xs                                ↔⟨ inverse $ Any-++ _ _ _ ⟩
  z ∈ filter p xs ++ filter q xs                                   □
  where
  inj : (b₁ b₂ : Bool) → T (b₁ ∨ b₂) ↣ T b₁ ⊎ T b₂
  inj true  true  = record { to = inj₁; injective = λ _ → refl    }
  inj true  false = record { to = inj₁; injective = ⊎.cancel-inj₁ }
  inj false true  = record { to = inj₂; injective = ⊎.cancel-inj₂ }
  inj false false = record { to = λ (); injective = λ {}          }

  lemma : ∀ {a} (A : Set a) (b₁ b₂ : Bool) →
          A × T (b₁ ∨ b₂) ↣ A × T b₁ ⊎ A × T b₂
  lemma A b₁ b₂ =
    A × T (b₁ ∨ b₂)      ↝⟨ id ×-cong inj b₁ b₂ ⟩
    A × (T b₁ ⊎ T b₂)    ↔⟨ ×-⊎-distrib-left ⟩
    A × T b₁ ⊎ A × T b₂  □

------------------------------------------------------------------------
-- The first two definitions of bag equivalence above are logically
-- equivalent

-- One direction follows from the following lemma, which states that
-- list membership can be expressed as "there is an index which points
-- to the element".
--
-- As an aside, note that the right-hand side is almost
-- lookup xs ⁻¹ z.

∈-lookup : ∀ {a} {A : Set a} {z}
           (xs : List A) → z ∈ xs ↔ ∃ λ i → z ≡ lookup xs i
∈-lookup {z = z} [] =
  ⊥                                ↔⟨ inverse $ ∃-Fin-zero _ ⟩
  (∃ λ (i : ⊥) → z ≡ lookup [] i)  □
∈-lookup {z = z} (x ∷ xs) =
  z ≡ x ⊎ z ∈ xs                     ↔⟨ id ⊎-cong ∈-lookup xs ⟩
  z ≡ x ⊎ (∃ λ i → z ≡ lookup xs i)  ↔⟨ inverse $ ∃-Fin-suc _ ⟩
  (∃ λ i → z ≡ lookup (x ∷ xs) i)    □

-- The index which points to the element.

index : ∀ {a} {A : Set a} {z} {xs : List A} → z ∈ xs → Fin (length xs)
index = proj₁ ∘ _↔_.to (∈-lookup _)

-- For the other direction a sequence of lemmas is used.

-- The first lemma states that ∃ λ z → z ∈ xs is isomorphic to Fin n,
-- where n is the length of xs. Thierry Coquand pointed out that this
-- is a generalisation of singleton-contractible.

Fin-length : ∀ {a} {A : Set a}
             (xs : List A) → (∃ λ z → z ∈ xs) ↔ Fin (length xs)
Fin-length xs =
  (∃ λ z → z ∈ xs)                   ↔⟨ ∃-cong (λ _ → ∈-lookup xs) ⟩
  (∃ λ z → ∃ λ i → z ≡ lookup xs i)  ↔⟨ ∃-comm ⟩
  (∃ λ i → ∃ λ z → z ≡ lookup xs i)  ↔⟨⟩
  (∃ λ i → Singleton (lookup xs i))  ↔⟨ ∃-cong (λ _ → contractible↔⊤ (singleton-contractible _)) ⟩
  Fin (length xs) × ⊤                ↔⟨ ×-right-identity ⟩
  Fin (length xs)                    □

-- From this lemma we get that lists which are bag equivalent have
-- related lengths.

Fin-length-cong : ∀ {a} {A : Set a} {xs ys : List A} →
                  xs ≈-bag ys → Fin (length xs) ↔ Fin (length ys)
Fin-length-cong {xs = xs} {ys} xs≈ys =
  Fin (length xs)   ↔⟨ inverse $ Fin-length xs ⟩
  ∃ (λ z → z ∈ xs)  ↔⟨ ∃-cong xs≈ys ⟩
  ∃ (λ z → z ∈ ys)  ↔⟨ Fin-length ys ⟩
  Fin (length ys)   □

abstract

  -- In fact, they have equal lengths.

  length-cong : ∀ {a} {A : Set a} {xs ys : List A} →
                xs ≈-bag ys → length xs ≡ length ys
  length-cong = _⇔_.to Fin.isomorphic-same-size ∘ Fin-length-cong

  -- All that remains (except for some bookkeeping) is to show that
  -- the isomorphism which Fin-length-cong returns relates the two
  -- lists.

  Fin-length-cong-relates :
    ∀ {a} {A : Set a} {xs ys : List A} (xs≈ys : xs ≈-bag ys) →
    xs And ys Are-related-by Fin-length-cong xs≈ys
  Fin-length-cong-relates {xs = xs} {ys} xs≈ys i =
    lookup xs i                                 ≡⟨ proj₂ $ to (∈-lookup _) $ to (xs≈ys _) (from (∈-lookup _) (i , refl)) ⟩
    lookup ys (proj₁ $ to (∈-lookup _) $
               to (xs≈ys _) $
               from (∈-lookup _) (i , refl))    ≡⟨ refl ⟩∎
    lookup ys (to (Fin-length-cong xs≈ys) i)    ∎
    where open _↔_

-- We get that the two definitions of bag equivalence are logically
-- equivalent.

≈⇔≈′ : ∀ {a} {A : Set a} {xs ys : List A} → xs ≈-bag ys ⇔ xs ≈-bag′ ys
≈⇔≈′ = record
  { to   = λ xs≈ys → record
             { bijection = Fin-length-cong xs≈ys
             ; related   = Fin-length-cong-relates xs≈ys
             }
  ; from = from
  }
  where
  equality-lemma : ∀ {a} {A : Set a} {x y z : A} →
                   y ≡ z → (x ≡ y) ↔ (x ≡ z)
  equality-lemma refl = id

  from : ∀ {xs ys} → xs ≈-bag′ ys → xs ≈-bag ys
  from {xs} {ys} xs≈ys z =
    z ∈ xs                     ↔⟨ ∈-lookup xs ⟩
    ∃ (λ i → z ≡ lookup xs i)  ↔⟨ Σ-cong (_≈-bag′_.bijection xs≈ys)
                                         (λ i → equality-lemma $
                                                  _≈-bag′_.related xs≈ys i) ⟩
    ∃ (λ i → z ≡ lookup ys i)  ↔⟨ inverse (∈-lookup ys) ⟩
    z ∈ ys                     □

------------------------------------------------------------------------
-- Left cancellation

-- We have basically already showed that cons is left cancellative for
-- the (first) alternative definition of bag equivalence.

∷-left-cancellative′ : ∀ {a} {A : Set a} {x : A} xs ys →
                       x ∷ xs ≈-bag′ x ∷ ys → xs ≈-bag′ ys
∷-left-cancellative′ {x = x} xs ys x∷xs≈x∷ys = record
  { bijection = Fin.cancel-suc (_≈-bag′_.bijection x∷xs≈x∷ys)
  ; related   = Fin.cancel-suc-preserves-relatedness x xs ys
                  (_≈-bag′_.bijection x∷xs≈x∷ys)
                  (_≈-bag′_.related x∷xs≈x∷ys)
  }

-- By the equivalence above we get the result also for the first
-- definition of bag equivalence, but we can show this directly, with
-- the help of some lemmas.

abstract

  -- The index function commutes with applications of certain
  -- inverses. Note that the last three equational reasoning steps do
  -- not need to be written out; I included them in an attempt to make
  -- it easier to understand why the lemma holds.

  index-commutes : ∀ {a} {A : Set a} {z : A} {xs ys} →
                   (xs≈ys : xs ≈-bag ys) (p : z ∈ xs) →
                   index (_↔_.to (xs≈ys z) p) ≡
                   _↔_.to (Fin-length-cong xs≈ys) (index p)
  index-commutes {z = z} {xs} {ys} xs≈ys p =
    (index $ to (xs≈ys z) p)                             ≡⟨ lemma z p ⟩
    (index $ to (xs≈ys _) $ proj₂ $
     from (Fin-length xs) $ to (Fin-length xs) (z , p))  ≡⟨ refl ⟩
    (index $ proj₂ $ Σ-map P.id (to (xs≈ys _)) $
     from (Fin-length xs) $ to (Fin-length xs) (z , p))  ≡⟨ refl ⟩
    (to (Fin-length ys) $ Σ-map P.id (to (xs≈ys _)) $
     from (Fin-length xs) $ index p)                     ≡⟨ refl ⟩∎
    (to (Fin-length-cong xs≈ys) $ index p)               ∎
    where
    open _↔_

    lemma : ∀ z p →
            (index $ to (xs≈ys z) p) ≡
            (index $
             to (xs≈ys (lookup xs (to (Fin-length xs) (z , p)))) $
             proj₂ $ from (Fin-length xs) $
             to (Fin-length xs) (z , p))
    lemma z p with to (Fin-length xs) (z , p)
                 | Σ-≡,≡←≡ (left-inverse-of (Fin-length xs) (z , p))
    lemma .(lookup xs i) .(from (∈-lookup xs) (i , refl))
          | i | refl , refl = refl

  -- Bag equivalence isomorphisms preserve index equality. Note that
  -- this means that, even if the underlying equality is proof
  -- relevant, a bag equivalence isomorphism cannot map two distinct
  -- proofs of z ∈ xs (say) to different positions.

  index-equality-preserved :
    ∀ {a} {A : Set a} {z : A} {xs ys} {p q : z ∈ xs}
    (xs≈ys : xs ≈-bag ys) →
    index p ≡ index q →
    index (_↔_.to (xs≈ys z) p) ≡ index (_↔_.to (xs≈ys z) q)
  index-equality-preserved {z = z} {p = p} {q} xs≈ys eq =
    index (_↔_.to (xs≈ys z) p)                ≡⟨ index-commutes xs≈ys p ⟩
    _↔_.to (Fin-length-cong xs≈ys) (index p)  ≡⟨ cong (_↔_.to (Fin-length-cong xs≈ys)) eq ⟩
    _↔_.to (Fin-length-cong xs≈ys) (index q)  ≡⟨ sym $ index-commutes xs≈ys q ⟩∎
    index (_↔_.to (xs≈ys z) q)                ∎

-- If x ∷ xs is bag equivalent to x ∷ ys, then xs and ys are bag
-- equivalent.

∷-left-cancellative : ∀ {a} {A : Set a} {x : A} {xs ys} →
                      x ∷ xs ≈-bag x ∷ ys → xs ≈-bag ys
∷-left-cancellative {A = A} {x} {xs} {ys} x∷xs≈x∷ys z =
  ⊎-left-cancellative
    (x∷xs≈x∷ys z)
    (lemma x∷xs≈x∷ys)
    (lemma (inverse ∘ x∷xs≈x∷ys))
  where
  abstract

    -- If the equality type is proof irrelevant (so that p and q are
    -- equal), then this lemma can be proved without the help of
    -- index-equality-preserved.

    lemma : ∀ {xs ys} (inv : x ∷ xs ≈-bag x ∷ ys) →
            Well-behaved (_↔_.to (inv z))
    lemma {xs} inv {b = z∈xs} {a = p} {a′ = q} hyp₁ hyp₂ = ⊎.inj₁≢inj₂ (
      inj₁ tt                              ≡⟨ refl ⟩
      index {xs = x ∷ xs} (inj₁ p)         ≡⟨ cong index $ sym $ to-from hyp₂ ⟩
      index {xs = x ∷ xs} (from (inj₁ q))  ≡⟨ index-equality-preserved (inverse ∘ inv) refl ⟩
      index {xs = x ∷ xs} (from (inj₁ p))  ≡⟨ cong index $ to-from hyp₁ ⟩
      index {xs = x ∷ xs} (inj₂ z∈xs)      ≡⟨ refl ⟩∎
      inj₂ (index {xs = xs} z∈xs)          ∎)
      where open _↔_ (inv z)

-- Cons is not left cancellative for set equivalence.

∷-not-left-cancellative :
  ¬ (∀ {A : Set} {x : A} {xs ys} →
     x ∷ xs ∼[ set ] x ∷ ys → xs ∼[ set ] ys)
∷-not-left-cancellative cancel =
  _⇔_.to (cancel (++-idempotent (tt ∷ [])) tt) (inj₁ refl)

-- _++_ is left and right cancellative (for bag equivalence).

++-left-cancellative : ∀ {a} {A : Set a} (xs : List A) {ys zs} →
                       xs ++ ys ≈-bag xs ++ zs → ys ≈-bag zs
++-left-cancellative []       eq = eq
++-left-cancellative (x ∷ xs) eq =
  ++-left-cancellative xs (∷-left-cancellative eq)

++-right-cancellative : ∀ {a} {A : Set a} {xs ys zs : List A} →
                        xs ++ zs ≈-bag ys ++ zs → xs ≈-bag ys
++-right-cancellative {xs = xs} {ys} {zs} eq =
  ++-left-cancellative zs (λ z →
    z ∈ zs ++ xs  ↔⟨ ++-comm zs xs z ⟩
    z ∈ xs ++ zs  ↔⟨ eq z ⟩
    z ∈ ys ++ zs  ↔⟨ ++-comm ys zs z ⟩
    z ∈ zs ++ ys  □)

------------------------------------------------------------------------
-- The third definition of bag equivalence is sound with respect to
-- the other two

-- _∷_ preserves bag equivalence.

infixr  _∷-cong_

_∷-cong_ : ∀ {a} {A : Set a} {x y : A} {xs ys} →
           x ≡ y → xs ≈-bag ys → x ∷ xs ≈-bag y ∷ ys
_∷-cong_ {x = x} {xs = xs} {ys} refl xs≈ys = λ z →
  z ≡ x ⊎ z ∈ xs  ↔⟨ id ⊎-cong xs≈ys z ⟩
  z ≡ x ⊎ z ∈ ys  □

-- We can swap the first two elements of a list.

swap-first-two : ∀ {a} {A : Set a} {x y : A} {xs} →
                 x ∷ y ∷ xs ≈-bag y ∷ x ∷ xs
swap-first-two {x = x} {y} {xs} = λ z →
   z ≡ x ⊎ z ≡ y  ⊎ z ∈ xs  ↔⟨ ⊎-assoc ⟩
  (z ≡ x ⊎ z ≡ y) ⊎ z ∈ xs  ↔⟨ ⊎-comm ⊎-cong id ⟩
  (z ≡ y ⊎ z ≡ x) ⊎ z ∈ xs  ↔⟨ inverse ⊎-assoc ⟩
   z ≡ y ⊎ z ≡ x  ⊎ z ∈ xs  □

-- The third definition of bag equivalence is sound with respect to
-- the first one.

≈″⇒≈ : ∀ {a} {A : Set a} {xs ys : List A} → xs ≈-bag″ ys → xs ≈-bag ys
≈″⇒≈ []                  = λ _ → id
≈″⇒≈ (x ∷ xs≈ys)         = refl ∷-cong ≈″⇒≈ xs≈ys
≈″⇒≈ swap                = swap-first-two
≈″⇒≈ (trans xs≈ys ys≈zs) = λ z → _ ↔⟨ ≈″⇒≈ xs≈ys z ⟩ ≈″⇒≈ ys≈zs z

-- The other direction should also be provable, but I expect that this
-- requires some work.